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Unique continuation for elliptic and parabolic equations with applications

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Unique continuation for elliptic and parabolic equations with applications

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Content UNIQUE CONTINUATION FOR ELLIPTIC AND PARABOLIC EQUATIONS WITH APPLICATIONS by Quinn Le A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) December 2023 Copyright 2023 Quinn Le Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Unique continuation of parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Pointwise Schauder estimates for parabolic and elliptic equations . . . . . . . . . . . . . . 4 Chapter 2: Unique continuation of parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Frequency smallness lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Settings and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 The main frequency lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 The order of vanishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Pointwise in time observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.8 The case R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.9 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.9.1 The case L q tL p x: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.9.2 The case L∞ t L n/2 x and L∞ t L n x : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.9.3 The cases L∞ t L p x and L∞ t L q x where 2n/3 > p > n/2 and q > 2n : . . . . . . . . . 56 Chapter 3: Pointwise Schauder estimates for elliptic and parabolic equations . . . . . . . . . . . . . 59 3.1 Schauder estimates for elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Interior Wm,p existence lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.3 Extended range for the interior existence lemma . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 The bootstrapping argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 Proofs of Schauder estimates for elliptic equations . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 Schauder estimates for parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.7 Interior W m,1 q existence estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.8 The bootstrap argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.9 Proofs of Schauder estimates for parabolic equations . . . . . . . . . . . . . . . . . . . . . 97 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 ii Abstract We consider the quantitative uniqueness properties for a parabolic type equation ut − ∆u = w(x, t) · ∇u + v(x, t)u, when v ∈ L p2 t L p1 x and w ∈ L q2 t L q1 x , with a suitable range for exponents p1, p2, q1, and q2. We prove a strong unique continuation property and provide a pointwise in time observability estimate. We also provide pointwise Schauder estimates for the general range of L p exponents, extending previous results from p > n/m to 1 < p < n/m, in space L p -type regularity for elliptic and parabolic equations of order m in R n . iii Chapter 1 Introduction This thesis delves into two distinct yet interconnected aspects of the regularity theory for elliptic and parabolic equations: unique continuation property and pointwise Schauder estimates. Both have a rich history and well-known applications, ranging from studying the nodal set, to the observability of solution. The first part of this thesis is about the strong unique continuation of parabolic equations, including the main results and proofs, as well as applications and extensions. The second part gives a pointwise Schauder estimate for both elliptic and parabolic equations, which covers the full range of L p that previous results did not include. 1.1 Unique continuation of parabolic equations The unique continuation for PDEs has a rich history (cf. the review papers by Kenig [K1, K2] and Vessella [V]), so we only mention several results pertaining to this thesis. In [JK], Jerison and Kenig proved that the second order elliptic equation has the strong unique continuation property (i.e., is identically zero if it vanishes to an infinite order at a point) if w = 0 and v ∈ L n/2 , with a sufficiently small L n/2 norm, which is a sharp result. The parabolic counterpart was obtained by Escauriaza and Vega [EV] (see also [E] for a previous unique continuation result when v ∈ L∞ t L p x with p > n/2). The difficult case when w is nonzero was addressed by Koch and Tataru for the elliptic case in [KT1] and the parabolic case in [KT2]. 1 In particular, they obtained the strong unique continuation for v ∈ L 1 tL∞ x + L∞ t L n/2 x with the norm sufficiently small and w ∈ L n+2 x,t . All the mentioned works rely on suitable Carleman type estimates and lead to observability type estimates on the space-time rectangles. For other works on the frequency approach to the unique continuation, see [A, AN, Ch, Ku1, Ku2], for the works related to Dirichlet quotients, see [A, CFNT, FS] and for some other related works, cf. [AE, AEWZ, ApE, AMRV, An, AV, BC, B, BK, CRV, D, DF1, DF2, DZ1, DZ2, EF, EFV, EVe, F, H1, H2, K3, KSW, L, M, Z]. We address the unique continuation of parabolic equations whose periodic domain and R n with bounded Dirichlet quotient. Let u be a solution of ut − ∆u = w(x, t) · ∇u + v(x, t)u, where v and w satisfy v ∈ L p2 t L p1 x (Ω × I) and w ∈ L q2 t L q1 x (Ω × I), respectively, Ω ⊆ R n , and I = [T0, T0 + T]. Here we have Ω = T n or Ω = R n . Assume that the Dirichlet quotient is bounded, i.e., q0 = sup t∈I ∥∇u(·, t)∥ 2 L2(Ω) ∥u(·, t)∥ 2 L2(Ω) < ∞. (1.1) We obtain explicit algebraic observability estimates for a fixed time (i.e., not only on space-time rectangles) under the assumptions on the coefficients v ∈ L∞ t L p x and w ∈ L∞ t L q x, where p > 2n/3 and q > 2n. More general conditions v ∈ L p2 t L p1 x and w ∈ L q2 t L q1 x under certain assumptions on p1, p2, q1, and q2 are addressed in Theorems 2.9.1. While the results cited in the first paragraph use the Carleman estimates, we rely on the frequency function approach, developed in [Al, GL] for the elliptic and [Kur, P] for the parabolic equations. The main idea of this approach for parabolic equations is the logarithmic convexity of the frequency function Q(t) = |t| R Rn |∇u(x, t)| 2G(x, t) dx R Rn u(x, t) 2G(x, t) dx for the heat equation. In (1.2), G is the (4π) n/2 -multiple of the backward Gaussian kernel, i.e., G(x, t) = 1 |t| n/2 e |x| 2/4t . (1.2) 2 Another reformulation of the idea is to use the similarity variables (cf. (2.42) and (2.43) below) and obtain a logarithmic convexity of the unweighted norm [C, Ku3]. In [CK], the approach was used to obtain an estimate for an order of vanishing C(∥w∥ 2 L∞x,t + ∥v∥ 2/3 L∞x,t ), which is, at least for complex valued coefficients, sharp [CKW1, CKW2]. Poon and Kurata showed that the frequency approach leads to the unique continuation property for p > n and q = ∞. In this chapter, we deduce the quantitative unique continuation statement and the observability estimate for p > 2n/3 and q > 2n. The improved range is obtained by three main devices. The first is to find the point in space where the frequency function is the smallest and translate the equation so it starts at that point (cf. Lemma 2.3.1 below); this idea has been introduced in [Ku4, CK]. The second device is to use the embedding theorems with Laplacian and use (2.77) below to bound the parts containing v and w. The third is to use the finiteness of the integral in (2.120) below, which then allows us to show the convergence of the quantity under the integral. Note that we obtain an explicit algebraic bound on the order of vanishing, which is a constant multiple of ∥v∥ 2/(3−2n/p) L∞t L p x + ∥w∥ 2/(1−2n/q) L∞t L q x + 1. (1.3) When setting p = q = ∞, the estimate reduces to the sharp bound from [CK]. We also provide a pointwise estimate in time for a better understanding of the behavior of solution u. In particular, for all δ0 ∈ (0, 1], we have ∥u(·, t)∥L2(Rn) ≲ e P ∥u(·, t)∥L2(Bδ0 (0)), (1.4) for all t ∈ [T0 + T /2, T0 + T], where P is a polynomial depending only on n, q, p, δ0, ∥v∥L∞t L p x , and ∥w∥L∞t L q x . The explicit formula for P can be found in Lemma 3.7.2 below. Note that the estimates of the type (1.4) are an essential ingredient when considering qualitative properties of solutions of evolutionary parabolic PDE. For instance, they are needed when considering the size of the zero set of solution at 3 time t or, more generally, complexity of a graph of a function at a time t ([Ku3, Ku4]). Note finally that compared to [CK], we reduce the necessary regularity for the solution u to a simple boundedness, so we do not require differentiability. We conclude this chapter by several general comments about the presented results. We believe that the restrictions p > 2n/3 and q > 2n are sharp from the perspective of the frequency approach (see however Theorem 3.6.2 and Section 2.9 for extensions); it would be interesting to know if they are optimal for obtaining the inequality of the type (2.10) pointwise in time. The restriction on p and q results from the Gronwall-type argument applied to (2.95) below. It is not clear if the approaches related to the Carleman estimates (as those in [KT2]) can be adapted to obtain pointwise in time observability estimates with the low regularity of v and w. It seems that approaches using the frequency require all Lebesgue exponents to be greater than or equal to 2 (cf. [DZ1, DZ2] where the exponents lower than two were obtained in the elliptic case when n = 2). 1.2 Pointwise Schauder estimates for parabolic and elliptic equations We provide the Schauder estimate for elliptic operator with arbitrary order. In particular, we obtain explicit estimates for ∥u−P∥Lq(Br) and ∥Di (u−P)∥Lq(Br) where 1 ≤ q < np/(n−mp) and P is a homogeneous polynomial. We also extract an estimation for [u]C d+l,α Lq for elliptic case, which reads [u]C d+l,α Lq (0) ≲ ∥u∥Lp(B1) + [f]C d−m Lp (0) + [f]C d−m+l,α Lp (0). (1.5) where [u]C d,α Lp (0) = inf P ∈Pd sup 0<r≤1 ∥u − P∥Lp(Br) r d+α+n/p . (1.6) The key components we use to obtain the results in this chapter contain several results on the bounds on the fundamental solutions of elliptic equation, cf. [B], as well as results on homogeneity of polynomials. 4 We address Schauder estimates for elliptic equations Lu ≡ X 0≤|ν|≤m aν(x)∂ νu = f, x ∈ B1 ⊂ R n (1.7) of an arbitrary even order m, where f ∈ L p (B1) and all p ∈ [1, ∞). In particular, we derive the vanishing order of solution u of (1.7) in L q by approximating with polynomials for a suitable range of q being determined in Theorem 3.1.3. Additionally, we derive the pointwise Schauder estimate of the form [u]C d,α Lq (0) ≲ ∥u∥Lp(B1) + ∥f∥Lp(B1) + [f]C d−m,α Lp (0), (1.8) where [u]C d,α Lp (0) = inf P ∈Pd sup 0<r<1 ∥u − P∥Lp(Br) r d+α+n/p (1.9) denotes a pointwise continuity norm [H1]. The employed method also works for a parabolic equation with maintaining the minimal requirements for d, m, p, and the coefficients aν(x) from the elliptic case, as shown in the last section. Schauder estimates are a cornerstone of the regularity theory for elliptic and parabolic equations. For the classical theory of pointwise Schauder estimates for the second order elliptic equations, see the classical textbook [GT]. In [Be], Bers considered an elliptic equation of a general order and established the existence of an asymptotic polynomial at a point where the solution vanishes. He namely determined the behavior of solution near 0 with an estimate for error term, i.e, u(x) = Pd(x) +O(|x| d+ϵ ) where Pd being a polynomial of degree d and ϵ being Hölder exponent of the leading coefficients of L. This in a sense generalizes the existence of a Taylor polynomial to solutions of elliptic equations with minimal requirements for coefficients (the leading coefficient is only Hölder). An asymptotic polynomial for solutions of the second order parabolic equations was obtained by Alessandrini and Vessella in [AV], where they 5 proved that a solution u either has a zero of infinite order or u can be approximated by some polynomial P of order d such that they can estimate u − P in L∞(Br) for small r > 0. The Schauder estimates in L q (rather then pointwise) were obtained by Han in [H1] in the elliptic and in [H2] for the parabolic case. He namely derived an estimation for ∥u−P∥L∞(Br) under the restriction p > n/m. Note that this restriction is natural for the purpose of Schauder estimates due to the Sobolev embedding Wm,p ,→ L∞ under this condition. We refer the reader to [Br, DK1, DK2, LGM, MY, T] for some other works on the Schauder estimates for elliptic and to [CK2, DH, S, W] for parabolic type equations. In this chapter, we address the remaining range p < n/m and introduce an interval of q where we can estimate ∥u − P∥Lq(Br) for a suitable asymptotic polynomial. Our results are in correspondence with Han’s result when p approaches n/m, in which the end point of the interval for q goes to infinity. We obtain explicit estimates for ∥u−P∥Lq(Br) and ∥Di (u−P)∥Lq(Br) where 1 ≤ q < np/(n−mp) and P is a homogeneous polynomial. We also obtain an upper bound for [u]C d+l,α Lq (0) where the Schauder pointwise norm [u]C d,α Lp (0) = inf P ∈Pd sup 0<r<1 ∥u − P∥Lp(Br) r d+α+n/p (1.10) was introduced in [H1]; note that, by translation, we may always restrict the analysis to the point 0 and thus all the statements apply to an arbitrary point in space. For elliptic case, we obtain [u]C d+k,α Lq (0) ≲ ∥u∥Lp(B1) + [f]C d−m Lp (0) + [f]C d−m+k,α Lp (0) + 1, (1.11) while for the parabolic case X d+k i=d [u]Ci Lq (0) + [u]C d+k,α Lq (0) ≲ ∥u∥Lp(Q1) + d−Xm+k i=d−m [f]Ci Lp (0) + [f]C d−m+k,α Lp (0), (1.12) 6 where the implicit constants does not depend on u or f. In our approach we use many ideas from the works [H1, H2], which in turn draw from [B] and [AV]. Throughout this chapter, we use several results on the bounds on the fundamental solutions. For elliptic equation, we use the upper bound established by Bers in [B] (for the second order parabolic equation see [AV]) as well as results on homogeneity of polynomials. 7 Chapter 2 Unique continuation of parabolic equations 2.1 Main results We address the quantitative uniqueness of a nontrivial solution u ∈ L∞(I, L2 (T n )) ∩ L 2 (I, H1 (T n )) of the equation ut − ∆u = w(x, t) · ∇u + v(x, t)u u(·, T0) = u0, (2.1) with the first equation defined for(x, t) ∈ R n×I where I = [T0, T0+T] is a given time interval, assuming T, T0 > 0 and n ≥ 2. The theorems are also valid for n = 1 with minor changes; cf. Remark 2.1.6 below. We assume that u, v, and w are 1-periodic (in all n directions) and that they satisfy ∥v(·, t)∥Lp(Tn) ≤ M0 (2.2) and ∥w(·, t)∥Lq(Tn) ≤ M1, (2.3) 8 for all t ∈ I. When we consider the periodic boundary conditions, we use the notation Ω for the set [−1/2, 1/2]n , while T n means R n/Z n , i.e., T n is the set of equivalence classes of points which are identified if the difference belongs to Z n . Let O(x0,t0) (u) be the vanishing order of u at (x0, t0), which is defined as the largest integer d such that ∥u∥L2(Qr(x0,t0)) = O(r d+(n+2)/2 ) as r → 0, (2.4) where Qr(x0, t0) = {(x, t) ∈ R n × R : |x − x0| < r, −r 2 < t − t0 < 0} (2.5) stands for the parabolic cylinder centered at(x0, t0) with the radius r. Note that, by the parabolic regularity and Hölder’s inequality, this definition of vanishing order is equivalent to the one stated in the introduction. For t ∈ I, denote by qD(t) = ∥∇u(·, t)∥ 2 L2(Tn) ∥u(·, t)∥ 2 L2(Tn) (2.6) the Dirichlet quotient of u at the time t ∈ I. We assume that ∥u(·, t)∥L2(Tn) is nonzero for all t ∈ I. We also suppose that q0 = sup t∈I qD(t) < ∞. (2.7) The following is the main result of this chapter; cf. also Theorems 2.9.1, and 2.9.2 (as well as Remark 2.9.3) for extensions. Here and in the sequel, we denote L q tL p x(T n × I) = L q (I, Lp (T n )) and L p x,t = L p tL p x, for p, q ∈ [1, ∞]. Theorem 2.1.1. Let u ∈ L∞ x,t(T n×I) be a solution of (2.1) with v ∈ L∞ t L p x(T n×I) and w ∈ L∞ t L q x(T n×I) such that (2.2) and (2.3) hold where p > 2n 3 (2.8) 9 and q > 2n. (2.9) Then, for all (x0, t0) ∈ T n × [T0 + T /2, T0 + T], the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ Ma 0 + Mb 1 + 1, (2.10) where a = 2 3 − 2n/p and b = 2 1 − 2n/q , (2.11) with the implicit constant in (2.10) depending on q0 and T. Note that the dimension n is considered fixed, so all constants and polynomials may depend on n without mention. Remark 2.1.2. The assumption (2.7) is necessary as the Dirichlet quotient controls oscillations of solutions. For instance, let ϕn be an λn-eigenfunction of −∆ with periodic boundary conditions, which vanishes of order n at 0. Then the solution u = ϕne −λnt of the heat equation does not satisfy (2.10) for n sufficiently large as v = w = 0. Note that the Dirichlet quotient for this solution equals λn. The eigenfunctions with an arbitrarily high order of vanishing can easily be constructed in bounded domains with Dirichlet boundary conditions; however we expect that it holds for periodic boundary conditions as well; cf. also [CKW1, CKW2] for constructions of solutions of elliptic equations with a high order of vanishing, including the periodic boundary conditions. In the next statement, we provide a pointwise in time observability property of solutions. Theorem 2.1.3. Under the conditions of Theorem 3.1.3, there exists a polynomial P such that ∥u(·, t)∥L2(Tn) ≤ e P(δ0,M0,M1) ∥u(·, t)∥L2(B(0,δ0)), (2.12) 10 for all t ∈ [T0 + T /2, T0 + T] and δ0 ∈ (0, 1/2], where the coefficients depend on p, q, T, and q0. Here and in the sequel, P denotes a generic nonnegative polynomial. Although in the proof we do not follow the dependence on q0 and T, it is easy to check that the dependence on these quantities is also polynomial. Theorems 3.1.3 and 3.1.2 allow extensions, some of which are stated in Section 2.9. Here we point out one, which allows the exponents p and q to belong to extended ranges p > n/2 and q > n, considered critical for unique continuation. Theorem 2.1.4. Let u ∈ L∞ x,t(T n×I) be a solution of (2.1) with v ∈ L∞ t L p x(T n×I) and w ∈ L∞ t L q x(T n×I) such that (2.2) and (2.3) hold where p > n 2 (2.13) and q > 2n. (2.14) If M0 is less than a constant depending on q0, then for all (x0, t0) ∈ T n × [T0 + T /2, T0 + T], the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ Mb 1 + 1, (2.15) where b is as in (2.11). Similarly, if p > 2n 3 (2.16) and q > n, (2.17) 11 and if M1 is less than a constant depending on q0, then for all (x0, t0) ∈ T n × [T0 + T /2, T0 + T], the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ Ma 0 + 1, (2.18) where a is as in (2.11). If p > n/2 and q > n, and if M0 and M1 are less than a constant depending on q0, then the same conclusion holds with (2.15) replaced by O(x0,t0) (u) ≲ 1, with all the implicit constants depending on q0 and T. Now, consider u, v, and w defined on R n instead of T n . Suppose that u satisfies a doubling type (or mild-growth) condition Z Rn u(x, t) 2 dx ≤ K Z B1 u(x, t) 2 dx, t ∈ [T0, T0 + T], (2.19) for some constant K. In this case, we obtain the following analogue of Theorems 3.1.3 and 3.1.2. Theorem 2.1.5. Let u ∈ L∞ x,t(R n × I), where I = [T0, T0 + T] and T0, T > 0, be a solution of (2.1) satisfying (2.19), with the coefficients verifying v ∈ L∞ t L p x(R n × I) and w ∈ L∞ t L q x(R n × I) with ∥v(·, t)∥Lp(Rn) ≤ M0 (2.20) and ∥w(·, t)∥Lq(Rn) ≤ M1, (2.21) for t ∈ I. Assume additionally that p > 2n 3 (2.22) 12 and q > 2n. (2.23) Then, for all (x0, t0) ∈ BR × [T0 + T /2, T0 + T], where R > 0, the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ Ma 0 + Mb 1 + 1, (2.24) where a = 2/(3 − 2n/p) and b = 2/(1 − 2n/q), with the implicit constant in (2.24) depending on q0, K, T, and R. Moreover, for δ0 ∈ (0, 1/2], we have ∥u(·, t)∥L2(Tn) ≤ e P(K,δ0,M0,M1) ∥u(·, t)∥L2(Bδ0 ) , (2.25) for all t ∈ [T0 + T /2, T0 + T], where P is a polynomial with coefficients depending on q0. The theorem is proven in Section 2.9 below. Remark 2.1.6. In the theorems above, we assumed n ≥ 2. For the case n = 1, we suppose additionally that p ≥ 2 and q ≥ 4. The reason for the restriction p ≥ 2 is that the methods are L 2 -based requiring the exponents to be at least 2. The reason for q ≥ 4 is technical; cf. the comment below (2.87). 2.2 Methodology We propose using frequency function method to study unique continuation of parabolic equation. The frequency function is defined by Q(t) = |t| R Rn |∇u(x, t)| 2G(x, t) dx R Rn u(x, t) 2G(x, t) dx , (2.26) 13 where G(x, t) = 1 |t| n/2 e |x| 2/4t is the (4π) n/2 - multiple of the backward Gaussian kernel. One important feature of the frequency function is that it controls the vanishing order (that we show in 2.6 below). 2.3 Frequency smallness lemma By a translation and rescaling, we may restrict ourselves, throughout the section, to I = [−1, 0] and (x0, t0) = (0, 0). The following lemma allows us to find a point −xϵ where the frequency Q(t) = |t| R Rn |∇u(x, t)| 2G(x, t) dx R Rn u(x, t) 2G(x, t) dx , (2.27) after being translated so it is centered at −xϵ, is small, at a small time t = −ϵ, where ϵ ∈ (0, 1]. Lemma 2.3.1. Let u ∈ L∞ x,t(Ω × I) be smooth and 1-periodic in x, for t ∈ I = [−1, 0]. For any ϵ ∈ (0, 1] such that u(·, −ϵ) is not identically zero, there exists xϵ ∈ Ω such that ϵ R Rn |∇u(xϵ + y, −ϵ)| 2G(y, −ϵ) dy R Rn u(xϵ + y, −ϵ) 2G(y, −ϵ) dy ≤ ϵqD (−ϵ), (2.28) where G(x, t) = 1 |t| n/2 e |x| 2/4t , x ∈ R n , t < 0. (2.29) The lemma is proven in [CK, Ku4]; we provide a short proof for the sake of completeness. Proof of Lemma 2.3.1. Assume, contrary to the assertion, that we have qD(−ϵ) Z Rn u(x + y, −ϵ) 2G(y, −ϵ) dy < Z Rn |∇u(x + y, −ϵ)| 2G(y, −ϵ) dy, (2.30) 14 for all x ∈ Ω, which by a simple change of variable reads Z Rn u(y, −ϵ) 2G(y − x, −ϵ) dy < 1 qD (−ϵ) Z Rn |∇u(y, −ϵ)| 2G(y − x, −ϵ) dy, x ∈ Ω. (2.31) We shall integrate both sides in x over Ω, obtaining a contradiction. The integral of the left-hand side over Ω equals Z Ω Z Rn u(y, −ϵ) 2G(y − x, −ϵ) dy dx = Z Ω X j∈Zn Z j+Ω u(y, −ϵ) 2G(y − x, −ϵ) dy dx = Z Ω X j∈Zn Z Ω u(y, −ϵ) 2G(y + j − x, −ϵ) dy dx = Z Ω u(y, −ϵ) 2   X j∈Zn Z Ω G(y + j − x, −ϵ) dx   dy = Z Ω u(y, −ϵ) 2 Z Rn G(y − x, −ϵ) dx dy = (4π) n/2 Z Ω u(y, −ϵ) 2 dy, (2.32) where we used R Rn G(y − x, −ϵ) dx = (4π) n/2 in the last equality. Similarly, we have Z Ω Z Rn |∇u(y, −ϵ)| 2G(y − x, −ϵ) dy dx = (4π) n/2 Z Ω |∇u(y, −ϵ)| 2 dy. (2.33) Combining (2.32) and (2.33), we obtain qD(−ϵ) Z Ω u(y, −ϵ) 2 dy < Z Ω |∇u(y, −ϵ)| 2 dy, (2.34) which is a contradiction with the definition (2.6) of the Dirichlet quotient. Therefore, (2.30) cannot hold for all x ∈ Ω, and the lemma follows. Note that the argument above does not require u to solve (2.1). 15 2.4 Settings and notation Let ϵ ∈ (0, 1/2] be a fixed parameter, to be chosen in the proof of Lemma 3.2.1 below; cf. (2.59). We proceed with a change of variables u(x, t) = ¯u x + xϵ ϵ t, t , (2.35) where xϵ is as in Lemma 2.3.1, so that u¯(x, −ϵ) = u(x + xϵ, −ϵ) (2.36) and u¯(x, 0) = u(x, 0), (2.37) for all x ∈ T n . By Lemma 2.3.1, we have ϵ R Rn |∇u¯(y, −ϵ)| 2G(y, −ϵ) dy R Rn u¯(y, −ϵ) 2G(y, −ϵ) dy ≤ ϵq0, (2.38) i.e., the frequency of u¯ at t = −ϵ is small. It is not difficult to check that u¯ solves the equation ∂tu¯ − ∆¯u = − xϵ ϵ · ∇u¯ + w · ∇u¯ + vu. ¯ (2.39) Since u¯ and u have the same order of vanishing at (0, 0), we write u instead of u¯. Denoting r = − xϵ ϵ (2.40) 16 throughout, the equation (2.39) becomes ∂tu − ∆u = r · ∇u + w · ∇u + vu. (2.41) We now proceed with a change of variables U(y, τ ) = e −|y| 2/8u(ye−τ /2 , −e −τ ), (y, τ ) ∈ R n × [τ0, ∞), (2.42) that is, u(x, t) = e |x| 2/8(−t)U x √ −t , − log(−t) , (x, t) ∈ R n × [−ϵ, 0), (2.43) with y = x/√ −t and τ = − log(−t), where τ0 = log 1 ϵ . (2.44) Also, let V (y, τ ) = v(ye−τ /2 , −e −τ ), (y, τ ) ∈ R n × [τ0, ∞) (2.45) and W(y, τ ) = w(ye−τ /2 , −e −τ ), (y, τ ) ∈ R n × [τ0, ∞). (2.46) Then (2.41) becomes ∂τU + HU = e −τ /2 1 4 rjyjU + rj∂jU + e −τ /2 1 4 yjWjU + Wj∂jU + e −τV U, (2.47) 17 where HU = −∆U + |y| 2 16 − n 4 U, (2.48) with the initial data U(y, τ0) = U y, log 1 ϵ = e −|y| 2/8u(y √ ϵ, −ϵ). (2.49) A short computation shows that ∥U(·, τ )∥ 2 L2(Rn) = Z Rn u(x, t) 2G(x, t) dx, (2.50) where τ = − log(−t) throughout, and (HU, U)L2(Rn) = |t| Z Rn |∇u(x, t)| 2G(x, t) dx. (2.51) Thus also Q(τ ) = (HU, U)L2(Rn) ∥U∥ 2 = |t| R Rn |∇u(x, t)| 2G(x, t) dx R Rn u(x, t) 2G(x, t) dx , (2.52) where we write ∥ · ∥ = ∥ · ∥L2(Rn) ; (2.53) also, if the domain of integration is not indicated, it is assumed to be R n . Denoting A(τ )U = HU − 1 4 e −τ /2 rjyjU = −∆U + |y| 2 16 − n 4 U − 1 4 e −τ /2 rjyjU (2.54) and Q¯(τ ) = (A(τ )U, U)L2(Rn) ∥U∥ 2 = Q(τ ) − e −τ /2 rj 4∥U∥ 2 Z Rn yjU 2 dy, (2.55) 18 we may rewrite (2.47) as ∂τU + A(τ )U = F(U), (2.56) where F(U) = e −τ /2 rj∂jU + e −τ /2 1 4 yjWjU + Wj∂jU + e −τV U. (2.57) For simplicity, denote U˜ = U ∥U∥ , (2.58) so that ∥U˜∥ = 1. 2.5 The main frequency lemma We now show that the modified frequency function Q¯ is bounded with an expression on the right-hand side of (2.10) for a suitable choice of ϵ. Lemma 2.5.1. Let ϵ = 1 C(Ma 0 + Mb 1 + 1), (2.59) where a and b are given in (2.11), and C is a sufficiently large constant depending on q0. Under the assumptions of Theorem 3.1.3, and assuming that u, v, and w are smooth, the modified frequency function satisfies Q¯(τ ) ≲ Ma 0 + Mb 1 + 1, τ ≥ τ0, (2.60) where τ0 is given in (2.44), with the implicit constant in (2.60) depending on q0. Note that ϵ ∈ (0, 1/2] by (2.59). 19 Proof of Lemma 3.2.1. Let ϵ ∈ (0, 1/2] first be arbitrary, with the choice (2.59) made before (2.101) below. Also, we use the notation from Section 2.4. A simple computation shows that (2.56) implies, with I denoting the identity matrix, 1 2 Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 = 1 2 (A ′ (τ )U, ˜ U˜) + F(U˜),(A(τ ) − Q¯(τ )I)U˜ , (2.61) where we used that A(τ ) is symmetric. Since A′ (τ )U˜ = 1 8 e −τ /2 rjyjU˜, we obtain 1 2 Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 = 1 16 e −τ /2 (rjyjU, ˜ U˜) + e −τ /2 rj ∂jU, ˜ (A(τ ) − Q¯(τ )I)U˜ + e −τ /2 (yjWjU˜ + Wj∂jU˜) + e −τV U, ˜ (A(τ ) − Q¯(τ )I)U˜ . (2.62) For the second term on the right-hand side of (2.62), we have e −τ /2 rj ∂jU, ˜ (A(τ ) − Q¯(τ )I)U˜ = −e −τ /2 rj Z ∆U ∂˜ jU˜ + e −τ /2 rj Z |y| 2 16 U ∂˜ jU˜ − n 4 e −τ /2 rj Z U ∂˜ jU˜ − 1 4 e −τ rjrk Z ykU ∂˜ jU˜ = 1 16 e −τ /2 rj Z |y| 2U ∂˜ jU˜ − 1 4 e −τ rjrk Z ykU ∂˜ jU˜ = − 1 16 e −τ /2 rj Z yjU˜ 2 + 1 8 e −τ |r| 2 , (2.63) where we used R U ∂˜ jU˜ = 0 and R ∆U ∂˜ jU˜ = 0 in the second step (since v, w, and u0 are assumed smooth, U˜ and its derivatives are smooth and decaying fast in the spatial variable) and ∥U˜∥ = 1 in the last. Recall that all the integrals with the domain not indicated are understood to be over R n . Note that the first term on the far right side of (2.63) cancels with the first term on the right-hand side of (2.62). Therefore, 1 2 Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 = 1 8 e −τ |r| 2 + e −τ /2 Z (yjWjU˜ + Wj∂jU˜ + e −τ /2V U˜)(A(τ ) − Q¯(τ )I)U dy ˜ = I1 + I2. (2.64) In order to estimate I2, we first claim ∥D2U∥ ≲ ∥HU∥ + ∥U∥. (2.65) To prove (2.65), we expand ∥HU∥ 2 as ∥HU∥ 2 = Z −∆U + |y| 2 16 − n 4 U 2 dy = Z (∆U) 2 + |y| 2 16 − n 4 2 U 2 ! dy − 2 Z |y| 2 16 − n 4 U∆U dy = Z (∆U) 2 + |y| 2 16 − n 4 2 U 2 ! dy + 2 Z ∂jU ∂jU |y| 2 16 − n 4 dy + 1 4 Z yjU ∂jU dy. (2.66) Since the last term equals (−n/8) R U 2 dy, we get ∥HU∥ 2 ≥ Z (∆U) 2 + |y| 2 16 − n 4 2 U 2 ! dy − n 2 Z |∇U| 2 dy − n 8 Z U 2 dy, (2.67) from where ∥∆U∥ 2 ≲ ∥HU∥ 2 + ∥U∥ 2 + ∥∇U∥ 2 . (2.68) By Sobolev’s and the Cauchy-Schwarz inequalities, we get ∥∇U∥ 2 ≤ ∥U∥∥∆U∥ ≤ ∥U∥ 2 + ∥∆U∥ 2 2 , (2.69) and then using ∥D2U∥ ≲ ∥∆U∥, (2.70) 21 we obtain (2.65); note that the inequality (2.70) follows by R ∆U∆U = R ∂iiU ∂jjU = R ∂ijU ∂ijU, due to fast decay of U and its spatial derivatives (note that u is smooth and periodic). To treat I2 in (2.64), we first estimate ∥V U˜∥. Observe that we cannot apply the Gagliardo-Nirenberg inequalities directly since ∥V (·, τ )∥Lp is infinite whenever v is not identically zero, due to periodicity of v. Noting that V is e τ /2 -periodic, we tile R n as R n = [ j∈Zn Ωj,τ , (2.71) where Ωj,τ = jeτ /2 + e τ /2Ω. Then we have ∥V U∥ 2 = X j∈Zn ∥V U∥ 2 L2(Ωj,τ ) ≲ X j∈Zn ∥V ∥ 2 Lp(Ωj,τ ) ∥U∥ 2 L2p/(p−2)(Ωj,τ ) . (2.72) Note that ∥V ∥Lp(Ωj,τ ) ≤ M0e ατ , j ∈ Z n , (2.73) where α = n 2p , (2.74) by periodicity and using a substitution. Hence, we obtain ∥V U∥ 2 ≲ X j∈Zn M2 0 e 2ατ ∥U∥ 2−2α L2(Ωj,τ ) (∥D2U∥L2(Ωj,τ ) + ∥U∥L2(Ωj,τ ) ) 2α ≲ M2 0 e 2ατ   X j∈Zn ∥U∥ 2 L2(Ωj,τ )   1−α   X j∈Zn ∥D2U∥ 2 L2(Ωj,τ ) + X j∈Zn ∥U∥ 2 L2(Ωj,τ )   α = M2 0 e 2ατ ∥U∥ 2−2α (∥D2U∥ + ∥U∥) 2α , (2.75) 22 where we used the discrete Hölder inequality in the second step. Therefore, taking the square root of (2.75) and dividing by ∥U∥, we get ∥V U˜∥ ≲ M0e ατ (∥D2U˜∥ + 1)α , (2.76) where we used ∥U˜∥ = 1. By (2.65), we have ∥D2U˜∥ ≲ ∥HU˜∥ + 1 ≲ ∥HU˜ − A(τ )U˜∥ + ∥(A(τ ) − Q¯(τ )I)U˜∥ + ∥Q¯(τ )U˜∥ + 1 ≲ e −τ /2 |r|∥yU˜∥ + ∥(A(τ ) − Q¯(τ )I)U˜∥ + |Q¯(τ )| + 1, (2.77) applying (2.54) in the last step. Since ∥∇U˜∥ + ∥yU˜∥ ≲ |Q¯(τ )| 1/2 + e −τ /2 |r| + 1, (2.78) ([Ku4]), as one may readily check, we get ∥D2U˜∥ ≲ e −τ /2 |r| |Q¯(τ )| 1/2 + e −τ /2 |r| + 1 + ∥(A(τ ) − Q¯(τ )I)U˜∥ + |Q¯(τ )| + 1. (2.79) Using (2.76) and (2.79), we obtain ∥V U˜∥ ≲ M0e ατ e −ατ /2 |r| α |Q¯(τ )| α/2 + e −ατ /2 |r| α + 1 + ∥(A(τ ) − Q¯(τ )I)U˜∥ α + |Q¯(τ )| α + 1 . (2.80) Next, we proceed to estimate ∥yjWjU˜ + Wj∂jU˜∥ by first bounding ∥Wj∂jU˜∥ and then ∥yjWjU˜∥. Analogously to (2.72)–(2.75), we have ∥Wj∂jU∥ ≲ M1e (β−1/2)τ ∥U∥ 1−β (∥D2U∥ + ∥U∥) β (2.81) 2 with β = n 2q + 1 2 , (2.82) where we also used ∥W∥Lq(Ωj,τ ) ≲ M1e nτ /2q = M1e (β−1/2)τ , j ∈ Z n . (2.83) Note that the exponents a and b in (2.11) satisfy a = 2 3 − 4α , b = 2 3 − 4β . (2.84) Dividing (2.81) by ∥U∥, we obtain, similarly to (2.80), that ∥Wj∂jU˜∥ ≲ M1e (β−1/2)τ ∥HU˜∥ β + 1 ≲ M1e (β−1/2)τ (e −βτ /2 |r| β (|Q¯(τ )| β/2 + e −βτ /2 |r| β + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ β + |Q¯(τ )| β + 1). (2.85) Next, ∥yjWjU∥ 2 = X j∈Zn ∥yjWjU∥ 2 L2(Ωj,τ ) ≲ X j∈Zn ∥W∥ 2 Lq(Ωj,τ ) ∥yU∥ 2 L2q/(q−2)(Ωj,τ ) ≲ X j∈Zn M2 1 e (2β−1)τ ∥y|U˜| 1/2 ∥ 2 L4(Ωj,τ ) ∥|U| 1/2 ∥ 2 L4q/(q−4)(Ωj,τ ) = X j∈Zn M2 1 e (2β−1)τ ∥|y| 2U∥L2(Ωj,τ )∥U∥L2q/(q−4)(Ωj,τ ) ≲ X j∈Zn M2 1 e (2β−1)τ ∥|y| 2U∥L2(Ωj,τ )∥U∥ 1−n/2q L2q/(q−4)(Ωj,τ ) (∥D2U∥L2(Ωj,τ ) + ∥U∥L2(Ωj,τ ) ) n/2q . (2.86) 24 Applying the discrete Hölder inequality, taking a square root, and dividing by ∥U∥ leads to ∥yjWjU˜∥ ≲ M1e (β−1/2)τ ∥|y| 2U˜∥ 1/2 (∥∆U˜∥ n/2q + 1). (2.87) Observe that in (2.86) we need q ≥ 4. (Note that for n = 1 we have q ≥ 4, needed in (2.86), by Remark 2.1.6.) Recall that HU = −∆U + (|y| 2/16 − n/4)U, from where ∥|y| 2U∥ ≲ ∥HU∥ + ∥∆U∥ + ∥U∥ ≲ ∥HU∥ + ∥U∥, (2.88) by using (2.77) in the last inequality. Applying (2.88) in (2.87), we get ∥yjWjU˜∥ ≲ M1e (β−1/2)τ (∥∆U˜∥ n/2q + 1) ∥HU˜∥ 1/2 + 1 ≲ M1e (β−1/2)τ ∥HU˜∥ n/2q + 1∥HU˜∥ 1/2 + 1 ≲ M1e (β−1/2)τ (∥HU˜∥ β + 1), (2.89) where we used (2.82). With (2.79), we then obtain ∥yjWjU˜∥ ≲ M1e (β−1/2)τ e −βτ /2 |r| β (|Q¯(τ )| β/2 + e −βτ /2 |r| β + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ β + |Q¯(τ )| β + 1 . (2.90) By (2.80), (2.85), and (2.90), we get an estimate for I2 from (2.64) which reads I2 ≲ ∥(A(τ ) − Q¯(τ )I)U˜∥ e (β−1)τM1 e −βτ /2 |r| β (|Q¯(τ )| β/2 + e −βτ /2 |r| β + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ β + |Q¯(τ )| β + 1 + ∥(A(τ ) − Q¯(τ )I)U˜∥ e (α−1)τM0 e −ατ /2 |r| α (|Q¯(τ )| α/2 + e −ατ /2 |r| α + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ α + |Q¯(τ )| α + 1 . (2.91 Using (2.91) in (2.64), we obtain 1 2 Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 ≲ e −τ |r| 2 + e (β−1)τM1∥(A(τ ) − Q¯(τ )I)U˜∥ e −βτ /2 |r| β |Q¯(τ )| β/2 + e −βτ |r| 2β + e −βτ /2 |r| β + ∥(A(τ ) − Q¯(τ )I)U˜∥ β + |Q¯(τ )| β + 1 + e (α−1)τM0∥(A(τ ) − Q¯(τ )I)U˜∥ e −ατ /2 |r| α |Q¯(τ )| α/2 + e −ατ |r| 2α + e −ατ /2 |r| α + ∥(A(τ ) − Q¯(τ )I)U˜∥ α + |Q¯(τ )| α + 1 . (2.92) We now apply Young’s inequality to the terms involving ∥(A(τ ) − Q¯(τ )I)U˜∥ on the right-hand side so that we may absorb them into the second term in the left hand side; namely, we use N∥(A(τ ) − Q¯(τ )I)U˜∥ γ ≤ ϵ0∥(A(τ ) − Q¯(τ )I)U˜∥ 2 + Cϵ0N 2/(2−γ) , (2.93) where ϵ0 ∈ (0, 1] is arbitrarily small, with γ = 1, α+ 1, β + 1 and corresponding expressions for N. Thus, we get Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 ≲ e −τ |r| 2 + M2 1 e (β−2)τ |r| 2β |Q¯(τ )| β + M2 1 e (2β−2)τ |Q¯(τ )| 2β + M2 1 e −2τ |r| 4β + M2 1 e (β−2)τ |r| 2β + M 2/(1−β) 1 e −2τ + M2 1 e (2β−2)τ + M2 0 e (α−2)τ |r| 2α |Q¯(τ )| α + M2 0 e (2α−2)τ |Q¯(τ )| 2α + M2 0 e −2τ |r| 4α + M2 0 e (α−2)τ |r| 2α + M 2/(1−α) 0 e −2τ + M2 0 e (2α−2)τ . (2.94) 26 The second term on the right-hand side of (2.94) may be absorbed into the third and the fourth by the Cauchy-Schwarz inequality. Similarly, the eight term is absorbed by the ninth and tenth. Using also |r| ≲ 1/ϵ, the last inequality implies Q¯′ (τ ) ≲ e −τ ϵ −2 + M2 1 e (2β−2)τ |Q¯(τ )| 2β + M2 1 e −2τ ϵ −4β + M2 1 e (β−2)τ ϵ −2β + M 2/(1−β) 1 e −2τ + M2 1 e (2β−2)τ + M2 0 e (2α−2)τ |Q¯(τ )| 2α + M2 0 e −2τ ϵ −4α + M2 0 e (α−2)τ ϵ −2α + M 2/(1−α) 0 e −2τ + M2 0 e (2α−2)τ . (2.95) To estimate Q¯(τ0), we now compare Q¯(τ ) and Q(τ ) for any τ ≥ τ0. Using (2.78), we have ∥|y| 1/2U˜∥ ≤ ∥|y|U˜∥ 1/2 ≲ |Q¯(τ )| 1/4 + e −τ /4 |r| 1/2 + 1, (2.96) where we used the Cauchy-Schwarz inequality in the first step. By (2.55) and (2.96), we get Q¯(τ ) ≤ Q(τ ) + e −τ /2 rj Z yjU˜ 2 dy ≲ Q(τ ) + e −τ /2 |r|∥|y| 1/2U˜∥ 2 ≲ Q(τ ) + e −τ /2 |r| |Q¯(τ )| 1/2 + e −τ /2 |r| + 1 , (2.97) from where, after absorbing the second term on the far right side, Q¯(τ ) ≲ Q(τ ) + e −τ |r| 2 + 1. (2.98) Note, in passing, that a similar derivation also leads to Q(τ ) ≲ Q¯(τ )+ + e −τ |r| 2 + 1. (2.99) 2 Using Lemma 2.3.1 and (2.98), we have Q¯(τ0) ≤ C0 ϵqD(−ϵ) + ϵ −1 + 1 ≤ C1 ϵ , (2.100) where C1 = C0(q0 + 2) and C0 ≥ 1 is the constant in the inequality (2.98). Denote by C2 ≥ 1 the implicit constant in the inequality (2.95). Up to this point, all the estimates hold for any fixed ϵ ∈ (0, 1/2]. Now, fix ϵ ∈ (0, 1/2] as in (2.59), denoting the constant in (2.59) by C¯. We claim that (2.95) and (2.100) imply Q¯(τ ) ≤ 2 C1 + C2 ϵ , τ ≥ τ0 = − log ϵ. (2.101) Assume, contrary to the assertion, that there exists τ1 ≥ τ0 such that Q¯(τ1) = 2(C1 + C2)/ϵ, and assume that τ1 is the first such time. Also, let τ ′ 0 = sup τ ∈ [τ0, τ1] : Q¯(τ ) = C1 ϵ (2.102) so that Q¯(τ ) ∈ C1 ϵ , 2(C1 + C2) ϵ , τ ∈ [τ ′ 0 , τ1). (2.103) (The purpose of introducing τ ′ 0 is to remedy the fact that Q¯ may be negative with a possibly large absolute value.) Integrating (2.95) between τ ′ 0 and τ1 and using (2.103), we arrive at Q¯(τ1) ≤ Q¯(τ ′ 0 ) + C2ϵ −1 + 2C2M2 1 2 2β (C1 + C2) 2β ϵ 2−4β + C2M2 1 ϵ 2−4β + C2M2 1 ϵ 2−3β + C2M 2/(1−β) 1 ϵ 2 + 2C2M2 1 ϵ 2−2β + 2C2M2 0 2 2α (C1 + C2) 2α ϵ 2−4α + C2M2 0 ϵ 2−4α + C2M2 0 ϵ 2−3α + C2M 2/(1−α) 0 ϵ 2 + 2C2M2 0 ϵ 2−2α , (2.104) 2 where we used 0 ≤ α, β ≤ 3/4. Suppose first that (2.22) and (2.23) hold. The third term on the right-hand side of (2.104) satisfies 2C2M2 1 2 2β (C1 + C2) 2β ϵ 2−4β = 2C2M2 1 2 2β (C1 + C2) 2β ϵ 3−4β ϵ −1 ≤ 2C22 2β (C1 + C2) 2β C¯3−4βϵ ≤ C1 + C2 20ϵ , (2.105) where in the second step we used M2 1 ϵ 3−4β ≤ 1/C¯3−4β , and this results from (2.59), while the last inequality holds if C¯ is sufficiently large. We proceed similarly for the rest of the terms in (2.104) and obtain Q¯(τ1) ≤ Q¯(τ ′ 0 ) + C1 + C2 ϵ ≤ C1 ϵ + C1 + C2 ϵ ≤ 2 C1 + C2 ϵ . (2.106) This is a contradiction with a choice of τ1, and thus we conclude that (2.101) holds for all τ ≥ τ0. Finally, by (2.59) and (2.101), we get Q¯(τ ) ≲ Ma 0 + Mb 1 + 1, (2.107) as desired. 2.6 The order of vanishing Now, we show that the modified frequency function Q¯ controls the vanishing order of u. We first prove the following lemma, which shows the convergence of Q¯(τ ) as τ → ∞ and provides the connection between the order of vanishing of u and the quantity R u 2 (x, t)G(x, t) dx, where G is defined in (2.29). Lemma 2.6.1. Under the assumptions of Theorem 3.1.3, and assuming that u, v, and w are smooth, the modified frequency function satisfies Q¯(τ ) → m/2 as τ → ∞ for some m ∈ N such that m ≲ Ma 0 +Mb 1 + 1, 29 where a and b are as in (2.11). Also, let ϵ be as in (2.59). Moreover, for all δ > 0, there exist t1 ∈ (− log(1/ϵ), 0) and A1(δ), A2(δ) > 0 such that A1(δ)|t| m+δ ≤ Z Rn u 2 (x, t)G(x, t) dx ≤ A2(δ)|t| m−δ , (2.108) for all t ∈ [t1, 0). We emphasize that the constants A1(δ) and A2(δ) are allowed to depend on u, but not on t. Proof of Lemma 3.2.2. First, note that, by Lemma 3.2.1, we have (2.60). Also, let xϵ be as in Lemma 2.3.1 and r as in (2.40). Taking the inner product of (2.56) with U, we obtain 1 2 d dτ ∥U∥ 2 + (A(τ )U, U) = f(τ ), (2.109) since (e −τ /2 rj∂jU, U) = 0, where we denoted f(τ ) = 1 4 e −τ /2 (yjWjU, U) + e −τ /2 (Wj∂jU, U) + e −τ (V U, U). (2.110) Now, we estimate |f(τ )| ≲ e (β−1)τM1∥yU2 ∥Lq/(q−1)(Rn) + e (β−1)τM1∥U∇U∥Lq/(q−1)(Rn) + e (α−1)τM0∥U 2 ∥Lp/(p−1)(Rn) ≲ e (β−1)τM1∥yU∥∥U∥L2q/(q−2)(Rn) + e (β−1)τM1∥U∥L2q/(q−2)(Rn) ∥∇U∥ + e (α−1)τM0∥U∥ 2 L2p/(p−1)(Rn) , (2.111) 30 where we used (2.73) and (2.83). We now use the Gagliardo-Nirenberg inequality to get |f(τ )| ≲ e (β−1)τM1∥yU∥∥U∥ 1−n/q∥∇U∥ n/q + e (β−1)τM1∥U∥ 1−n/q∥∇U∥ n/q+1 + e (α−1)τM0∥U∥ 2−n/p∥∇U∥ n/p . (2.112) Therefore, by (2.78), we may estimate |f(τ )| ∥U(·, τ )∥ 2 ≲ e (β−1)τM1 |Q¯(τ )| + e −τ |r| 2 + 1β + e (α−1)τM0 |Q¯(τ )| + e −τ |r| 2 + 1α , (2.113) where α and β are as in (2.74) and (2.82), and thus, allowing all constants in this proof to depend on M0 and M1 (and thus also on ϵ and r), we obtain Z τ τ0 f(s) ∥U(·, s)∥ 2 ds ≲ e −τ /4 . (2.114) Integrating the equation 1 2∥U∥ 2 d dτ ∥U∥ 2 + Q¯(τ ) = f(τ ) ∥U∥ 2 , (2.115) from τ0 to τ , we get 1 2 log ∥U(·, τ )∥ 2 − 1 2 log ∥U(·, τ0)∥ 2 = − Z τ τ0 Q¯(s) ds + Z τ τ0 f(s) ∥U(·, s)∥ 2 ds ≤ − m 2 (τ − τ0) − Z τ τ0 Q¯(s) − m 2 ds + C(e −τ0/4 + e −τ /4 ). (2.116) To show that Q¯(τ ) → m/2 as τ → ∞ for some m as in the statement, we use Q¯(τ ) ≲ 1 in (2.94) (note that all constants are allowed to depend on M0 and M1) and integrate between τ0 and τ , where 0 ≤ τ0 ≤ τ obtaining Q¯(τ ) − Q¯(τ0) + Z τ τ0 ∥ A(s) − Q¯(s)I U˜∥ 2 ds ≲ e −τ0/4 + e −τ /4 , (2.117) from where, by Q¯(τ0) ≲ 1, Z τ τ0 ∥ A(s) − Q¯(s)I U˜∥ 2 ds ≲ 1 + e −τ0/4 + e −τ /4 . (2.118) We also have ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 ≳ 1 2 ∥(H − Q¯(τ )I)U˜∥ 2 − e −τ |r| 2Q¯ − e −τ |r| 2 ≳ ∥(H − Q¯(τ )I)U˜∥ 2 − 1, (2.119) by the definitions of A(τ ) and H. Therefore, applying (2.119) in (2.118), Z τ τ0 ∥(H −Q¯(s)I)U˜∥ 2 ds ≲ Z τ τ0 ∥ A(s)−Q¯(s)I U˜∥ 2 ds+e −τ ≲ 1+e −τ0/4 +e −τ /4 < ∞, τ0 ≤ τ < ∞, (2.120) where we used the boundedness of Q¯ and a in the first inequality. Combining (2.120) with dist(Q¯(τ ),sp(H)) ≲ ∥(H − Q¯ϵ (τ )I)U˜∥ (2.121) and recalling that ∥U˜∥ = 1, we get Z τ τ0 dist(Q¯(s),sp(H))2 ds < ∞; (2.122) note that the inequality (2.121) follows since H is a self-adjoint positive operator whose inverse is a bounded compact operator. It is well-known that sp(H) = nm 2 : m ∈ N0 o , (2.123) (cf. [CK]), whence Z τ τ0 dist(Q¯(s), 2 −1N0) 2 ds < ∞. (2.124) This, together with Q¯′ (τ ) ≲ 1 for τ ≥ τ0 easily implies that dist(Q¯(τ ), 2 −1N0) → 0 as τ → ∞. (To justify this, simply observe that if dist(Q¯(τ1), 2 −1N0) ≥ ϵ0 for some large τ1 ≥ τ0 and some ϵ0 ∈ (0, 1], then dist(Q¯(τ ), 2 −1N0) ≥ ϵ0/2 for τ ∈ (τ1 − ϵ0/C, τ1), where C is a sufficiently large constant, due to Q¯′ (τ ) < 1.) Thus we obtain limτ→∞ Q¯(τ ) → m 2 , (2.125) for some m ∈ N0 as in the statement. Now, from (2.116) and (2.125), we deduce that for every δ > 0 there exists τ1 > 0, depending on δ, such that −δ(τ − τ1) ≤ log ∥U(τ )∥ 2 − log ∥U(τ1)∥ 2 + m(τ − τ1) ≤ δ(τ − τ1), τ ≥ τ1. (2.126) Then, e −δ(τ−τ1) ≤ e τm∥U(τ )∥ 2 e τ1m∥U(τ1)∥ 2 ≤ e δ(τ−τ1) . (2.127) Therefore, there exists A1(δ), A2(δ) > 0 such that A1(δ)e −δτ ≤ e τm∥U(τ )∥ 2 ≤ A2(δ)e δτ . (2.128) Finally, recalling (2.50), we obtain (2.108). The following lemma provides control on R BR P(x, t)G(x, t) dx, with P a homogeneous polynomial of degree d. We use this lemma in the proof of Theorem 3.1.3. 33 Lemma 2.6.2. Let P(x, t) = Σ|µ|+2l=dCµ,lx µ t l be a homogeneous polynomial of degree d ∈ N. Then, Z Rn P(x, t)G(x, t) dx ≲ |t| d/2 , (2.129) where the constant in (2.129) depends on the polynomial only. Moreover, if all the coordinates of µ = (µ1, µ2, . . . , µn) are even, then for all R > 0, Z BR x µ t lG(x, t) dx ≲ |t| l+|µ|/2 , (2.130) as t → 0 −. If µi is an odd integer for some i ∈ {1, . . . , n}, then Z BR x µ t lG(x, t) dx = 0. (2.131) For the proof of Lemma 3.3.1, cf. [CK]. Proof of Theorem 3.1.3. Without loss of generality, let I = [−1, 0] and (x0, t0) = (0, 0). First, we assume that u, v, and w are smooth. Let ϵ be as in (2.59), and let m be as in the statement of Lemma 3.2.2. Denote by d the vanishing order of u at (0, 0). We claim that d ≤ m. Since the degree of vanishing of u at (0, 0) is d, we have |u(x, t)| ≲ (|x| 2 + |t|) d/2 , (2.132) for all (x, t) ∈ Q1(0, 0) with Q1(0, 0) defined in (2.5); from (2.132) to (2.139) below, the constants are allowed to depend on u. Let δ ∈ (0, 1] be arbitrary. By Lemma 3.2.2, there exists t1 ∈ (− log(1/ϵ), 0) and A1(δ), A2(δ) > 0 such that A1(δ)|t| m+δ ≤ Z u 2 (x, t)G(x, t) dx ≤ A2(δ)|t| m−δ , t ∈ [t1, 0). (2.133) 34 Let R = 1/4. Note that we have Z Rn\BR u 2 (x, t)G(x, t) dx ≲ ∥u∥ 2 L∞(Tn) Z ∞ R e −ρ 2/4|t| |t| n/2 ρ n−1 dρ ≤ ∥u∥ 2 L∞(Tn) e −R2/8|t| Z ∞ R e −ρ 2/8|t| |t| n/2 ρ n−1 dρ ≲ ∥u∥ 2 L∞(Tn) e −R2/8|t| Z ∞ 0 e −ρ 2/8|t| |t| n/2 ρ n−1 dρ ≲ ∥u∥ 2 L∞(Tn) e −R2/8|t| . (2.134) Using (2.134) in (2.133), we may increase t1 < 0 to obtain 1 2 A1(δ)|t| m+δ ≤ Z BR u 2 (x, t)G(x, t) dx ≤ 2A2(δ)|t| m−δ , t ∈ [t1, 0). (2.135) Moreover, by Lemma 3.3.1, Z BR u 2 (x, t)G(x, t) dx ≲ Z BR (|x| 2 + |t|) dG(x, t) dx ≲ |t| d . (2.136) Combining (2.135) with (2.136), we get A1(δ)|t| m+δ ≲ |t| d , t ∈ [t1, 0), (2.137) which yields d ≤ m + δ. (2.138) Letting δ → 0, we conclude that d ≤ m, (2.139) as desired. Thus we have proven Theorem 3.1.3 under the additional assumption that v, w, and u0 are smooth. 35 Now, consider the general case. Recall that u (which still represents u¯ from (2.35)) satisfies the equation (2.41), and it is defined for all t ∈ [−1, 0], even though its frequency was studied only for t ≥ log ϵ. Note that by the parabolic regularity u is locally Hölder continuous and u ∈ L∞ t H1 x∩L 2 t H2 x on (−1+δ, 0)×T n . Therefore, we may assume, without loss of generality, that u0 ∈ C(T n ) ∩ H1 (T n ), just by adjusting the initial time and that u ∈ L∞ t H1 x ∩ L 2 t H2 x on (−1, 0) × T n . Recall that n ≥ 2; cf. Remark 2.1.6 for n = 1. Now, approximate v, w, and u0 by smooth functions v η , w η , and u η 0 so that v η , w η , and u η 0 converge in L s tL p x, L s tL q x, and L∞ x H1 x to v, w, and u0 respectively for any s ∈ [1, ∞) as η → 0. In other words, we have lim η→0 (∥v − v η ∥Ls tL p x + ∥w − w η ∥Ls tL q x + ∥u0 − u η 0 ∥H1(Tn) + ∥u0 − u η 0 ∥L∞) = 0, s ∈ [1, ∞). (2.140) In the rest of the proof, the space-time Lebesgue spaces are understood to be over T n × (−1, 0); also, we may assume that ∥v η∥L∞t L p x , ∥w η∥L∞t L q x , ∥u η 0 ∥L∞(Tn) , and ∥u η 0 ∥H1(Tn) are uniformly bounded by ∥v∥L∞t L p x , ∥w∥L∞t L q x , ∥u0∥L∞(Tn) , and ∥u0∥H1(Tn) , respectively. For convenience, we allow all constants until (2.163) below to depend on these four quantities, as well as on supt∈[−1,0) ∥u(·, t)∥H1(Tn) and ∥u∥L2 t L2 x . Let u η be a solution of the equation ∂tu η − ∆u η = r · ∇u η + w η · ∇u η + v ηu η u η (·, −1) = u η 0 . (2.141) Subtracting (2.141) from (2.1), we get ∂tu˜ − ∆˜u = r · ∇u˜ + w η · ∇u˜ + ˜w · ∇u + v ηu˜ + ˜vu u˜(·, −1) = u η 0 − u0, (2.142) 36 where u˜ = u η −u, v˜ = v η −v, and w˜ = w η −w. First, we have u, u˜ ∈ L∞ t L 2 x∩L 2 t H1 x . Using v ∈ L∞ t L 3n/2 x and w ∈ L∞ t L 2n x , we also get D2u, D2u˜ ∈ L 2 tL 2 x . (2.143) Taking the inner product of (2.142) with u˜, we obtain 1 2 d dt∥u˜∥ 2 L2(Tn) + ∥∇u˜∥ 2 L2(Tn) = Z Tn rju∂˜ ju˜ + Z Tn w η j u∂˜ ju˜ + Z Tn w˜ju∂˜ ju + Z Tn v ηu˜ 2 + Z Tn vu˜ u˜ ≲ |r|∥u˜∥L2(Tn)∥∇u˜∥L2(Tn) + ∥w η ∥Lq(Tn)∥u˜∥L2q/(q−2)(Tn) ∥∇u˜∥L2(Tn) + ∥w˜∥Lq(Tn)∥u˜∥L2q/(q−2)(Tn) ∥∇u∥L2(Tn) + ∥v η ∥Lp(Tn)∥u˜∥ 2 L2p/(p−1)(Tn) + ∥v˜∥Lp(Tn)∥u∥L2p/(p−1)(Tn) ∥u˜∥L2p/(p−1)(Tn) , (2.144) whence 1 2 d dt∥u˜∥ 2 L2(Tn) + ∥∇u˜∥ 2 L2(Tn) ≲ ∥u˜∥L2(Tn)∥∇u˜∥L2(Tn) + ∥u˜∥L2q/(q−2)(Tn) ∥∇u˜∥L2(Tn) + ∥w˜∥Lq(Tn)∥u˜∥L2q/(q−2)(Tn) + ∥u˜∥ 2 L2p/(p−1)(Tn) + ∥v˜∥Lp(Tn)∥u∥L2p/(p−1)(Tn) ∥u˜∥L2p/(p−1)(Tn) . (2.145) Using ∥u˜∥L2q/(q−2)(Tn) ≲ ∥u˜∥ 1−n/q L2(Tn) ∥∇u˜∥ n/q L2(Tn) + ∥u˜∥L2(Tn) and ∥u˜∥L2p/(p−1)(Tn) ≲ ∥u˜∥ 1−n/2p L2(Tn) ∥∇u˜∥ n/2p L2(Tn) + ∥u˜∥L2(Tn) and then applying Young’s inequality to absorb the terms involving the gradient, we get d dt∥u˜∥ 2 L2(Tn) ≤ ∥u˜∥ 2 L2(Tn) + ∥w˜∥ 2 Lq(Tn) + ∥v˜∥ 2 Lp(Tn) . (2.146) 37 Let ϵ0 ∈ (0, 1]. Then, for η sufficiently small, we have ∥u˜(·, −1)∥ 2 H1(Tn) , ∥w˜∥ 2 L2 t L q x , ∥v˜∥ 2 L2 t L p x ≤ ϵ0, (2.147) where, recall, the mixed space-time norms are taken over T n ×(−1, 0). Applying (2.147) and the Gronwall inequality to (2.146), we get ∥u˜∥ 2 H1(Tn) ≲ ϵ0, t ∈ [−1, 0]. (2.148) Since ϵ0 was arbitrary, we get lim η→0 sup t∈[−1,0] ∥u˜(·, t)∥L2(Tn) = 0. (2.149) In order to obtain the analog of (2.149) for the H1 -norm, we test the first equation in (2.142) with −∆˜u, which we may by (2.143), and obtain 1 2 d dt∥∇u˜(·, t)∥ 2 L2(Tn) + ∥∆˜u(·, t)∥ 2 L2(Tn) = − Z Tn rj∂ju˜∆˜u − Z Tn w η j ∂ju˜∆˜u − Z Tn w˜j∂ju∆˜u − Z Tn v ηu˜∆˜u − Z Tn vu˜ ∆˜u. (2.150) By Hölder’s and Sobolev’s inequalities, we get 1 2 d dt∥∇u˜(·, t)∥ 2 L2(Tn) + ∥∆˜u(·, t)∥ 2 L2(Tn) ≲ |r|∥∇u˜∥L2(Tn)∥∆˜u∥L2(Tn) + ∥w η ∥Lq(Tn)∥∇u˜∥L2q/(q−2)(Tn) ∥∆˜u∥L2(Tn) + ∥w˜∥Lq(Tn)∥∇u∥L2q/(q−2)(Tn) ∥∆˜u∥L2(Tn) + ∥v η ∥Lp(Tn)∥u˜∥L2p/(p−2)(Tn) ∥∆˜u∥L2(Tn) + ∥v˜∥Lp(Tn)∥u∥L2p/(p−2)(Tn) ∥∆˜u∥L2(Tn) . (2.151) 38 For the second term on the right-hand side, we have, using the agreement on constants from above (2.141), ∥w η ∥Lq(Tn)∥∇u˜∥L2q/(q−2)(Tn) ∥∆˜u∥L2(Tn) ≤ ∥∇u˜∥L2q/(q−2)(Tn) ∥∆˜u∥L2(Tn) ≲ ∥∇u˜∥ (q−n)/q L2(Tn) (∥D2u˜∥L2(Tn) + ∥u˜∥L2 ) n/q∥∆˜u∥L2(Tn) ≲ ∥∇u˜∥ (q−n)/q L2(Tn) (∥∆˜u∥L2(Tn) + ∥u˜∥L2 ) n/q∥∆˜u∥L2(Tn) ≲ ϵ0∥∆˜u∥ 2 L2(Tn) + Cϵ0 (∥∇u˜∥ 2 L2(Tn) + ∥u˜∥ 2 L2(Tn) ), (2.152) where ϵ0 ∈ (0, 1] is arbitrary. An analogous estimate also holds for the fourth term on the right-hand side of (2.151). For the third term, we have similarly to (2.152) ∥w˜∥Lq(Tn)∥∇u∥L2q/(q−2)(Tn) ∥∆˜u∥L2(Tn) ≲ ∥w˜∥Lq(Tn)∥∇u∥ (q−n)/q L2(Tn) (∥D2u∥L2 + ∥u∥L2 ) n/q∥∆˜u∥L2(Tn) ≲ ∥w˜∥Lq(Tn) (∥D2u∥L2(Tn) + 1)n/q∥∆˜u∥L2(Tn) ≲ ∥w˜∥Lq(Tn) (∥∆u∥L2(Tn) + 1)n/q∥∆˜u∥L2(Tn) ≲ ϵ0∥∆˜u∥ 2 L2(Tn) + ϵ0∥∆u∥ 2 L2(Tn) + Cϵ0 ∥w˜∥ 2 Lq(Tn) , (2.153) while for the last term in (2.151), we estimate similarly ∥v˜∥Lp(Tn)∥u∥L2p/(p−2)(Tn) ∥∆˜u∥L2(Tn) ≲ ∥v˜∥Lp(Tn)∥u∥ (q−n)/q L2(Tn) (∥Du∥L2 + ∥u∥L2 ) n/q∥∆˜u∥L2(Tn) ≲ ∥v˜∥Lq(Tn)∥∆˜u∥L2(Tn) ≲ ϵ0∥∆˜u∥ 2 L2(Tn) + Cϵ0 ∥v˜∥ 2 Lq(Tn) . (2.154) Using all the bounds on the terms on the right hand side of (2.151) and absorbing the terms involving ∥∆˜u∥ 2 L2(Tn) , we obtain d dt∥∇u˜(·, t)∥ 2 L2(Tn) + ∥∆˜u(·, t)∥ 2 L2(Tn) ≲ ∥u˜∥ 2 L2(Tn) + ∥∇u˜∥ 2 L2(Tn) + ∥v˜∥ 2 Lq(Tn) + ∥w˜∥ 2 Lq(Tn) . (2.155) 39 Now, applying the Gronwall inequality, along with (2.140) and (2.149), we get lim η→0 sup t∈[−1,0] ∥∇u˜(·, t)∥L2(Tn) = 0. (2.156) Denote by Q¯η (τ ) and Q¯(τ ) the modified frequency functions corresponding to u η and u, respectively. Recalling that Q¯(τ ) = (HU, U) ∥U∥ 2 − e −τ /2 rj ∥U∥ 2 Z yjU 2 dy, (2.157) and similarly Q¯(τ ) = (HUη , Uη ) ∥Uη∥ 2 − e −τ /2 rj ∥Uη∥ 2 Z yj (U η ) 2 dy, (2.158) we now claim that Q¯η (τ ) → Q¯(τ ) as η → 0 for all τ ∈ [0, ∞); here, U is given in (2.42), while U η (y, τ ) = e −|y| 2/8u τ (ye−τ /2 , −e −τ ) for (y, τ ) ∈ R n × [τ0, ∞). Note that, for η ≥ 0, we have ∥U η (τ )∥ 2 − ∥U(τ )∥ 2 = Z ((u η ) 2 − u 2 )G(x, t) dx ≲ 1 t n/2 Z |(u η ) 2 − u 2 | dx ≲ 1 t n/2 ∥u˜(·, t)∥L2(Tn) (∥u η (·, t)∥L2(Tn) + ∥u(·, t)∥L2(Tn) ), (2.159) and thus lim η→0 ∥U η (τ )∥ 2 = ∥U(τ )∥ 2 , τ ≥ 0, (2.160) by (2.149). Also, for (HUη , Uη ) − (HU, U), we have |(HUη , Uη ) − (HU, U)| ≤ |t| Z |∇u η | 2 − |∇u| 2 G(x, t) dx ≤ 1 |t| n/2−1 ∥∇u˜∥L2(Tn)∥∇(u η + u)∥L2(Tn) , (2.161) from where lim η→0 (HUη , Uη ) = (HU, U), τ ≥ 0, (2.162) 40 by (2.156). Lastly, we examine the convergence of R yj (U η ) 2 dy. For this purpose, we estimate, for every τ ≥ 0, Z yj (U η ) 2 dy − Z yjU 2 dy ≤ Z |y||(U η ) 2 − U 2 | ≲ ∥y(U η − U)∥∥U η + U∥ ≲ (H(U η − U), Uη − U) 1/2 + ∥U η − U∥ ∥U η + U∥ = |t| Z Rn |∇u˜(x, t)| 2G(x, t) dx + ∥U η − U∥ ∥U η + U∥ ≲ (|t|∥∇(u η − u)∥∥∇(u η − u)G∥) 1/2 + ∥U η − U∥ ∥U η + U∥, (2.163) where we used ∥yU∥ 2 ≲ (HU, U) + ∥U∥ 2 in the third inequality, which in turn follows from the identity (HU, U) = ∥∇U∥ 2 + 1 16 ∥yU∥ 2 − n 4 ∥U∥ 2 ; (2.164) also, in the last inequality of (2.163), we applied (2.51). By (2.160), (2.161), and (2.163), we get lim τ→0 Z y(U η ) 2 dy = Z yU2 dy. (2.165) Using (2.160), (2.162), and (2.165) in (2.157) and (2.158), we obtain that Q¯η (τ ) → Q¯(τ ) as η → 0 for all τ ∈ [0, ∞). Note that we have Q¯η (τ ) ≲ Ma 0 + Mb 1 + 1, (2.166) uniformly in τ and η ∈ (0, 1], where the constant depends only on q0. Passing to the limit, we get Q¯(τ ) ≲ Ma 0 + Mb 1 + 1, τ ≥ 0. (2.167) Next, by (2.120) and (2.121), we obtain, for all η ∈ (0, 1], Z τ τ0 dist(Q¯η (τ ),sp(H))2 ≲ Ma 0 + Mb 1 + 1 + e −τ0/4 + e −τ /4 ≲ Ma 0 + Mb 1 + 1, (2.168) where we used M0, M1 ≥ 1. Observe that by (2.166) we have Q¯η (τ ) ∈ [0, M¯ ], where M¯ is independent of η and τ . Thus letting η → 0 in (2.168) yields Z τ τ0 dist(Q¯(τ ),sp(H))2 ≲ Ma 0 + Mb 1 + 1, (2.169) with a and b as in (2.10). Combining (2.167), (2.169), and (2.123), we may use the same argument as in the proof of Lemma 3.2.2, to obtain Q¯(τ ) → m/2 for some m ∈ N0 such that m ≲ Ma 0 + Mb 1 + 1. The rest of the proof is similar to the case when v, w, and u0 are smooth. 2.7 Pointwise in time observability In this section,we use the notation from Section 2.4 again. We fix ϵ as in (2.59), where a and b are given in (2.11). Since we are interested in observability, it is advantageous to slightly generalize the function u¯ from (2.35) as follows. Let t1 ∈ [−ϵ/2, 0] be arbitrary but fixed, and let τ1 = − log t1. Then, define u(x, t) = ¯u x − xϵ t1 + ϵ (x − t1), t , (2.170) where xϵ is as in Lemma 2.3.1, so that, instead of (2.36) and (2.37), we have u¯(x, −ϵ) = u(x + xϵ, −ϵ) (2.171) 42 and u¯(x, t1) = u(x, t1), (2.172) for all x ∈ T n . The equation (2.41) continues to hold with (2.40) replaced by r = − xϵ t1 + ϵ . Since |r| ≲ ϵ −1 , due to t1 ∈ [−ϵ/2, 0], all the estimates from previous sections continue to hold for u¯ in (2.170). The following lemma provides a comparison between the unweighted and weighted L 2 norms of u¯. Lemma 2.7.1. Under the assumptions of Theorem 3.1.3, with u¯ defined in (2.170), and ϵ in (2.59), we have ∥u¯(·, t)∥ 2 L2(Tn) ≲ e K |t|M Z Rn u¯(x, t) 2G(x, t) dx, (2.173) for all t ∈ [−ϵ, 0), where K = C(M2 1 + M0 + M1M2β−1 + M0M2α−1 ) and M = Ma 0 + Mb 2 + 1 with a and b as in (2.11). Proof of Lemma 3.7.2. In this proof, we use the convention from the proof of Lemma 3.2.1 by writing u instead of u¯. We start with the identity (2.109), where f is defined in (2.110). From (2.113), we get f(τ ) ∥U∥ 2 ≳ −e (β−1)τM1 |Q¯(τ )| + e −τ |r| 2 + 1β − e (α−1)τM0 |Q¯(τ )| + e −τ |r| 2 + 1α , (2.174) where α and β are as in (2.74) and (2.82). Therefore, as in the first equality in (2.116), we have 1 2 log ∥U(·, τ )∥ 2 − 1 2 log ∥U(·, τ0)∥ 2 = − Z τ τ0 Q¯(s) ds + Z τ τ0 f(s) ∥U(·, s)∥ 2 ds, (2.175) from where we obtain, using Q¯(τ ) ≲ M = Ma 0 + Mb 1 + 1 for τ ≥ τ0, that log ∥U(·, τ0)∥ 2 ∥U(·, τ )∥ 2 ≲ (τ − τ0)M + e (β−1)τ0M1(M + e −τ0 |r| 2 + 1)β + e (α−1)τ0M0(M + e −τ0 |r| 2 + 1)α ≲ (τ − τ0)M + ϵ 1−βM1(M + ϵ|r| 2 + 1)β + ϵ 1−αM0(M + ϵ|r| 2 + 1)α , (2.176) where we used ϵ = e −τ0 from (2.44). We now derive a similar estimate for ∥u(·, t)∥ 2 L2(Tn) /∥u(·, −ϵ)∥ 2 L2(Tn) . Taking the inner product of (2.41) with u and using ∥∇u∥ 2 L2(Tn) = qD (t)∥u∥ 2 L2(Tn) , where qD is defined in (2.6), we obtain 1 2 d dt∥u∥ 2 L2(Tn) + qD(t)∥u(·, t)∥ 2 L2(Tn) ≲ ∥w∥Lq(Tn)∥u∥L2q/(q−2)(Tn) ∥u∥L2(Tn)qD(t) 1/2 + ∥v∥Lp(Tn)∥u∥ 2 L2p/(p−1)(Tn) ≲ M1∥u∥ 2−2β L2(Tn) ∥∇u∥ 2β−1 L2(Tn) ∥u∥L2(Tn)qD(t) 1/2 + M1∥u∥ 2 L2(Tn) qD(t) 1/2 + M0∥u∥ 2−2α L2(Tn) ∥∇u∥ 2α L2(Tn) + M0∥u∥ 2 L2(Tn) , (2.177) where the quantities are evaluated at t, since R Tn rju∂ju dx = 0. Dividing both sides of (2.177) by ∥u∥ 2 L2(Tn) , we get 1 ∥u∥ 2 L2(Tn) d dt∥u∥ 2 L2(Tn) + qD(t) ≲ M1qD(t) β + qD(t) 1/2 + M0qD(t) α + M0 ≲ M0 + M1, (2.178) allowing the last implicit constant to depend on q0. Integrating (2.178) from −ϵ to t leads to log ∥u(·, t)∥ 2 L2(Tn) ∥u(·, −ϵ)∥ 2 L2(Tn) ≲ (M1 + M0)ϵ, t ∈ (−ϵ, 0). (2.179) 44 By the definition of U in (2.42), we have ∥U(·, τ0)∥ 2 ∥u(·, −ϵ)∥ 2 L2(Tn) = R Rn u(x, −ϵ) 2 e −|x| 2/4ϵ dx ϵ n/2 R Tn u(x, −ϵ) 2 dx ≥ e −n/4ϵ ϵ n/2 , (2.180) since |x| ≤ √ n for x ∈ Ω. We then take the logarithm of (2.180) to get log ∥u(·, −ϵ)∥ 2 L2(Tn) ∥U(·, τ0)∥ 2 ≤ − n 2 log 1 ϵ + n 4ϵ ≲ 1 ϵ . (2.181) Adding (2.176), (2.179), and (2.181), we obtain log ∥u(·, t)∥ 2 L2(Tn) ∥U(·, τ )∥ 2 ≲ (M1 + M0)ϵ + 1 ϵ + (τ − log ϵ −1 )M + ϵ 1−βM1(M + ϵ|r| 2 + 1)β + ϵ 1−αM0(M + ϵ|r| 2 + 1)α ≲ M0 + M1 M + M + τM + (M0 + M1)M2β−1 , (2.182) where we used ϵ ≲ 1/M and |r| ≲ ϵ −1 ≲ M, and thus log ∥u(·, t)∥ 2 L2(Tn) ∥U(·, τ )∥ 2 ≲ M2 1 + M0 + Mτ + M1M2β−1 + M0M2α−1 , (2.183) whence ∥u(·, t)∥ 2 L2(Tn) ≲ exp(C(M2 1 + M0 + M1M2β−1 + M0M2α−1 )) 1 |t|M Z Rn u(x, t) 2G(x, t) dx, (2.184) as claimed. Lemma 3.7.2 is combined below with the next statement from [Ku4]. 45 Lemma 2.7.2. Let K ≥ 0 and t0 ∈ [−1/4, 0). If a 1-periodic function u satisfies ∥u∥ 2 L2(Tn) ≤ e K Z Rn u(x) 2G(x, t) dx (2.185) with t ∈ [t0, 0) such that 1 |t| ≥ C |t0| log 1 |t| + C(K + 1) |t0| , (2.186) for a sufficiently large constant C > 0, then ∥u∥ 2 L2(Tn) ≤ CeK |t| n/2 ∥u∥ 2 L2(B√ |t0| ) . (2.187) For the proof, cf. [Ku4]. Proof of Theorem 3.1.2. We first assume that u0, v, and w are smooth as in the proof of Theorem 3.1.3. By Lemma 3.7.2, we have (2.173), where K and M are given in the statement. In order to apply Lemma 3.4.1, we need t ∈ [−ϵ/2, 0), along with 1 |t| ≥ C ϵ log 1 |t| + C(K + 1) ϵ M log 1 |t| . (2.188) One may readily check that the sufficient condition for (2.188) is |t| ≤ ϵ C(K + 1)M(log(1/ϵ))2(log((K + 2)(M + 1)))2 . (2.189) Let t1 ∈ [−ϵ/2, 0] be such that |t1| equals the right-hand side of (2.189). Using this t1 in (2.170) and applying Lemma 3.4.1 leads to ∥u(·, t1)∥L2(Tn) ≤ e P(M) ∥u(·, t1)∥L2(B√ |t1| ) , (2.190) 46 where P is a polynomial, and we obtain the conclusion of the theorem for the time t1. For other times, we simply translate in time. (Note that it sufficient to obtain the observability estimate (2.12) for a sufficiently small δ0, as it is then automatic for larger ones.) For the general case, we approximate u0, v, and w by smooth functions u η 0 , v η , and w η respectively. We then have ∥u η (·, t)∥ 2 L2(Tn) → ∥u(·, t)∥ 2 L2(Tn) and ∥u η (·, t)∥ 2 L2(Bδ0 ) → ∥u(·, t)∥ 2 L2(Bδ0 ) as η → 0 by (2.149). 2.8 The case R n We now address the case of R n and provide the proof for analogous theorems. Proof of Theorem 3.6.3. Note that we now assume the growth condition (2.19) instead of periodicity. Again, without loss of generality, we may consider I = [−1, 0] and (x0, t0) = (0, 0). The proof is similar to the periodic case with small modifications. With ϵ ∈ [0, 1/2], we have ϵ R Rn |∇u(xϵ + y, −ϵ)| 2G(y, −ϵ) dy R Rn u(xϵ + y, −ϵ) 2G(y, −ϵ) dy ≤ 4KϵqD(−ϵ), (2.191) for some xϵ ∈ B2. This was proven in [CK], but since the argument is short, we present it here. Assume that λ Z Rn u(x + y, −ϵ) 2G(y, −ϵ) dy ≤ Z Rn |∇u(x + y, −ϵ)| 2G(y, −ϵ) dy, x ∈ B2, (2.192) where λ = 4KqD (−ϵ). Integrating (2.192) over B2, we have Z Rn G(y, −ϵ) dy Z B2 u(x + y, −ϵ) 2 dx ≤ 1 λ Z Rn G(y, −ϵ) dy Z B2 |∇u(x + y, −ϵ)| 2 dx ≤ 1 λ ∥∇u(·, −ϵ)∥ 2 Z Rn G(y, −ϵ) dy ≤ (2π) n/2 λ ∥∇u(·, −ϵ)∥ 2 , (2.193) 47 where we continue using the convention (2.53). On the other hand, we have the lower bound for the far-left side of (2.193), which reads Z Rn G(y, −ϵ) dy Z B2 u(x + y, −ϵ) 2 dx = Z Rn G(y, −ϵ) dy Z B2(y) u(x, −ϵ) 2 dx ≥ Z B1/2 G(y, −ϵ) dy Z B2(y) u(x, −ϵ) 2 dx ≥ Z B1/2 G(y, −ϵ)dy Z B1 u(x, −ϵ) 2 dx ≥ 1 2 (2π) n/2 Z B1 u(x, −ϵ) 2 dx ≥ 1 2K (2π) n/2 Z Rn u(x, −ϵ) 2 dx, (2.194) where the last inequality holds by the doubling type condition (2.19). Thus we obtain ∥u(x, −ϵ)∥ 2 ≤ 2K λ ∥∇u(·, −ϵ)∥ 2 ≤ 1 2qD (−ϵ) ∥∇u(·, −ϵ)∥ 2 . (2.195) We have a contradiction since qD (−ϵ) = ∥∇u(·, −ϵ)∥ 2/∥u(·, −ϵ)∥ 2 . Therefore, (2.192) cannot hold for all x ∈ B2, i.e., there exists xϵ ∈ B2 such that (2.191) holds. Theorem 3.6.3 then follows as in the proofs of Theorems 3.1.3 and 3.1.2. 2.9 Extensions We now extend Theorems 3.1.3 and 3.1.2 to two settings. In the first (cf. Theorem 2.9.1 below), we allow for v and w to have a lower integrability in time, while in the second, the space exponents for v and w are lowered below 2n/3 and 2n requiring a degree of vanishing of the norms. Here we adopt the notation from Section ??; in particular, I = [T0, T + T0]. 2.9.1 The case L q tL p x : The methods we provided above allow us to consider the case v ∈ L p2 t L p1 x (T n×I) and w ∈ L q2 t L q1 x (T n×I) with p2 and q2 finite. For simplicity, we restrict ourselves to the case of periodic boundary conditions; 48 however, the same holds for the case of R n using the approach from the previous section. When limiting p2 → ∞ and q2 → ∞, the next theorem reduces to the results in Section ??. Theorem 2.9.1. Let n ≥ 2 and I = [T0, T0 +T]. Assume that u ∈ L∞ x,t(T n ×I) is a solution of (2.1) where v ∈ L p2 t L p1 x (T n × I) and w ∈ L q2 t L q1 x (T n × I) such that p1 > 2n/3 and q1 > 2n with p2 > max 2 1 − α , 2 3 − 4α (2.196) and q2 > max 2 1 − β , 2 3 − 4β (2.197) where α = n/2p1 and β = 1/2 + n/2q1. Then, for all (x0, t0) ∈ T n × [T0 + T /2, T0 + T], the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ ∥v∥ a L p2 t L p1 x (Tn×I) + ∥w∥ b L q2 t L q1 x (Tn×I) + 1, (2.198) where a = 2 3 − 4α − 2/p2 (2.199) and b = 2 3 − 4β − 2/q2 . (2.200) Proof of Theorem 2.9.1. Without loss of generality, we may assume I = [−1, 0] and (x0, t0) = (0, 0). For simplicity, we only consider the case w = 0 and v ∈ L p2 t L p1 x (T n × I), as for a nonzero w the proof is similar. Here we define M0(τ ) = ∥v(·, t)∥Lp1 (Tn) , 49 where τ = − log(−t). As in (2.95), we have Q¯′ (τ ) ≲ e −τ ϵ −2 + M0(τ ) 2 e (2α−2)τ |Q¯(τ )| 2α + M0(τ ) 2 e −2τ ϵ −4α + M0(τ ) 2 e (α−2)τ ϵ −2α + M0(τ ) 2/(1−α) e −2τ + M0(τ ) 2 e (2α−2)τ , (2.201) where α = n/2p1, i.e., (2.95) holds with M0 replaced by M0(τ ) and M1 set to zero as we assumed that w = 0. Let ϵ = 1 C¯(∥v∥ a L p2 t L p1 x (Tn×I) + 1), (2.202) with a as in (2.199) and C¯ sufficiently large determined in the Gronwall argument bellow. Denote by C2 ≥ 1 the implicit constant in (2.201). Under the condition (2.100), we claim that Q¯(τ ) ≤ 2 C1 + C2 ϵ , τ ≥ τ0, (2.203) where C1 = C0(q0 + 2) and C0 ≥ 1 is the constant in (2.98) and where τ0 is given in (2.44). Assume, contrary to the assertion, that there exists τ1 ≥ τ0 such that Q¯(τ1) = 2(C1 + C2)/ϵ, and suppose that τ1 is the first time with this property. Also let τ ′ 0 be as in (2.102) so that, in particular, (2.103) holds. Then we have Q¯′ (τ ) ≤ C2e −τ ϵ −2 + C2M0(τ ) 2 2 2α e (2α−2)τ (C1 + C2) 2α ϵ −2α + C2M0(τ ) 2 e −2τ ϵ −4α + C2M0(τ ) 2 e (α−2)τ ϵ −2α + C2M0(τ ) 2/(1−α) e −2τ + C2M0(τ ) 2 e (2α−2)τ (2.204) 50 for τ ∈ [τ ′ 0 , τ1]. We integrate (2.204) in τ from τ ′ 0 to τ1 and obtain Q¯(τ1) ≤ Q¯(τ ′ 0 ) + C2ϵ −1 + C22 2α (C1 + C2) 2α ϵ −2α Z τ1 τ0 M0(τ ) 2 e (2α−2)τ dτ + C2ϵ −4α Z τ1 τ0 M0(τ ) 2 e −2τ dτ + C2ϵ −2α Z τ1 τ0 M0(τ ) 2 e (α−2)τ dτ + C2 Z τ1 τ0 M0(τ ) 2/(1−α) e −2τ dτ + C2 Z τ1 τ0 M0(τ ) 2 e (2α−2)τ dτ, (2.205) since τ0 ≤ τ ′ 0 . By Hölder’s inequality, the third term on the right-hand side satisfies C22 2α (C1 + C2) 2α ϵ −2α Z τ1 τ0 M0(τ ) 2 e (2α−2)τ dτ ≤ C22 2α (C1 + C2) 2α ϵ −2α ∥M0(τ ) 2 e −2τ /p2 ∥Lp2/2(τ0,∞) ∥e (2α−2+2/p2)τ ∥Lp2/(p2−2)(τ0,∞) ≤ C22 2α (C1 + C2) 2α ϵ −2α Z ∞ τ0 M0(τ ) p2 e −τ dτ2/p2 ϵ 2−2α−2/p2 ≤ C22 2α (C1 + C2) 2α ϵ 3−4α−2/p2 ∥v∥ 2 L p2 t L p1 x (Tn×I) ϵ −1 ≤ C1 + C2 20ϵ ; (2.206) in the last step, we used p2 > 2/(3 − 4α) from (2.196) in addition to ϵ 3−4α−2/p2 ∥v∥ 2 L p2 t L p1 x (Tn×I) ≤ 1/C¯3−4α−2/p2 which is due to (2.202) with C¯ sufficiently large. Other terms in (2.205) are estimated similarly, except for the sixth one, for which we write C2 Z τ1 τ0 M0(τ ) 2/(1−α) e −2τ dτ ≤ C2∥M0(τ ) 2/(1−α) e −2τ /(1−α)p2 ∥Lp2(1−α)/2(τ0,∞) ∥e (−2+2/(1−α)p2)τ ∥Lp2(1−α)/((1−α)p2−2)(τ0,∞) ≤ C2ϵ 2−2/(1−α)p2 ∥M0(τ )e −τ /p2 ∥ 2/(1−α) Lp2 (τ0,∞) ≤ C2ϵ 3−2/(1−α)p2 ∥v∥ 2/(1−α) L p2 t L p1 x (Tn×I) ϵ −1 ≤ C1 + C2 20ϵ , (2.207) In order to use Hölder’s inequality, we need p2(1−α)/2 ≥ 1, which is guaranteed by (2.196). Also, the last step requires ϵ 3−2/(1−α)p2 ∥v∥ 2/(1−α) L p2 t L p1 x (Tn×I) ≤ 1/C¯3−2/(1−α)p2 , cf. (2.202), in addition to p2 > 2/3(1 − α), 51 which is satisfied due to (2.196). We proceed similarly estimating all the terms in (2.205) from the far right side of (2.206), obtaining Q¯(τ1) ≤ Q¯(τ ′ 0 ) + C1 + C2 2ϵ ≤ 3 2 C1 + C2 ϵ . (2.208) This is a contradiction since Q¯(τ1) = 2(C1 + C2)/ϵ, showing that (2.203) indeed holds for all τ ≥ τ0. The general case, when both v ∈ L p2 t L p1 x (T n × I) and w ∈ L q2 t L q1 x (T n × I) are present, is obtained analogously. 2.9.2 The case L ∞ t L n/2 x and L ∞ t L n x : We assume t −α0 v ∈ L∞L p (T n × I) and t −β0w ∈ L∞L q (T n × I) in the interval n/2 ≤ p ≤ 2n/3 and n ≤ q ≤ 2n with α0 and β0 indicated in the statement. We assume that n ≥ 2 throughout this section and that M0 and M1 are constants such that ∥(t − (T + T0))−α0 v(·, t)∥Lp(Tn) ≤ M0, (2.209) and ∥(t − (T + T0))−β0w(·, t)∥Lq(Tn) ≤ M1, (2.210) for t ∈ I. Theorem 2.9.2. Let n/2 ≤ p ≤ 2n/3 and n ≤ q ≤ 2n. Assume that u ∈ L∞ t L∞ x (T n × I) is a solution of (2.1) with t −α0 v ∈ L∞ t L p x(T n × I) and t −β0w ∈ L∞ t L q x(T n × I) satisfying (2.209) and (2.210) for t ∈ I such that α0 > 2n/p − 3 2 (2.211) and β0 > 2n/q − 1 2 . (2.212) 52 Then, for all (x0, t0) ∈ T n × [T0 + T /2, T0 + T], the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ Ma 0 + Mb 1 + 1, (2.213) where a = 2/(3 + 2α0 −2n/p) and b = 2/(1 + 2β0 −2n/q), with the implicit constant in (2.213) depending on q0 and T. Proof of Theorem 2.9.2. Without loss of generality, I = [−1, 0] and (x0, t0) = (0, 0). Using the notation (2.74) and (2.82), the conditions (2.211) and (2.212) read α0 > 4α − 3 2 (2.214) and β0 > 4β − 3 2 . (2.215) We also have a = 2/(3+2α0−4α), and b = 2/(3+2β0−4β). It suffices to show that Q¯(τ ) ≲ Ma 0 +Mb 1+1, where Q¯ and τ are defined in the proof of Theorem 3.1.3. Let V and W be as in (2.45) and (2.46), and note that V and W are e τ /2 -periodic. As in (2.71), we write R n = [ j∈Zn Ωj,τ , (2.216) where Ωj,τ = jeτ /2 + e τ /2Ω. After a simple change of variable, we then have ∥V ∥Lp(Ωj,τ ) ≤ M0e ατ e −τα0 = M0e (α−α0)τ , (2.217) and ∥W∥Lq(Ωj,τ ) ≤ M1e (β−1/2)τ e −τβ0 ≤ M1e (β−β0−1/2)τ . (2.218) 53 from where, similarly to (2.75) (but with U˜ = U/∥U∥ instead of U), ∥V U˜∥ 2 = X j∈Zn ∥V U˜∥ 2 L2(Ωj,r) ≲ X j∈Zn ∥V ∥ 2 Lp(Ωj,r) ∥U˜∥L2p/(p−2)(Ωj,r) ≲ M2 0 e 2(α−α0)τ ∥U∥ 2−2α (∥D2U˜∥ + 1)2α , (2.219) where we used (2.217) in the last inequality. Similarly, we have the modified version of (2.86), which is ∥yjWjU˜∥ 2 = X j∈Zn ∥yjWjU˜∥ 2 L2(Ωj,r) ≲ X j∈Zn ∥W∥ 2 Lq(Ωj,r) ∥yU˜∥ 2 L2q/(q−2)(Ωj,r) ≲ X j∈Zn M2 1 e (2β−2β0−1)τ ∥|y| 2U˜∥∥U˜∥ 1−n/2q L2q/(q−4)(Ωj,r) (∥D2U˜∥L2(Ωj,r) + ∥U˜∥L2(Ωj,r) ) n/2q . (2.220) and then, applying Hölder’s inequality, ∥yjWjU˜∥ ≲ M1e (β−β0−1/2)τ ∥|y| 2U˜∥ 1/2 (∥∆U˜∥ n/2q + 1). (2.221) Other estimates are the same as in the proof of Lemma 3.2.1, leading to a bound for Q¯′ (τ ), which reads 1 2 Q¯′ (τ ) + ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 ≲ e −τ |r| 2 + ∥(A(τ ) − Q¯(τ )I)U˜∥ e (β−β0−1)τM1 e −βτ /2 |r| β (|Q¯(τ )| β/2 + e −βτ /2 |r| β + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ β + Q¯(τ ) β + 1 + ∥(A(τ ) − Q¯(τ )I)U˜∥ e (α−α0−1)τM0 e −ατ /2 |r| α (|Q¯(τ )| α/2 + e −ατ /2 |r| α + 1) + ∥(A(τ ) − Q¯(τ )I)U˜∥ α + Q¯(τ ) α + 1 ; (2.222) cf. (2.92). Using Young’s inequality, we absorb the terms involving ∥(A(τ ) − Q¯(τ )I)U˜∥ on the right-hand side of (2.222) into ∥(A(τ ) − Q¯(τ )I)U˜∥ 2 on the left-hand side to obtain Q¯′ (τ ) ≲ e −τ |r| 2 + e (β−2β0−2)τM2 1 |r| 2β |Q¯(τ )| β + e (2β−2β0−2)τM2 1 |Q¯(τ )| 2β + M2 1 e (−2β0−2)τ |r| 4β + M2 1 e (β−2β0−2)τ |r| 2β + M 2/(1−β) 1 e 2(β−β0−1)τ /(1−β) + M2 1 e (2β−2β0−2)τ + M2 0 e (α−2α0−2)τ |r| 2α |Q¯(τ )| α + M2 0 e (2α−2α0−2)τ |Q¯(τ )| 2α + M2 0 e (−2α0−2)τ |r| 4α + M2 0 e (α−2α0−2)τ |r| 2α + M 2/(1−α) 0 e 2(α−α0−1)τ /(1−α) + M2 0 e (2α−2α0−2)τ . (2.223) Using |r| ≲ ϵ −1 and absorbing the second and eight terms, the inequality (2.223) implies Q¯′ (τ ) ≲ e −τ ϵ −2 + e (2β−2β0−2)τM2 1 |Q¯(τ )| 2β + M2 1 e (−2β0−2)τ ϵ −4β + M2 1 e (β−2β0−2)τ ϵ −2β + M 2/(1−β) 1 e 2(β−β0−1)τ /(1−β) + M2 1 e (2β−2β0−2)τ + M2 0 e (2α−2α0−2)τ |Q¯(τ )| 2α + M2 0 e (−2α0−2)τ ϵ −4α + M2 0 e (α−2α0−2)τ ϵ −2α + M 2/(1−α) 0 e 2(α−α0−1)τ /(1−α) + M2 0 e (2α−2α0−2)τ . (2.224) Let C2 be the implicit constant in (2.224) and C1 be defined as in (2.100). We can now use the barrier argument to prove that Q¯(τ ) ≤ 2 C1 + C2 ϵ , τ ≥ τ0 = − log ϵ, (2.225) where ϵ = 1 C¯(Ma 0 + Mb 1 + 1), (2.226) with C¯ sufficiently large. The concluding Gronwall argument is identical to that in the proof of Lemma 3.2.1, and thus we omit the details. 55 Remark 2.9.3. A minor modification in the proof allows us to consider also L p2 t L p1 x (R n × I) and L q2 t L q1 x (R n × I) where n/2 ≤ p1 ≤ 2n/3 and n ≤ q1 ≤ 2n. To avoid repetition, we only state the result, which is as follows. Let α = n/2p1 and β = 1/2 + n/2q1. Assume that u ∈ L∞ t L∞ x (T n × I) is a solution of (2.1) with t −α0 v ∈ L p2 t L p1 x (T n × I) and t −β0w ∈ L q2 t L q1 x (T n × I) where p2 > 2/(1 − α) and q2 > 2/(1 − β) are such that α0 > max 2/p2 + 4α − 3 2 , 2/p2 + 2α − 2 2 (2.227) and β0 > max 2/q2 + 4β − 3 2 , 2/q2 + 2β − 2 2 . (2.228) Then, for all (x0, t0) ∈ T n × I, the vanishing order of u at (x0, t0) satisfies O(x0,t0) (u) ≲ ∥t −α0 v∥ a L p2 t L p1 x (Tn×I) + ∥t −β0w∥ b L q2 t L q1 x (Tn×I) + 1 (2.229) where a = 2 3 + 2α0 − 2/p2 − 4α (2.230) and b = 2 3 + 2β0 − 2/q2 − 4β , (2.231) with the implicit constant in (2.229) depending only on q0 and T. 2.9.3 The cases L ∞ t L p x and L ∞ t L q x where 2n/3 > p > n/2 and q > 2n : The result for these cases is the Theorem 3.6.2 with the proof provided below. 56 Proof of Theorem 3.6.2. To avoid repetition, we only provide the estimate (2.15) under the smallness assumption on M0. Note that the proof in Section 2.5 holds until (2.101), but we use ϵ = 1 C¯(Mb 1 + 1), (2.232) for C¯ sufficiently large. We use the barrier argument to obtain Q¯(τ ) ≤ 2 C1 + C2 ϵ , τ ≥ τ0. (2.233) Assume that there exists τ ≥ τ0 such that (2.233) does not hold and let τ1 be the first such time. Define τ ′ 0 by (2.102), and observe that (2.103) holds. Integrating (2.95) in τ between τ ′ 0 and τ1 yields Q¯(τ1) ≤ Q¯(τ ′ 0 ) + C2ϵ −1 + C2M2 1 2 β (C1 + C2) β ϵ 2−4β + C2 2 − 2β M2 1 2 2β (C1 + C2) 2β ϵ 2−4β + C2M2 1 ϵ 2−4β + C2M2 1 ϵ 2−3β + C2M 2/(1−β) 1 ϵ 2 + C2 2 − 2β M2 1 ϵ 2−2β + C2M2 0 2 α (C1 + C2) α ϵ 2−4α + C2 2 − 2α M2 0 2 2α (C1 + C2) 2α ϵ 2−4α + C2M2 0 ϵ 2−4α + C2M2 0 ϵ 2−3α + C2M 2/(1−α) 0 ϵ 2 + C2 2 − 2α M2 0 ϵ 2−2α , (2.234) where we use 0 ≤ α, β ≤ 1. We now claim that each term on the right-hand side of (2.234) is bounded by (C1 + C2)/20ϵ. We only estimate the terms involving M0 since others are estimated same as (2.104) and since the condition on q is the same as in Theorem 3.6.3. Starting with the ninth term, we have C2M2 0 2 α (C1 + C2) α ϵ 2−4α ≤ (C1 + C2)M2 0 ϵ 2−4α ≤ (C1 + C2)M2 0 C 4α−3 (1 + Mb 1 ) 4α−3 ϵ −1 ≤ C1 + C2 20ϵ , (2.235) 57 where we use C1 is sufficiently large in the first inequality and M0 is sufficiently small in the last inequality. Note that −1 < 3 − 4α < 0 since n/2 < p < 2n/3. Similarly, C2 2 − 2α M2 0 2 2α (C1 + C2) 2α ϵ 2−4α ≤ C2 2 − 2α M2 0 2 2α (C1 + C2) 2αC 4α−3 (1 + Mb 1 ) 4α−3 ϵ −1 ≤ C1 + C2 20ϵ , (2.236) given M0 is sufficiently small. Proceeding similarly with the other terms, we conclude that Q¯(τ ) ≤ 2(C1+ C2)/ϵ, which is a contradiction. Therefore, (2.233) holds for all τ ≥ τ0, i.e., Q¯(τ ) ≲ Mb 1 + 1, for all τ ≥ τ0. 58 Chapter 3 Pointwise Schauder estimates for elliptic and parabolic equations 3.1 Schauder estimates for elliptic equations We derive the Schauder estimate for the general order elliptic equation Lu = f, (3.1) where L is an m-th order homogeneous elliptic linear operator in B1 ⊆ R n given by L = X |ν|≤m aν(x)∂ ν , (3.2) where we assume m < n. (The case m ≥ n is covered by the results in [H1].) Note that m needs to be an even integer. Suppose that the coefficients of L satisfy the ellipticity condition X |ν|=m aν(x)ξ ν ≥ 1 K , ξ ∈ ∂B1, x ∈ B1, (3.3) the boundedness X |ν|≤m |aν(x)| ≤ K, x ∈ B1, (3.4) 59 and the Hölder continuity of the leading coefficients X |ν|=m |aν(x) − aν(0)| ≤ K|x| α , x ∈ B1, (3.5) for some positive constants K and α ∈ (0, 1). In the sequel, C denotes a positive constant, which may change from line to line and is allowed to depend on the space dimension n and the order of the differential operator m. We also allow all implicit constants to depend on K and α without mention. We write a ≲ b when a ≤ Cb for some constant C. Denote by Pd the set of polynomials in n variables of degree at most d and P˙ d the set of homogeneous polynomials of degree exactly d, with an addition of the zero polynomial. Note that for d = 0, both P0 and P˙ 0 contain only constants. Definition 3.1.1. Let u ∈ L p (B1) for p ∈ [1, ∞]. For an integer d ≥ 1, we say that u ∈ C d Lp (0) if there exists P ∈ Pd−1 satisfying sup r≤1 ∥u − P∥Lp(Br) r d+n/p < ∞. (3.6) For α ∈ (0, 1), we say that u ∈ C d,α Lp (0) if there exists P ∈ Pd such that sup r≤1 ∥u − P∥Lp(Br) r d+α+n/p < ∞. (3.7) We also introduce the corresponding semi-norms as [u]Cd Lp (0) = inf P ∈Pd−1 sup 0<r<1 ∥u − P∥Lp(Br) r d+n/p (3.8) and [u]C d,α Lp (0) = inf P ∈Pd sup 0<r<1 ∥u − P∥Lp(Br) r d+α+n/p . (3.9) 60 The following is the main result of this chapter; it provides a Schauder estimate for the elliptic equation (3.1). See Section 3.6 for the analogous results for parabolic type equations. Theorem 3.1.2. Let u ∈ Wm,p, where 1 < p < n m , (3.10) be a solution of Lu = f in B1 for f ∈ L p (B1) and L an elliptic operator (3.2) satisfying (3.3)–(3.5) in B1. Suppose that c = sup r≤1 ∥u∥Lp(Br) r d+n/p < ∞ (3.11) and cf = sup r≤1 ∥f∥Lp(Br) r d−m+n/p < ∞, (3.12) for some integer d ≥ m + 1. Assume additionally that f ∈ C d−m+k,α Lp (0) and aν ∈ C k−m+|ν|,α Lp (0) for some α ∈ (0, 1) and k ∈ N, for any |ν| ≥ max{m − k, 0}. Then for all q ∈ [1, np/(n − mp)), we have u ∈ C d+k,α Lq (0), and [u]C d+k,α Lq (0) ≲ ∥u∥Lp(B1) + [f]C d−m Lp (0) + [f]C d−m+k,α Lp (0) + c, (3.13) where the implicit constant in (3.13) depends only on p, d, k, and [aν] C k−m+|ν|,α Lp (0) for |ν| ≥ m − k. The theorem extends the main result in [H1] to the case p < n/m. It also asserts the continuity for the range q ∈ [1, np/(n − mp)). Observe that if a function u satisfies |u| ≲ |x| d on B1, where d ≥ 0 and n ≥ 2, then ∥u∥Lp(Br) ≲ r d+n/p. Thus the assumption (3.11) may be interpreted as u vanishes at 0 with of order at least d in the L p sense. 61 In the following auxiliary statement, we estimate the high L p -Hölder type norms around the points where u vanishes of order d in the L p sense and f vanishes of order d − m in the L p sense. The result we obtain applies to the L q -norm where q belongs to [1, np/(n − mp)), which includes p itself. Theorem 3.1.3. Let u ∈ Wm,p(B1), with p as in (3.10), be a solution of Lu = f, where L is the elliptic operator (3.2) satisfying (3.3)–(3.5) and f ∈ L p (B1). Suppose that, with an integer d ≥ m + 1, we have ∥f − Q∥Lp(Br) ≤ Mrd−m+α+n/p, r ∈ (0, 1], (3.14) for some Q ∈ P˙ d−m and M > 0. Then if c0 = sup r≤1 ∥u∥Lp(Br) r d−1+η+n/p < ∞, (3.15) for some η ∈ (0, 1], there exists P ∈ P˙ d solving X |ν|=m aν(0)∂ νP = Q (3.16) in R n such that ∥P∥Lp(Br) ≲ (M + c0 + ∥Q∥Lp(B1) )r d+n/p, r > 0 (3.17) and X |ν|≤m r |ν| ∥∂ ν (u − P)∥Lq(Br) ≲ M + c0 + ∥Q∥Lp(B1) r d+α+n/q, r ∈ (0, 1/2], (3.18) for all q ∈ [1, np/(n − mp)). Moreover, ∥u∥Lp(Br) ≲ M + c0 + ∥Q∥Lp(B1) r d+n/p, r ∈ (0, 1]. (3.19) The implicit constants depend only on p, q, and d. Theorem 3.1.3 provides control on the leading polynomial and the error terms, given control on the source term f and the assumption that u vanishes of order slightly less than d at 0. It is illustrative to consider the case when f ≡ Q ≡ 0, in which case we are looking at a solution of Lu = 0. Then the assumption (3.15) requires that u vanishes of a fractional order strictly greater than d − 1 in the L p sense and (3.19) asserts that u vanishes of order d (in the L p sense). The proofs of Theorems 3.1.2 and 3.1.3 rely on the following interior existence theorem and a bootstrapping argument in Lemma 3.4.1 below. 3.2 Interior Wm,p existence lemma The following statement asserts the existence of a solution in a small neighborhood which vanishes of high degree provided the same holds for the source term. Lemma 3.2.1. Assume that L = P |ν|≤m aν(x)∂ ν , defined in B1, satisfies the conditions (3.3)–(3.5). Suppose that f ∈ L p (B1), where (3.10) holds, is such that ∥f∥Lp(Br) ≤ Mrd−m+γ+n/p, for r ∈ (0, 1], where γ ∈ (0, 1) and d ≥ m. Then there exist R > 0, depending on α, and u ∈ Wm,p(BR) solving Lu = f in BR such that X |ν|≤m r |ν| ∥∂ νu∥Lp(Br) ≲ Mrd+γ+n/p, r ∈ (0, R]. (3.20) In particular, if the source term vanishes of order at least d − m + γ, then there exists a local solution u which vanishes of order d + γ. Before the proof, we need to estimate R (|f(y)|/|y| b ) dy over Br and B1\Br. Denote by p ′ the Hölder conjugate of p. 63 Lemma 3.2.2. Assume that f ∈ L p (B1), where p ∈ [1, ∞], satisfies ∥f∥Lp(Br) ≤ Mra , r ∈ (0, 1], (3.21) where a ∈ R. (i) If b < a + n/p′ , we have Z Br |f(y)| dy |y| b ≲ Mra−b+n/p′ , r ∈ (0, 1], (3.22) where the implicit constant depends on a, b, and p. (ii) If b > a + n p ′ , (3.23) we have Z B1\Br |f(y)| dy |y| b ≲ Mra−b+n/p′ , r ∈ (0, 1], (3.24) where the implicit constant depends on a, b, and p. Proof of Lemma 3.2.2. (i) Dividing Br into dyadic shells, we have Z Br |f(y)| dy |y| b = X∞ m=0 Z B2−mr \B2−m−1r |f(y)| dy |y| b ≲ X∞ m=0 ∥f∥Lp(B2−mr )∥|y| −b ∥Lp′ (B2−mr \B2−m−1r ) ≲ X∞ m=0 M(2−mr) a−b+n/p′ ≲ Mra−b+n/p′ , (3.25) under the condition b < a + n/p′ . To obtain the second inequality in (3.25), we need to separate the cases bp′ = n and bp′ ̸= n; however, it is easy to check that in both cases we obtain the same result. 64 (ii) Choose m0 ∈ N0 such that 2 −m0 < r ≤ 2 −m0+1. Then we have Z B1\Br |f(y)| dy |y| b ≤ Xm0 m=0 Z B2−m\B2−m−1 |f(y)| dy |y| b ≲ Xm0 m=0 ∥f∥Lp(B2−m)∥|y| −b ∥Lp′ (B2−m\B2−m−1 ) ≲ Xm0 m=0 (2−m) a−b+n/p′ = Xm0 m=0 (2m) −a+b−n/p′ ≲ (2m0 ) −a+b−n/p′ ≲ r a−b+n/p′ , (3.26) again checking separately the case bp′ = n. Throughout this section, we use the notation L(0) = X |ν|=m aν(0)∂ ν , (3.27) while Γ denotes the fundamental solution of L(0); in particular, L(0)Γ = δ0. Since m < n, it satisfies the estimate |∂ β xΓ(x)| ≲ 1 |x| n+|β|−m , β ∈ N d 0 , (3.28) where the constant depends on |β|; see [B]. Proof of Lemma 3.2.1. We start with the case when the coefficients are constant and the lower order terms are not present, i.e., aν ≡ aν(0) for |ν| = m and aν ≡ 0 for |ν| < m. The convolution w(x) = Z |y|<1 Γ(x − y)f(y) dy (3.29) satisfies L(0)w = f in B1. Also, define Pw(x) = X |β|≤d x β β! Z |y|<1 ∂ β xΓ(−y)f(y) dy. (3.30) 65 To obtain the necessary integrability, observe that |∂ β xΓ(−y)f(y)| ≲ |f(y)| |y| n+|β|−m , (3.31) by (3.28), and apply Lemma 3.2.2 (i); note that the condition b < a + n/p′ becomes |β| < d + γ, which indeed holds. Thus, Pw is a polynomial of degree less than or equal to d such that L(0)Pw = 0. Now, we prove that the function u(x) = w(x) − Pw(x) = Z |y|<1 Γ(x − y) − X 0≤|β|≤d ∂ β xΓ(−y) x β β! f(y) dy, (3.32) which satisfies L(0)u = f, verifies (3.20). For any r ∈ (0, 1/2], we have ∥u∥Lp(Br) ≤ Z |y|≤2r Γ(x − y)f(y) dy Lp(Br) + Z |y|≤2r X 0≤|β|≤d ∂ β xΓ(−y) x β β! f(y) dy Lp(Br) + Z 2r≤|y|<1 Γ(x − y) − X 0≤|β|≤d ∂ β xΓ(−y) x β β! f(y) dy Lp(Br) = I1 + I2 + I3. (3.33) For the first term, we use Young’s inequality to bound I1 = Z Γ(x − y)f(y)χB2r (y) dy Lp(Br) ≤ ∥Γ∥L1(Rn)∥f∥Lp(Br) ≤ ∥Γ∥L1(B3r)∥f∥Lp(B2r) ≲ Mrd−m+γ+n/p Z 3r 0 s n−1 s n−m ds ≲ Mrd+γ+n/p , (3.34) where we used (3.28) in the fourth step; note that the L p (Br) norm is taken in the x variable. To estimate I2, we write Z |y|≤2r ∂ β xΓ(−y) x β β! f(y) dy ≲ |x| |β| Z |y|≤2r |f(y)| |y| n+|β|−m dy ≲ r |β| Z |y|≤2r |f(y)| |y| n+|β|−m dy, (3.35) 66 for any β such that |β| ≤ d. Applying Lemma 3.2.2 (i) with a = d − m + γ + n/p and b = n + |β| − m, we have Z |y|≤2r ∂ β xΓ(−y) x β β! f(y) dy ≲ Mr|β| r d+γ−|β| ≲ Mrd+γ . (3.36) Note that the condition b < a + n/p′ becomes γ > 0. Therefore, it follows that I2 ≲ Mrd+γ+n/p as desired. To estimate I3, we first claim that I(x) ≲ Mrd+γ , |x| ≤ r, (3.37) where I(x) = Z 2r≤|y|<1 Γ(x − y) − X |β|≤d ∂ β xΓ(−y) x β β! f(y) dy . (3.38) By Taylor’s theorem Γ(x − y) − X |β|≤d ∂ β xΓ(−y) x β β! ≲ |x| d+1 |y − θx| n+d+1−m ≲ |x| d+1 |y| n+d+1−m , (3.39) for some θ ∈ (0, 1), where we used (3.31) in the first inequality and |x| ≤ r ≤ |y|/2 in the second. Hence, we obtain I(x) ≲ |x| d+1 Z 2r≤|y|<1 |f(y)| |y| n+d+1−m dy. We apply Lemma 3.2.2 (ii) to get I(x) ≲ Mrd+1r γ−1 ≲ Mrd+γ ; (3.40) note that the condition (3.23) becomes γ < 1, which indeed holds. Then, by (3.40), I3 ≲ Mrd+γ+n/p as desired. Therefore, ∥u∥Lp(Br) ≲ Mrd+γ+n/p, r ≤ 1 2 . (3.41) 67 The inequality (3.20) then holds by applying the elliptic regularity to Lu = f, leading to r |ν|∥∂ νu∥Lp(Br) ≲ ∥u∥Lp(B2r) + r m∥f∥Lp(B2r) ≲ Mrd+γ+n/p for any r ≤ 1/4 and |ν| ≤ m. For the variable coefficient case, we follow the approach in [B] and [H1]. Assume first that the coefficients of L are uniformly close to the case considered first, i.e., we have X |ν|=m |aν(x) − aν(0)| + X |ν|<m |aν(x)| ≤ ϵ, (3.42) for some small positive ϵ depending on p, d, and γ only. We rewrite Lu = f as L(0)u = f + (L(0) − L)u, (3.43) where L(0) is introduced in (3.27). Consider the convex set S = u ∈ Wm,p(B1) : ∥u∥W m,p(B1) ≤ M1, X |ν|≤m r |ν| ∥∂ νu∥Lp(Br) ≤ M2r d+γ+n/p , ∀r < 1 , (3.44) and define T : S → Wm,p(B1) as T(u) = Z |y|<1 Γ(x − y) f(y) + (L(0) − L)u(y) dy. (3.45) We claim T(S) ⊆ S and that T has a fixed point is S; the fixed point solves (3.43) and satisfies the condition (3.20). We first claim T(S) ⊆ S for appropriate M1 and M2. For any u ∈ S, there exists T(u) in Wm,p such that L(0)(T(u)) = f + (L(0) − L)u, (3.46) 6 by the previous case. Moreover, ∥T(u)∥W m,p(B1) ≲ ∥f∥Lp(B1) + ∥(L(0) − L)u∥Lp(B1) ≲ ∥f∥Lp(B1) + ϵ∥u∥W m,p(B1) , (3.47) where we used (3.42) in the last inequality. Letting C0 be the implicit constant in (3.47), we choose F, M1, and ϵ such that C0F = M1/2 and C0ϵ ≤ 1/2. By (3.47), we have ∥T(u)∥W m,p(B1) ≤ M1 for ∥u∥Lp(B1) ≤ M1 and ∥f∥Lp(B1) ≤ F. Also, choosing M2 = 2C0M, for any r ≤ 1 we may write X |ν|≤m r |ν| ∥∂ νT(u)∥Lp(Br) ≤ C0(M + ϵM2)r d+γ+n/p ≤ Nrd+γ+n/p (3.48) if P |ν|≤m r |ν|∥∂ νu∥Lp(Br) ≤ M2r d+γ+n/p and ∥f∥Lp(Br) ≤ Mrd−m+γ+n/p. Hence, T(S) ⊆ S for M1, M2, and ϵ chosen above. We now claim that T is a contraction. For any u and v in S, we have T(u) − T(v) = Z |y|<1 Γ(x − y)(L(0) − L)(u − v)(y) dy. (3.49) Therefore, by the global Wm,p estimates and expression of T(u) − T(v), we obtain ∥T(u) − T(v)∥W m,p(B1) ≤ C0ϵ∥u − v∥W m,p(B1) ≤ 1 2 ∥u − v∥W m,p(B1) . (3.50) Thus, T is a contraction from S to S. Let u be a fixed point of T. Then u solves (3.43) and satisfies (3.20), as desired. For the general case, when (3.42) is not required, we consider the transformation x → Rx. For R sufficiently small (depending on K and α, considered fixed), the condition (3.42) is satisfied so the result of the previous case can be applied. 69 Note that in the proof of Lemma 3.2.1, we have also obtained the following statement. Lemma 3.2.3. Assume that w(x) = R |y|<1 Γ(x − y)f(y) dy, where ∥f∥Lp(Br) ≤ Mrd−m+γ+n/p , (3.51) for some γ ∈ (0, 1) and for p satisfying (3.10), and let Pw(x) be its Taylor polynomial of degree d, i.e., Pw(x) = X |β|≤d x β β! Z |y|<1 ∂ β xΓ(−y)f(y) dy. (3.52) Then ∥w − Pw∥Lp(Br) ≤ Mrd+γ+n/p, r ∈ (0, 1/2]. (3.53) 3.3 Extended range for the interior existence lemma The interior Wm,p estimate may be extended to the L q norm for 1 ≤ q < np/(n − mp) as follows. Lemma 3.3.1. Assume that L in (3.2) satisfies the conditions (3.3)–(3.5). Suppose that f ∈ L p (B1), where p satisfies (3.10), and that there exist γ ∈ (0, 1) and d ≥ m such that ∥f∥Lp(Br) ≤ Mrd−m+γ+n/p, for all r ≤ 1. Then there exists a positive constant R, depending on α, and a solution u ∈ Wm,q(BR) of Lu = f in BR (3.54) such that X |ν|≤m r |ν| ∥∂ νu∥Lq(Br) ≲ Mrd+γ+n/q, r ≤ R, (3.55) 70 for 1 ≤ q < np n − mp , (3.56) where the constant in (3.55) depends on p, q, d, and α. Proof of Lemma 3.3.1. The proof is similar to that of Lemma 3.2.1, except for the estimates of I1, I2, and I3, defined in (3.33), with L p (Br) replaced by L q (Br). For I1, we have, assuming r ≤ 1/2, I1 = Z |y|<2r Γ(x−y)f(y) dy Lq(Br) ≲ ∥Γ∥Lp˜(B3r)∥f∥Lp(B2r) ≲ Z 3r 0 s (m−n)˜p+n−1 ds1/p˜ ∥f∥Lp(B2r) , (3.57) where 1/p˜+ 1/p = 1 + 1/q. Since m − n + n/p >˜ 0 by (3.56), we obtain I1 ≲ Mrm−n+n/p˜ r d−m+γ+n/p ≲ Mrd+γ+n/q , (3.58) where we used 1/p˜+1/p = 1+1/q in the last inequality. Since R |y|<2r Pd k=0 ∂ k xΓ(−y)f(y) dy ≲ Mrd+γ for all r ≤ 1/2, as shown (3.36), we have I2 = Z |x|<rZ |y|<2r X d k=0 ∂ k xΓ(−y)f(y) dyq dx1/q ≲ Z |x|<r (Mrd+γ ) q dx1/q ≲ Mrd+γ+n/q . (3.59) Similarly, with I defined in (3.38), I3 = Z |x|<r I(x) q dx1/q ≲ Mrd+γ+n/q (3.60) since I(x) ≲ Mrd+γ , as proven in (3.39). 71 3.4 The bootstrapping argument Before proving Theorem 3.1.3, we demonstrate the following bootstrapping argument, which works under the assumptions of Theorem 3.1.3. Lemma 3.4.1. Under the assumptions of Theorem 3.1.3, suppose that there exists k ∈ N0 such that (k + 1)α + η < 1. If u vanishes of L p -order d − 1 + η + kα, i.e., ck = sup r≤1 ∥u∥Lp(Br) r d−1+η+kα+n/p < ∞, (3.61) then u vanishes of L p -order d − 1 + η + (k + 1)α, and moreover ck+1 = sup r≤1 ∥u∥Lp(Br) r d−1+η+(k+1)α+n/p ≲ ck + M + ∥Q∥Lp(B1) < ∞ (3.62) holds, where M and Q ∈ P˙ d−m are defined in Theorem 3.1.3, and the constant depends on d, p, and q. Recall that all constants are allowed to depend on m, n, α, and K (which are considered fixed). Proof of Lemma 3.4.1. Assuming that (3.61) holds, we first show that ck+1 = sup r≤1 ∥u∥Lp(Br) r d−1+η+(k+1)α+n/p < ∞. (3.63) Applying the classical elliptic interior estimate to (3.54), we have X |ν|≤m r |ν| ∥∂ νu∥Lp(Br) ≲ ∥u∥Lp(B2r) + r m∥f∥Lp(B2r) ≲ ∥u∥Lp(B2r) + r m∥f − Q∥Lp(B2r) + r m∥Q∥Lp(B2r) ≲ ck + M + ∥Q∥Lp(B1) r d−1+η+kα+n/p, r ∈ (0, 1/2], (3.64) 7 where we used (3.14), (3.61), and that Q is a homogeneous polynomial of degree d−m in the last inequality. By Lu = f, we have L(0)u − Q = X |ν|=m aν(0) − aν ∂ νu − X |ν|<m aν∂ νu + (f − Q). (3.65) Then taking the L p -norm of (3.65), we obtain ∥L(0)u − Q∥Lp(Br) ≲ X |ν|=m ∥aν(0) − aν(x)∥L∞(Br)∥∂ νu∥Lp(Br) + X |ν|<m |aν(x)|∥∂ νu∥Lp(Br) + ∥f − Q∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d−1+η+(k+1)α−m+n/p, r ∈ (0, 1/2], (3.66) where we used (3.4), (3.5), and (3.64) in the last inequality. Similarly to Lemma 3.2.1, we write u = v + w, where w(x) = Z |y|≤1/2 Γ(x − y) L(0)u(y) − Q(y) dy, (3.67) and approximate v and w by Pv and Pw, respectively (with Pw defined in (3.70) below and Pv after (3.73)). Note that w solves the equation L(0)w = L(0)u − Q. (3.68) Since η + (k + 1)α ∈ (0, 1), we apply Lemma 3.2.3 with degree d − 1 instead of d to (3.67) and use (3.66) to get ∥w − Pw∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d−1+η+(k+1)α+n/p, r ∈ (0, 1/2], (3.69) where Pw(x) = X |β|≤d−1 x β β! Z |y|≤1/2 ∂ β xΓ(−y) L(0)u(y) − Q(y) dy. (3.70 Moreover, by (3.67), ∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2) + ∥Q∥Lp(B1) . (3.71) Thus, the function v = u − w satisfies ∥v∥Lp(B1/2) ≲ ∥u∥Lp(B1/2) + ∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2) + ∥Q∥Lp(B1) . (3.72) Note that from (3.64), we have ∥∂ νu∥Lp(B1/2) ≲ ck + M + ∥Q∥Lp(B1) for all |ν| ≤ m, which implies ∥u∥W m,p(B1/2) ≲ ck + M + ∥Q∥Lp(B1) . Therefore, ∥v∥Lp(B1/2) ≲ ck + M + ∥Q∥Lp(B1) . (3.73) Let Pv be the Taylor polynomial of v with the degree d − 1, then for |x| = r ≤ 1/4, we may bound |v(x) − Pv(x)| ≲ ∥∂ d v∥L∞(Br)r d ≲ ∥∂ d v∥L∞(B1/4)r d ≲ ∥Q∥Lp(B1/2) + ∥v∥Lp(B1/2) r d ≲ ck + M + ∥Q∥Lp(B1) r d , (3.74) where we used the Taylor theorem in the first step and the elliptic regularity for L(0)v = Q in the second. Therefore, ∥v − Pv∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d+n/p, r ∈ (0, 1/4]. (3.75) Letting P = Pv + Pw, we have u − P = (v − Pv) + (w − Pw). Thus, for all r ≤ 1/4, we may bound ∥u − P∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d+min((k+1)α+η−1,0)+n/p = ck + M + ∥Q∥Lp(B1) r d+(k+1)α+η−1+n/p , (3.76 where we used (k + 1)α + η < 1 in the last equality. We now claim that P ≡ 0. If, contrary to the assertion, we have P ̸≡ 0, then using (3.76) we obtain ∥P∥Lp(Br) ≤ ∥u − P∥Lp(Br) + ∥u∥Lp(Br) ≲ r d−1+η+(k+1)α+n/p + r d−1+η+kα+n/p ≲ r d−1+η+kα+n/p . (3.77) On the other hand, since P is a nontrivial polynomial of degree less than or equal to d − 1, we have ∥P∥Lp(Br) ≳ r d−1+n/p, for all r > 0. Combining with (3.77), we get a contradiction when letting r → 0 since α, η > 0. This proves that P ≡ 0. Substituting P ≡ 0 in (3.76), we get ∥u∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d+min((k+1)α+η−1,0)+n/p, r ∈ (0, 1/4]. (3.78) Since (k + 1)α + η < 1, we have proven ∥u∥Lp(Br) ≲ ck + M + ∥Q∥Lp(B1) r d+η−1+n/p+(k+1)α , r ∈ (0, 1/4], (3.79) so (3.62) holds. 3.5 Proofs of Schauder estimates for elliptic equations Proof of Theorem 3.1.3. We first reduce η slightly so that η/k0 ∈/ Q. Let k0 ∈ N0 be such that k0α + η < 1 ≤ (k0 + 1)α + η. Since η/k0 ∈/ Q, the second equality is strict. We first show that ∥u∥Lp(Br) ≲ (c0 + M + ∥Q∥Lp(B1) )r d+n/p . (3.80) By (3.15), we have c0 = supr≤1 ∥u∥Lp(Br)/rd−1+η+n/p < ∞. Applying Lemma 3.4.1 k0 times, we obtain ck0 = sup r≤1 ∥u∥Lp(Br) r d−1+η+k0α+n/p ≲ c0 + M + ∥Q∥Lp(B1) . (3.81) As in (3.64), we have X |ν|≤m r |ν| ∥∂ νu∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d−1+η+k0α+n/p, r ≤ 1 2 , (3.82) while as in (3.66) we may bound ∥L(0)u − Q∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d−1+η+(k0+1)α−m+n/p, r ≤ 1 2 . (3.83) Letting w(x) = Z |y|≤1/2 Γ(x − y) L(0)u(y) − Q(y) dy, (3.84) it is clear that L(0)w = L(0)u − Q. (3.85) Since η + (k0 + 1)α − 1 ∈ (0, 1), we apply Lemma 3.2.3 to (3.84) and use (3.83) to get ∥w − Pw∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d−1+η+(k0+1)α+n/p , (3.86) where Pw(x) = X |β|≤d x β β! Z |y|≤1/2 ∂ β xΓ(−y) L(0)u(y) − Q(y) dy. (3.87) Moreover, by (3.84), ∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2) + ∥Q∥Lp(B1) . (3.88 Thus the function v = u − w satisfies L(0)v = Q and ∥v∥Lp(B1/2) ≲ ∥u∥Lp(B1/2) + ∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2) + ∥Q∥Lp(B1) ≲ c0 +M + ∥Q∥Lp(B1) , (3.89) where we used (3.82) in the last inequality. Let Pv be the Taylor polynomial of v of degree d. Then L(0)Pv = Q and for |x| = r ≤ 1/4, we have |v(x) − Pv(x)| ≲ ∥∂ d+1v∥L∞(Br)r d+1 ≲ ∥∂ d+1v∥L∞(B1/4)r d+1 ≲ ∥Q∥Lp(B1/2) + ∥v∥Lp(B1/2) r d+1 ≲ c0 + M + ∥Q∥Lp(B1) r d+1 , (3.90) where we used the Taylor theorem in the first step. Therefore, ∥v − Pv∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+1+n/p . (3.91) Letting P˜ = Pv + Pw, we have u − P˜ = (v − Pv) + (w − Pw). Thus, by (3.86) and (3.91), ∥u − P˜∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+min((k0+1)α+η−1,1)+n/p ≲ c0 + M + ∥Q∥Lp(B1) r d+(k0+1)α+η−1+n/p , (3.92) where we used (k0 + 1)α + η − 1 < 1. Since the degree of P˜ is less than or equal to d, we may combine (3.15) with (3.92) and write ∥P˜∥Lp(Br) ≤ ∥u − P˜∥Lp(Br) + ∥u∥Lp(Br) ≲ r d+(k0+1)α+η−1+n/p , (3.93) implying P˜ ≡ 0. Therefore, from (3.92), ∥u∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+(k0+1)α+η−1+n/p ≲ c0 + M + ∥Q∥Lp(B1) r d+n/p . (3. We now construct the homogeneous polynomial P. Applying the classical elliptic interior estimate to (3.54) and using (3.94), we have X |ν|≤m r |ν| ∥∂ νu∥Lp(Br) ≲ ∥u∥Lp(B2r) + r m∥f∥Lp(B2r) ≲ ∥u∥Lp(B2r) + r m∥f − Q∥Lp(B2r) + r m∥Q∥Lp(B2r) ≲ c0 + M + ∥Q∥Lp(B1) r d+n/p, r ≤ 1 2 , (3.95) where we used (3.14), (3.94), and that Q is a homogeneous of degree d − m in the last inequality. Taking the L p -norm of (3.65) and combining with (3.95), we obtain ∥L(0)u − Q∥Lp(Br) ≲ X |ν|=m |aν(0) − aν(x)|∥∂ νu∥Lp(Br) + X |ν|<m |aν(x)|∥∂ νu∥Lp(Br) + ∥f − Q∥Lp(B2r) ≲ c0 + M + ∥Q∥Lp(B1) r d+α+n/p, r ≤ 1 2 , (3.96) where we used (3.4) and (3.5) in the last inequality. Letting w(x) = Z |y|≤1/2 Γ(x − y) L(0)u(y) − Q(y) dy, (3.97) it is clear that L(0)w = L(0)u − Q. (3.98) Since α ∈ (0, 1), we may apply Lemma 3.2.3 for (3.98) and use (3.96) to get ∥w − Pw∥Lp(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+α+n/p , (3.99) where Pw is defined in (3.87). Moreover, by (3.97), ∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2) + ∥Q∥Lp(B1) . (3.100) Thus v = u − w satisfies L(0)v = Q and ∥v∥Lp(B1/2) ≲ ∥u∥Lp(B1/2)+∥w∥Lp(B1/2) ≲ ∥u∥W m,p(B1/2)+∥Q∥Lp(B1) ≲ c0+M +∥Q∥Lp(B1) , (3.101) where we used (3.95) in the last inequality. Letting Pv be the Taylor polynomial of v with the degree d, we have L(0)Pv = Q and for |x| = r ≤ 1/4, and we may bound |v(x) − Pv(x)| ≲ ∥∂ d+1v∥L∞(Br)r d+1 ≲ ∥∂ d+1v∥L∞(B1/4)r d+1 ≲ ∥Q∥Lp(B1/2) + ∥v∥Lp(B1/2) r d+1 ≲ c0 + M + ∥Q∥Lp(B1) r d+1 , (3.102) and thus ∥v − Pv∥Lq(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+1+n/q . (3.103) Letting P = Pv + Pw, we have u − P = (v − Pv) + (w − Pw), and we obtain ∥u − P∥Lq(Br) ≲ c0 + M + ∥Q∥Lp(B1) r d+α+n/q, r ≤ 1 4 . (3.104) Combining (3.104) with (3.15), we conclude that P is a homogeneous polynomial of degree d. We now prove that u and P satisfy (3.17) and (3.18). Without loss of generality, we assume that X |ν|=m |aν(x) − aν(0)| + X |ν|<m |aν(x)| ≤ ϵ, (3.105) for some small positive ϵ depending on p, d, and α only, since the general case follows immediately by a dilation x → Rx for some R ∈ (0, 1] sufficiently small. Let c˜ = sup r≤1 ∥u − P∥Lp(Br) r d+α+n/p , (3.106) which is finite due to (3.76). Using L(0)P = Q, we rewrite Lu = f as L(u − P) = f − LP = (f − Q) + (L(0) − L)P. (3.107) Applying the interior elliptic estimate to (3.107) yields X |ν|≤m r |ν| ∥∂ ν (u − P)∥Lp(Br) ≲ ∥u − P∥Lp(B2r) + r m∥f − Q∥Lp(B2r) + r m∥(L(0) − L)P∥Lp(B2r) ≲ c˜+ M + ∥P∥Lp(B1) r d+α+n/p, r ≤ 1 2 . (3.108) We now rewrite the equation (3.107) as L(0)(u − P) = X |ν|=m (aν(0) − aν)∂ ν (u − P) − X |ν|<m aν∂ ν (u − P) + (f − Q) + (L(0) − L)P. (3.109) Denoting the right hand side of (3.109) by F, we have ∥F∥Lp(Br) ≲ X |ν|=m ∥aν(0) − aν(x)∥L∞(Br)∥∂ ν (u − P)∥Lp(Br) + X |ν|<m ∥aν(x)∥L∞(Br)∥∂ ν (u − P)∥Lp(Br) + ∥f − Q∥Lp(Br) + X |ν|=m ∥aν(0) − aν(x)∥L∞(Br)∥∂ νP∥Lp(Br) + X |ν|<m ∥aν(x)∥L∞(Br)∥∂ νP∥Lp(Br) . (3.110) 8 Now we employ (3.105) and (3.5) to obtain ∥F∥Lp(Br) ≲ ϵ X |ν|≤m ∥∂ ν (u − P)∥Lp(Br) + ∥f − Q∥Lp(Br) + ϵrα ∥∂ mP∥Lp(Br) + ϵ X |ν|<m ∥∂ νP∥Lp(Br) ≲ ϵc˜+ M + ∥P∥Lp(B1) r d−m+α+n/p , (3.111) where we used (3.108) and that P is homogeneous polynomial of degree d in the last inequality. Applying Lemma 3.2.1, there exists v in Wm,p(B1/2 ) such that L(0)v = F in B1/2 and ∥v∥Lp(Br) ≲ ϵc˜+ M + ∥P∥Lp(B1) r d+α+n/p . (3.112) By (3.106) and (3.112), the Taylor series of (u − P) − v begins with a homogeneous polynomial of degree at least d + 1. By the interior elliptic estimate, for any r ≤ 1/2, ∥(u − P) − v∥Lp(Br) ≲ ∥(u − P) − v∥Lp(B1/2)r d+α+n/p . (3.113) Thus, ∥u − P∥Lp(Br) ≲ ϵc˜+ M + ∥u − P∥Lp(B1) + ∥P∥Lp(B1) r d+α+n/p, r ≤ 1 2 . (3.114) Note that (3.114) also holds for 1/2 ≤ r ≤ 1. Taking the supremum over (0, 1], we get c˜ ≲ ϵc˜+ M + ∥u − P∥Lp(B1) + ∥P∥Lp(B1) . (3.115) Denote by C the implicit constant in (3.115) and choose ϵ such that Cϵ = 1/2 to get c˜ ≤ C(M + ∥u − P∥Lp(B1) + ∥P∥Lp(B1) ), (3.116) i.e., ∥u − P∥Lp(Br) ≤ C(M + ∥u − P∥Lp(B1) + ∥P∥Lp(B1) )r d+α+n/p, r ≤ 1. (3.117) Then by Sobolev’s embedding we have the estimate ∥u − P∥Lq(Br) ≲ (M + ∥u − P∥Lp(B1) + ∥P∥Lp(B1) )r d+α+n/q . (3.118) We now bound P as ∥P∥Lp(Br) ≤ ∥u∥Lp(Br) + ∥u − P∥Lp(Br) ≲ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/p + (M + ∥u∥Lp(B1) + ∥P∥Lp(B1) )r d+α+n/p ≲ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/p + ∥P∥Lp(B1) |x| d+α+n/p . (3.119) By the homogeneity of P, we have ∥P∥Lp(Br) ≲ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/p + ∥P∥Lp(Br)r α . (3.120) We denote by C ′ the implicit constant in (3.120) and choose r sufficiently small such that C ′ r α ≤ 1/2 to get ∥P∥Lp(Br) ≤ C ′ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/p + 1 2 ∥P∥Lp(Br) . (3.121) Thus, ∥P∥Lp(Br) ≲ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/p , (3.122) 82 which is (3.17). Moreover, by (3.118) and (3.122), we obtain ∥u − P∥Lq(Br) ≲ (M + ∥u − P∥Lp(B1) + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/q ≲ (M + ∥P∥Lp(B1) + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/q ≲ (M + ∥u∥Lp(B1) + ∥Q∥Lp(B1) )r d+n/q , (3.123) where we used (3.122) in the last inequality. Therefore, (3.17) and (3.18) follow. The next lemma is a consequence of Theorem 3.1.3. Lemma 3.5.1. Under the same conditions as in Theorem 3.1.3, assume that f ∈ C d−m+1,α Lp (0) and aν ∈ C 1−m+|ν|,α for |ν| = 0, 1. Then there exists Pd ∈ P˙ d such that sup r≤1 ∥u − Pd∥Lp(Br) r d+α+n/p < ∞. (3.124) Moreover, u ∈ C d+1,α Lq (0) for any q ∈ [1, np/(n − mp)), and [u]C d+1,α Lq (0) ≲ c0 + [f]C d−m Lp (0) + [f]C d−m+1,α Lp (0), (3.125) where the implicit constant in (3.125) depends on p, q, d, and aν. Proof of Lemma 3.5.1. Let Pd and Qd−m be P and Q in the proof of Theorem 3.1.3 (the subindices are added to emphasize the orders of P and Q). Then (3.124) follows immediately from (3.108). Recalling that Qd−m = P |ν|=m aν(0)∂ νPd, we rewrite Lu = f as L(u − Pd) = (f − Qd−m) + X |ν|=m (aν(0) − aν)∂ νPd − X |ν|<m aν∂ νPd. (3.126) 83 Denoting the right hand side of (3.126) by F˜, our goal is to show ∥F˜ − Q˜∥Lp(Br) ≲ r d−m+1+α+n/p , (3.127) for some Q˜ ∈ P˙ d−m+1. Since f ∈ C d−m+1,α Lp , there exists Qd−m+1 ∈ P˙ d+1−m such that ∥f − Qd−m − Qd−m+1∥Lp(Br) ≤ [f]C d−m+1,α Lp (0)r d−m+1+α+n/p . (3.128) Therefore, F˜ = (f − Qd−m − Qd−m+1) + Qd−m+1 + X |ν|=m (aν(0) − aν)∂ νPd − X |ν|<m aν∂ νPd. (3.129) Let Q = P |ν|=m(aν(0) − aν)∂ νPd − P |ν|<m aν∂ νPd. The assumptions on coefficients aν allow us to approximate Q by a homogeneous polynomial of order d − m + 1. Indeed, we have ∥aν − aν(0) − p (1) ν ∥Lp(Br) ≤ [aν]C 1,α Lp (0)r 1+α+n/p , |ν| = m, (3.130) and ∥aν − p (0) ν ∥Lp(Br) ≤ [aν]C 0,α Lp (0), |ν| = m − 1, (3.131) for some p (1) ν ∈ P˙ 1 and p (0) ν ∈ P˙ 0. Then P˜ = − P |ν|=m p (1) ν ∂ νPd− P |ν|=m−1 p (0) ν ∂ νPd is a homogeneous polynomial of order d − m + 1, so is P˜ + Qd−m+1. Combining (3.128), (3.130), and (3.131), we have ∥F˜−P˜−Qd−m+1∥Lp(Br) ≤ ([f]C d−m+1,α Lp (0)+[aν]C 1,α Lp (0)∥Pd∥Lp(B1)+[aν]C 0,α Lp (0))r d−m+1+α+n/p . (3.132) 84 Conditions (3.124) with (3.132) allow us to apply Theorem 3.1.3 with d replaced by d+ 1. Then there exists a homogeneous polynomial Pd+1 of degree d + 1 satisfying X |ν|=m aν(0)∂ νPd+1 = P˜ + Qd−m+1 = Q. ˜ (3.133) Moreover, we have the bound ∥u∥Lp(Br) ≲ ([f]C d−m+1,α Lp (0) + c0 + ∥Q˜∥Lp(B1/2(0)))r d+1+α+n/p , (3.134) and, for any q ∈ [1, np/(n − mp)), ∥u − Pd − Pd+1∥Lq(Br) ≲ ([f]C d−m+1,α Lp (0) + c0 + ∥Q˜∥Lp(B1/2(0)))r d+1+α+n/q , (3.135) for all r ≤ 1/4. By definition, ∥Q˜∥Lp(Br) ≲ [aν] C 1−m+|ν| Lp (0) + [f]C d−m+1,α Lp (0), and then (3.125) follows. Remark 3.5.2. Lemma 3.5.1 demonstrates how we obtain the by one order higher Hölder norm for solution u given the one order higher norm for f and aν. To prove Theorem 3.1.2, we apply the argument in Lemma 3.5.1 multiple times as long as f and aν are controlled. Proof of Theorem 3.1.2. Theorem 3.1.3 and Lemma 3.5.1 are special cases of Theorem 3.1.2 with k = 0 and k = 1 respectively. We complete the proof of Theorem 3.1.2 by an induction argument on k. Repeating the argument in Lemma 3.5.1 k times, from (3.134) we have sup r≤1 ∥u∥Lp(Br) r d+k+α+n/p < ∞. (3.136) 85 On the other hand, (3.133) and (3.135) implies there exists Pd+k ∈ Pd+k and Q˜ d−m+k ∈ Pd−m+k such that Q˜ d−m+k = P |ν|=m aν(0)∂ νPd+k, and L(u − Pd+k) = (f − Qd+k−m) + X |ν|=m (aν(0) − aν)∂ νPd+k − X |ν|<m aν∂ νPd+k. (3.137) Therefore, we proceed similarly as Lemma 3.5.1 and get the conclusion for k + 1, which completes the induction argument. 3.6 Schauder estimates for parabolic equations The method used in the previous section may be extended to parabolic equations, as shown here. First, we introduce the corresponding terminology. For any (x0, t0) ∈ R n × R and any r > 0, we denote the parabolic ball centered at (x0, t0) with radius r by Qr(x0, t0) = {(x, t) ∈ R n × R : |x − x0| < r, −r m < t − t0 < 0}, (3.138) and write Q1 for the parabolic ball centered at (0, 0) with radius 1. With m ∈ 2N fixed, consider a general order parabolic equation Lu = f, (3.139) where Lu ≡ ut − X |ν|≤m aν∂ νu (3.140) is an m-th order parabolic operator in Q1, which satisfies the parabolicity condition (−1)m/2−1 X |ν|=m aνξ ν ≥ 1 K , ξ ∈ S n−1 , (x, t) ∈ Q1, (3.141) 86 the boundedness X |ν|≤m |aν(x, t)| ≤ K, (x, t) ∈ Q1, (3.142) and the Hölder continuity of the leading coefficients X |ν|=m |aν(x, t) − aν(0, 0)| ≤ K|(x, t)| α , (x, t) ∈ Q1, (3.143) for some positive constants K, and α ∈ (0, 1), where |(x, t)| is the parabolic norm of (x, t), i.e., |(x, t)| = (|x| m + |t|) 1/m. We allow all constants to depend on the space dimension n, the order of the differential operator m, as well as on K and α without mention. Denote by W m,1 p (Q1) the Sobolev space of functions whose x-derivatives up to m-th order and the t-derivative of order 1 belong to L p (Q1). Similarly to the elliptic case, let Qd the set of polynomials in n variables of degree at most d and Q˙ d be the set of homogeneous polynomials of degree exactly d, with an addition of the zero polynomial. The homogeneity and the polynomial degree in this section are in parabolic sense, which are defined as follows. A function f(x, t) is homogeneous of degree d if for any λ > 0 and (x, t) ∈ R n × R\{(0, 0)}, we have f(λx, λmt) = λ d f(x, t). (3.144) A polynomial P(x, t) is of degree at most d if it can be decomposed into a sum of homogeneous polynomials, whose degree is at most d. Definition 3.6.1. Let u ∈ L p (Q1) for p ∈ [1, ∞]. For an integer d ≥ 1 and α ∈ (0, 1), we say that u ∈ C d+α,(d+α)/m Lp (0) if there exists P ∈ Qd satisfying sup r≤1 ∥u − P∥Lp(Qr) r d+α+(m+n)/p < ∞. (3.145) 87 We also introduce the corresponding semi-norms as [u] C i,i/m Lp (0) = X |ν|+ml=i |∂ ν x∂ l tP(0)|, (3.146) for i = 0, 1, · · · , d, and [u] C d+α,(d+α)/m Lp (0) = sup 0<r≤1 ∥u − P∥Lp(Qr) r d+α+(m+n)/p . (3.147) The following is a parabolic analog of Theorem 3.1.2, asserting an L p Schauder estimate for the parabolic equation (3.139). Theorem 3.6.2. Let u ∈ W m,1 p (Q1), where 1 < p < 1 + n m , (3.148) be a solution of Lu = f in Q1, where f ∈ L p (Q1) and L a parabolic operator (3.140) satisfying (3.141)– (3.143). Suppose that c = sup r≤1 ∥u∥Lp(Qr) r d+(m+n)/p < ∞ (3.149) and cf = sup r≤1 ∥f∥Lp(Qr) r d−m+(m+n)/p < ∞, (3.150) for some integer d ≥ m+ 1. Assume additionally that f ∈ C d−m+k,α Lp (0) and aν ∈ C k−m+|ν|+α Lp (0) for some α ∈ (0, 1) and k ∈ N, for any |ν| ≥ max{m − k, 0}. Then for all q ∈ [1,(m + n)p/(m + n − mp)), we have u ∈ C d+k+α Lq (0), and X d+k i=d [u]Ci Lq (0) + [u]C d+k,α Lq (0) ≲ ∥u∥Lp(Q1) + d−Xm+k i=d−m [f]Ci Lp (0) + [f]C d−m+k,α Lp (0) + c, (3.151) 88 where the implicit constant in (3.151) depends only on p, d, k, and [aν]Ci Lp (0) for d − m ≤ i ≤ d − m + k and [aν] C l−m+|ν|,α Lp (0) for |ν| ≥ m − k. Next, we estimate the high L p -Hölder type norms around the points where u vanishes of order d in the L p sense. As in the elliptic case, the result holds for the L q -norm where q ∈ [1,(m + n)p/(m + n − mp)). Theorem 3.6.3. Let u ∈ W m,1 p (Q1), with p as in (3.148), be a solution of Lu = f, where L is the parabolic operator (3.140) satisfying (3.141)–(3.143) and f ∈ L p (Q1). Suppose that ∥f − Q∥Lp(Qr) ≤ Mrd−m+α+(m+n)/p, r ∈ (0, 1], (3.152) where d ≥ m + 1, for some Q ∈ Q˙ d−m and M > 0. If c0 = sup r≤1 ∥u∥Lp(Qr) r d−1+η+(m+n)/p < ∞, (3.153) where η ∈ (0, 1], there is P ∈ Q˙ d solving Pt − X |ν|=m aν(0, 0)∂ νP = Q (3.154) in R n × R such that ∥P∥Lp(Qr) ≲ (M + c0 + ∥Q∥Lp(Q1) )r d+(m+n)/p (3.155) and X |ν|≤m r |ν| ∥∂ ν (u − P)∥Lq(Qr) ≲ (M + c0 + ∥Q∥Lp(Q1) )r d+α+(m+n)/q, r ∈ (0, 1/2] (3.156) 89 for q ∈ [1,(m + n)p/(m + n − mp)). Also, ∥u∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) )r d+(m+n)/p , (3.157) where all the implicit constants depend only on p, q, and d. Throughout this section, we use the notation L(0) = ∂t − X |ν|=m aν(0, 0)∂ ν , (3.158) and Γ is the fundamental solution of L(0), which satisfies |∂ β x ∂ l tΓ(x, t)| ≲ 1 |(x, t)| n+|β|+ml , β ∈ N n 0 , l ∈ N0. (3.159) 3.7 Interior Wm,1 q existence estimate We now state an analog of Lemma 3.3.1. The proof is similar to the elliptic case except that the fundamental solution Γ now satisfies (3.159). Lemma 3.7.1. Assume that L = ∂t − P |ν|≤m aν(x)∂ ν , defined in Q1, satisfies the conditions (3.141)−(3.143). Suppose that f ∈ L p (Q1) is such that there exist M > 0 and γ ∈ (0, 1) satisfying ∥f∥Lp(Qr) ≤ Mrd−m+γ+(m+n)/p, r ∈ (0, 1]. (3.160) Then there exist R > 0, depending on α, and u ∈ W m,1 q (Q1) solving Lu = f in QR such that X |ν|≤m r |ν| ∥∂ νu∥Lq(Qr) ≲ Mrd+γ+(m+n)/q, r ≤ R, (3.161) 90 for q ∈ [1,(m + n)p/(m + n − mp)). To prove Lemma 3.7.1, we first need an estimate for the integral R (|f(y, s)|/|(y, s)| b ) dy ds over Qr and Q1\Qr. Lemma 3.7.2. Assume that f ∈ L p (Q1), where p ∈ [1, ∞], satisfies ∥f∥Lp(Qr) ≤ Mra , r ∈ (0, 1], (3.162) where a ∈ R. (i) If for some b ∈ R, we have b < a + m + n p ′ , (3.163) then Z Qr |f(y, s)| dy ds |(y, s)| b ≲ Mra−b+(m+n)/p′ , r ∈ (0, 1], (3.164) where the implicit constant depends on a, b, and p. (ii) If b > a + m + n p ′ , (3.165) we have Z Q1\Qr |f(y, s)| dy ds |(y, s)| b ≲ Mra−b+(m+n)/p′ , r ∈ (0, 1], (3.166) where the implicit constant depends on a, b, and p. Proof of Lemma 3.7.2. We prove the statement when p ∈ (1, ∞); it is easy to check that the same holds if p = 1 or p = ∞. 91 (i) We write Z Qr |f(y, s)| dy ds |(y, s)| b = X∞ m=0 Z Q2−mr \Q2−m−1r |f(y, s)| dy ds |(y, s)| b ≲ X∞ m=0 ∥f∥Lp(Q2−mr )∥|(y, s)| −b ∥Lp′ (Q2−mr \Q2−m−1r ) ≲ X∞ m=0 M(2−mr) a−b+(m+n)/p′ ≲ Mra−b+(m+n)/p′ , (3.167) under the condition (3.163). (ii) With m ∈ N0 such that, we have Z Q1\Qr |f(y, s)| dy ds |(y, s)| b ≤ Xm0 m=0 Z Q2−m\Q2−m−1 |f(y, s)| dy ds |(y, s)| b ≲ Xm0 m=0 ∥f∥Lp(Q2−m)∥|(y, s)| −b ∥Lp′ (Q2−m\Q2−m−1 ) ≲ Xm0 m=0 (2−m) a−b+(m+n)/p′ ≲ (2−m0 ) a−b+(m+n)/p′ ≲ r a−b+(m+n)/p′ , (3.168) and the proof is concluded. Proof of Lemma 3.7.1. We only argue for the case where aν ≡ aν(0, 0) for |ν| = m and aν ≡ 0 for |ν| < m. The other cases follow by the contraction mapping argument and rescaling analogously to Lemma 3.2.1. Without loss of generality, we may assume f(x, t) = 0 for |(x, t)| > 1. For the integral w(x, t) = Z |(y,s)|<1 Γ(x − y, t − s)f(y, s) dy ds (3.169) we have L(0)w = f in Q1 and ∥w∥W m,1 p (Q1) ≲ ∥f∥Lp(Q1) ≲ M. (3.170) 92 Since the polynomial Pw(x, t) = X 0≤|β|+ml≤d x β t l β!l! Z |(y,s)|<1 ∂ β x ∂ l tΓ(−y, −s)f(y, s) dy ds (3.171) satisfies L(0)Pw = 0 the difference u = w − Pw solves L(0)u = f. We now show that u satisfies (3.161). For any positive r ≤ 1/2 and q ∈ [1,(m + n)p/(m + n − mp)), we have ∥u∥Lq(Qr) ≤ Z |(y,s)|≤2r Γ(x − y, t − s)f(y, s) dy ds Lq(Qr) + Z |(y,s)|≤2r X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds Lq(Qr) + Z 2r≤|(y,s)|<1 Γ(x − y, t − s) − X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds Lq(Qr) = I1 + I2 + I3. (3.172) Using Young’s inequality and (3.159), we have I1 ≤ Z Γ(x − y, t − s)f(y, s)χQ2r (y, s) dy ds Lq(Qr) ≤ ∥Γ∥L1(Q3r)∥f∥Lq(Q2r) ≲ Mrd−m+γ+(m+n)/q Z |(x,t)|≤3r 1 |(x, t)| n dx dt ≲ Mrd+γ+(m+n)/q . (3.173) To estimate I2, we first claim that Z |(y,s)|≤2r ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds ≲ Mrd+γ , (3.174) 93 for all |(x, t)| ≤ r. Indeed, by (3.159), we have Z |(y,s)|≤2r ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds ≲ |(x, t)| |β|+ml Z |(y,s)|≤2r |f(y, s)| |(y, s)| n+|β|+ml dy ds ≲ r |β|+ml Z |(y,s)|≤2r |f(y, s)| |(y, s)| n+|β|+ml dy ds ≲ Mr|β|+mlr d+γ−|β|−ml ≲ r d+γ , (3.175) where we used Lemma 3.7.2 (i) with a = d − m + γ + (m + n)/p and b = n + |β| + ml in the third inequality. Therefore, I2 = Z |(x,t)|≤r Z |(y,s)|≤2r X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds q dx dt1/q ≲ Z |(x,t)|≤r Mq r (d+γ)q dx dt1/q ≲ Mrd+γ+(m+n)/q . (3.176) We now estimate I3. Similarly as for I2, it suffices to show that for all |x| ≤ r, we have I(x) ≲ Mrd+γ , where I(x) = Z 2r≤|(y,s)|<1 Γ(x − y, t − s) − X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! f(y, s) dy ds . (3.177) By Taylor’s theorem, there exists ξ and K in (0, 1) such that Γ(x − y, t − s) − X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! ≲ X d i=0 X |β|+l=i,|β|+ml≥d+1 |∂ β x ∂ l tΓ(−y, −s)| |x| β |t| l β!l! + X |β|+l=d+1 |∂ β x ∂ l tΓ(ξx − y, Kt − s)| |x| β |t| l β!l! ≲ X d i=0 X |β|+l=i,|β|+ml≥d+1 1 |(y, s)| n+|β|+ml |x| β |t| l β!l! + X |β|+l=d+1 1 |(ξx − y, Kt − s)| n+|β|+ml |x| β |t| l β!l! . (3.178) 94 Using |(y, s)| ≤ 2|(ξx − y, Kt − s)|, we obtain Γ(x − y, t − s) − X 0≤|β|+ml≤d ∂ β x ∂ l tΓ(−y, −s) x β t l β!l! ≲ mX (d+1) j=d+1 1 |(y, s)| n+j + 1 |(ξx − y, Kt − s)| n+j |(x, t)| j ≲ mX (d+1) j=d+1 |(x, t)| j |(y, s)| n+j . (3.179) Therefore, I(x) ≲ mX (d+1) j=d+1 |(x, t)| j Z 2r≤|(y,s)|<1 |f(y, s)| |(y, s)| n+j dy ds ≲ mX (d+1) j=d+1 r j N X−1 i=1 Z 2 ir<|(y,s)|<2 i+1r |f(y, s)| |(y, s)| n+j dy ds, (3.180) where N is an integer satisfying 2 N−1 r ≤ 1 ≤ 2 N r. Applying Lemma 3.7.2 (ii), we get I(x) ≲ Mrd+γ . (3.181) Similarly to I2, we obtain that I3 ≲ Mrd+γ+(m+n)/q. Therefore, ∥u∥Lq(Qr) ≲ Mrd+γ+(m+n)/q, for r ∈ (0, 1/2], and (3.161) follows. As in the elliptic case, we point out that the proof of Lemma 3.7.1 also gives the next statement. Lemma 3.7.3. Suppose that f ∈ L p (Q1) satisfies ∥f∥Lp(Qr) ≤ Mrd−m+α+(m+n)/p, r ≤ 1, (3.182) for some integer d ≥ m and α ∈ (0, 1). For any solution u ∈ W m,1 p (Q1) of (3.139), there exists a polynomial Pd of degree at most d such that L(0)Pd = 0, (3.183) 95 and for any q such that 1 ≤ q < (m + n)p/(m + n − mp), we have ∥u − Pd∥Lq(Qr) ≲ (M + ∥u∥Lp(Q1) )r d+α+(m+n)/q , (x, t) ∈ Q1/2 , (3.184) where the implicit constants depending only on p, q, and d. 3.8 The bootstrap argument The following bootstrapping argument is needed in the proof of Theorem 3.6.3. Since the argument is similar to Lemma 3.4.1, we omit the proof. Lemma 3.8.1. Under the assumptions of Theorem 3.6.3, suppose that there exists k ∈ N0 such that (k + 1)α + η < 1. Then if u vanishes of order d − 1 + η + kα + (m + n)/p, i.e., ck = sup r≤1 ∥u∥Lp(Qr) r d−1+η+kα+(m+n)/p < ∞, (3.185) then u vanishes of L p -order d − 1 + η + (k + 1)α + (m + n)/p, and moreover, ck+1 = sup r≤1 ∥u∥Lp(Qr) r d−1+η+(k+1)α+(m+n)/p ≲ ck + M + ∥Q∥Lp(Q1) < ∞ (3.186) also holds, where M and Q are defined in Theorem 3.6.3, and the implicit constant in (3.186) depends on d, p, and q. We point out that, as in the elliptic case, we have the inequalities X |ν|≤m r |ν| ∥∂ νu∥Lp(Qr) ≲ c0 + M + ∥Q∥Lp(Q1) r d−1+η+kα+(m+n)/p (3.187) 9 and ∥L(0)u − Q∥Lp(Qr) ≲ c0 + M + ∥Q∥Lp(Q1) r d−m+η+(k+1)α−1+(m+n)/p . (3.188) 3.9 Proofs of Schauder estimates for parabolic equations Proof of Theorem 3.6.3. Let k0 ∈ N0 be such that k0α + η < 1 ≤ (k0 + 1)α + η. As above, we can assume that the second inequality is strict. We first show that ∥u∥Lp(Qr) ≤ (c0 + M + ∥Q∥Lp(Qr) )r d+(m+n)/p . (3.189) Applying Lemma 3.8.1 k0 times, we obtain ck0 = sup r≤1 ∥u∥Lp(Qr) r d−1+η+k0α+(m+n)/p ≲ c0 + M + ∥Q∥Lp(Q1) , (3.190) where c0 is defined in (3.153). As in (3.187), we have X |ν|≤m r |ν| ∥∂ νu∥Lp(Qr) ≲ c0 + M + ∥Q∥Lp(Q1) r d−1+η+k0α+(m+n)/p, r ∈ (0, 1/2], (3.191) while as in (3.188), we may bound ∥L(0)u − Q∥Lp(Qr) ≲ c0 + M + ∥Q∥Lp(Q1) r d−m+η+(k0+1)α−1+(m+n)/p, r ∈ (0, 1/2]. (3.192) Defining w(x) = Z |(y,s)|≤1/2 Γ(x − y, t − s)(L(0)u(y, s) − Q(y, s)) dy ds, (3.193) we have L(0)w = L(0)u − Q. So we apply Lemma 3.7.3 and use (3.192) to get ∥w − Pw∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) )r d+η+(k0+1)α−1+(m+n)/p , (3.194) where Pw is the d-degree Taylor polynomial of w. Thus v = u − w satisfies L(0)v = Q and ∥v∥Lp(Q1/2) ≲ ∥u∥Lp(B1/2)+∥w∥Lp(B1/2) ≲ ∥u∥W m,1 p (B1/2)+∥Q∥Lp(Q1) ≲ c0+M +∥Q∥Lp(Q1) , (3.195) where we used (3.191) in the last inequality. We now let Pv be the Taylor polynomial of v with the degree d. Then L(0)Pv = Q and that ∥v − Pv∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) )r d+1+(m+n)/p . (3.196) Therefore, we let P¯ = Pv + Pw and combine (3.194) with (3.196) to obtain ∥u − P¯∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) )r d+min(η+(k0+1)α−1,1)+(m+n)/p ≲ (c0 + M + ∥Q∥Lp(Q1) )r d+η+(k0+1)α−1+(m+n)/p . (3.197) Let L˜ = ∂t − P |ν|=m aν∂ ν . Denoting ϕ = P |ν|<m aν∂ νu + (f − Q), we get by (3.191) that ∥ϕ∥Lp(Qr) ≲ r 1−m∥u∥Lp(Q2r) + r∥f∥Lp(Qr) + ∥f − Q∥Lp(Q2r) ≲ (c0 + M + ∥Q∥Lp(Q1) )r d−m+min(η+k0α,α)+(m+n)/p, r ∈ (0, 1/2]. (3.198) We rewrite the equation Lu = f as L˜(u − P¯) = (L(0) − L˜)P¯ + ϕ = X |ν|=m (aν − aν(0, 0))∂ νP¯ + ϕ. (3.199) 98 Applying Hölder’s inequality yields X |ν|=m ∥(aν − aν(0, 0))∂ νP¯∥Lp(Qr) ≲ X |ν|=m ∥aν − aν(0, 0)∥L∞(Qr)∥∂ νP¯∥Lp(Qr) ≲ ∥P¯∥Lp(Q1)r d−m+α+(m+n)/p , (3.200) for all r ≤ 1. By combining (3.198) and (3.200), we have the estimate for the right hand side of (3.199) which reads X |ν|=m (aν −aν(0, 0))∂ νP¯ +ϕ Lp(Qr) ≲ (c0 +M +∥Q∥Lp(Q1) +∥P¯∥Lp(Q1) )r d−m+min(η+k0α,α)+(m+n)/p , (3.201) for all r ∈ (0, 1/2]. We now rewrite the equation as L(0)(u − P¯) = (L(0) − L˜)(u − P¯) + X |ν|=m (aν − aν(0, 0))∂ νP¯ + ϕ = X |ν|=m (aν − aν(0, 0))∂ ν (u − P¯) + X |ν|=m (aν − aν(0, 0))∂ νP¯ + ϕ. (3.202) The first term on the right hand side of (3.202) is estimated as X |ν|=m ∥(aν − aν(0, 0))∂ ν (u − P¯)∥Lp(Qr) ≲ X |ν|=m ∥aν − aν(0, 0)∥L∞(Qr)∥∂ ν (u − P¯)∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d−m−1+η+α+(m+n)/p . (3.203) Let ϕ˜ be the right hand side of (3.202). By combining (3.201) and (3.203), we obtain ∥ϕ˜∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d−m−1+η+α+(m+n)/p (3.204) 99 and L(0)(u − P¯) = ϕ˜. Thus there exists a polynomial P0 of degree less than d such that ∥u(x, t) − P¯(x, t) − P0(x, t)∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) + ∥u∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d−1+η+(k0+1)α+(m+n)/q , (3.205) and then ∥u(x, t) − P0(x, t)∥Lp(Qr) ≲ (c0 + M + ∥Q∥Lp(Q1) + ∥u∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d−1+η+(k0+1)α+(m+n)/p , (x, t) ∈ QR. (3.206) Arguing similarly as in Theorem 3.1.3, we have P0 ≡ 0 by (3.153) and therefore, sup r≤1 ∥u∥Lp(Qr) r d+(m+n)/p < ∞, (3.207) since 1 ≤ (k0 + 1)α + η. Then, instead of (3.203), we have X |ν|=m ∥(aν − aν(0, 0))∂ ν (u − P)∥Lp(Qr) ≲ (ck + M + ∥Q∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d−m−1+η+kα+(m+n)/p, r ≤ R. (3.208) By Lemma 3.7.3, there exists a polynomial P2 of degree strictly less than d + 1 such that L(0)P2 = 0 and ∥u(x, t) − P¯(x, t) − P2(x, t)∥Lq(Qr) ≲ (c0 + M + ∥u∥Lp(Q1) + ∥Q∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d+α+(m+n)/q , (3.209) for all (x, t) ∈ QR. Let P = P¯ + P2. Then P is homogeneous polynomial of degree d and L(0)P = Q. Letting c˜ = sup r≤0 ∥u∥Lp(Qr) r d+(m+n)/p < ∞, (3.210) 100 we have ∥u − P∥Lq(Qr) ≲ (˜c + M + ∥u∥Lp(Q1) + ∥Q∥Lp(Q1) + ∥P¯∥Lp(Q1) )r d+α+(m+n)/q , (3.211) for any (x, t) ∈ QR. The estimation of u−P and P then follows similarly as in the proof of Theorem 3.1.3. The proof of Theorem 3.6.2 is analogous to Theorem 3.1.2, and is thus omitted. 101 Bibliography [A] S. 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Abstract (if available)

Abstract We consider the quantitative uniqueness properties for a parabolic type equation ut − Δu = w(x, t) · ∇u + v(x, t)u, when v ∈ Lp2Lp1 and w ∈ Lq2Lq1, with a suitable range for exponents p1, p2, q1, and q2. We prove a strong unique continuation property and provide a pointwise in time observability estimate. We also provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p < n/m, in space Lp-type regularity for elliptic and parabolic equations of order m in Rn.

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Creator Le, Quinn (author)

Core Title Unique continuation for elliptic and parabolic equations with applications

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Degree Conferral Date 2023-12

Publication Date 10/25/2023

Defense Date 10/18/2023

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,parabolic equations,Schauder estimates,unique continuation

Format theses (aat)

Language English

Advisor Kukavica, Igor (committee chair)

Creator Email nguyet.t.t.le@gmail.com,ntle@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113761000

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