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Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces

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Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces

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Content ASYMPTOTIC EXPANSION FOR SOLUTIONS OF THE NAVIER-STOKES EQUATIONS WITH POTENTIAL FORCES by Ednei Reis A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2011 Copyright 2011 Ednei Reis Table of Contents Abstract iii Chapter 1: Introduction 1 Chapter 2: Notation and the main result 4 Chapter 3: Lower order asymptotics (jjn + 3) 13 Chapter 4: Higher order asymptotics (jjn + 4) 20 Chapter 5: Asymptotics using the vorticity moments 23 Bibliography 25 ii Abstract We derive an asymptotic expansion for smooth solutions of the Navier-Stokes equations in weighted spaces. This result removes previous restrictions on the number of terms of the asymptotics, as well as on the range of the polynomial weights. We also write the expansion in terms of expressions involving non-linear quantities. iii Chapter 1 Introduction In this thesis, we nd a complete asymptotic expansion of solution of the initial value problem for the Navier-Stokes equations @ t u u +uru +rp = 0 ru = 0 (1.0.1) inR n . We provide a development around the spatial innity and obtain a precise tem- poral rate of decay of the error. In the case of periodic boundary conditions, Foias and Saut established a description of the large time behavior of solutions of the Navier-Stokes equations with potential forc- ing They proved that there exist a sequence of manifolds with increasing codimension such that the solutions belonging to the scale of manifolds have a faster rate of expo- nential decay [FS1, FS2]. They moreover found the corresponding normal form which provides a complete description of the large time behavior of solutions [FS3, FS4]. In recent decades, there has been a great deal of interest dedicated to the large time behavior of solutions of the Navier-Stokes equations in the case of unbounded domains. The main dierence from the periodic case is the lack of the Poincar e inequality, and the algebraic decay of solutions was obtained with more sophisticated techniques. The decay of solutions of the 2D Navier-Stokes system was established by Kato using density and the xed point technique [K]. The open problem of decay of solutions of the Navier-Stokes system in three space dimension, raised by Leray in [Le], was then 1 solved by Schonbek using a new Fourier splitting technique. In [S2], she proved that if the initial datum u 0 2L 2 (R 3 )\L r (R 3 ), where 1r< 2, has a nonzero average, then 1 C(1 +t) 3(1=r1=2)=2 ku(;t)k L 2 C (1 +t) 3(1=r1=2)=2 : For well-localized data, a result by Wiegner [W] states that for a suciently localized initial velocity, i.e., the velocity which decays suciently fast to zero at innity, we have an upper bound ku(;t)k L 2 =O(t (n+2)=4 ): (1.0.2) This rate of decay is sharp in the sense that if a solution decays with a faster rate, it has to satisfy a set of algebraic identities which most of the solutions do not [MS] (the codimension of such solutions was estimated from above by Gallay and Wayne in [GW]). It is well-known that in even space dimension, there exist explicit solutions which decay to zero exponentially fast. More recently, it was shown by Brandolese in [B1, B2] that there exist solutions in odd space dimensions which decay with a certain algebraic rate faster than (1.0.2). The space time decay for the Navier-Stokes equations was addressed by Takahashi [T] and Amrouche et al [AGSS]. The pressure causes an interesting phenomenon if we start with initial data inS: The solution immediately decays only algebraically at spatial innity [B2, L]. If a solution decays in the space variable any faster than a generic rate, it satises a certain set of algebraic identities [DS]. Brandolese and Meyer studied this problem and showed that non-generic data satisfying special symmetries exist [BM]. We note that for well-localized datum, the sharp rate of decay was obtained by Kukavica and Torres in [KT1, KT2]. For other important results on decay of solutions of the Navier-Stokes system, cf. [BJ, Ku1, KM, M, S1, S3, SW]. Regarding the asymptotic proles foru, Carpio obtained an asymptotic development for time decay of u up to second order [C]. Fujigaki and Miyakawa extended this result by considering a Taylor expansion for the Oseen kernel; the the number of terms of the 2 sum is bounded by the dimension n. Choe and Jin proved that the expansion in [FM] can be used to obtain an asymptotic development in polynomial weighted spaces with the number of terms bounded by n. Asymptotic proles were also obtained by Gallay and Wayne in [GW] using similarity variables, and by Brandolese and Vigneron in [BV], who proved that the solution u behaves asymptotically as a potential eld asjxj!1. In this thesis we provide a complete asymptotic expansion for the velocity eld. This development is easy to calculate, and as opposed to previous results, the number of terms is not restricted. We also emphasize that the the degree of the polynomial weight increases as we increase the number of terms of the expansion. Moreover, the results in this thesis can be generalized to obtain expansions for higher order derivatives of u. The techniques used rely on expanding the vorticity in the Fourier variables and splitting in lower and upper frequencies, an approach introduced in [KT2]. As a result, the terms are computed linearly with respect to the vorticity averages. It is possible, and we do so in Chapters 3{5 below, to express the linear quantities in terms of quadratic expressions and quadratic moments as used in [MS]. Since the velocityu does not decay fast enough asjxj!1 (see [L, Chapter 25]), we need to use a dierent method to treat the higher order terms of the asymptotics (jjn + 4) as shown in Chapters 4 and 5. 3 Chapter 2 Notation and the main result It is well-known that for every given initial datum u 0 2 L 2 (R n ), there exists a global weak solution u satisfying a form of energy inequality (cf. [Le] for n 3 and [KM] for n 4). In this thesis, we assume that the initial datum is smooth and well-localized; for simplicity, we take u 0 2S satisfying divu 0 = 0. The sharpest decay result is due to Wiegner [W] and asserts that for such data, the weak solution satises ku(;t)k L 2 =O(t (n+2)=4 ): This is a generic rate of decay for well-localized data; the solutions which decay quicker satisfy a set of algebraic identities, which most solutions do not. While our main focus is on the solutions with generic decay t (n+2)=4 , we can also treat solutions which decay with a quicker rate. Hence, we may assume ku(;t)k L 2 =O(t 0 ) (2.0.1) where f(t) =O(t ) means sup t>0 t jf(t)j < +1. Due to [W], we can assume that without loss of generality 0 n + 2 4 : Our method is based on considerations based on vorticity; to avoid issues concerning smoothness, we assume that the solution is strong, i.e., u2C([0;1);H 1 (R n )). This is automatically true in space dimension 2 if u 0 2 H 1 (R 2 ), but it has to be assumed in higher space dimensions (the set of such solutions is not empty; if the L n norm of u 0 is suciently small, the smoothness condition is satised [K]). 4 First we note that by a result in [Ku2] we have kjxj a u(;t)k L p =O(t 0 +a=2(n=2)(1=21=p) ) for a2 [0;n=p 0 + 1). Consider the vorticity tensor ! = (! ij ) i;j=1;2;:::;n which is dened by ! ij =@ i u j @ j u i ; i;j = 1;:::;n: It satises the equation @ t ! ij ! ij +@ i (u k ! kj )@ j (u k ! ki ) = 0; i;j = 1;:::;n: (2.0.2) (The summation convention on repeated indices is used throughout.) From [KT2, Ku2], recall the estimate kjxj a !(;t)k L p =O t 0 1=2+a=2(n=2)(1=21=p) (2.0.3) for p2 [2; +1] and a 0. By the inequalitykvk L 1Ckvk 1=2 L 2 kjxj n vk 1=2 L 2 (cf. [GK]), we obtain that (2.0.3) holds also for p = 1, and by H older's inequality also for p2 [1; 2]. In the thesis, we use the Fourier transform ^ f() = R f(x)e ix dx. The following results gives a complete space-time asymptotic of the solution. Theorem 2.0.1 Let 2S(R n ) be such that (1())=jj 2 is a C 1 function. Let u be a smooth solution of the Navier-Stokes equations with initial data u 0 2S such that divu 0 = 0. Also, assume that u satisesku(;t)k L 2 =O (t 0 ) where 0 (n + 2)=4. Then jxj a 0 @ u j (;t) X 2jjm 1 ! (i) jj+1 ( p t) k jj 2 ! _ Z y ! kj (y;t)dy 1 A L p =O t 0 +a=2(n=2)(1=21=p) (2.0.4) 5 for any p2 [2;1], m2N, and a2 [0;m +nn=p). In particular, for all m2N and 0a<m +n, we have the space-time asymptotic expansion u j (x;t) X 2jjm 1 ! (i) jj+1 ( p t) k jj 2 ! _ Z y ! kj (y;t)dy C jxj a t 0 a=2+n=4 where the constant C does not depend on x or t. For = ( 1 ;:::; n )2N n 0 nf0g, denote by N() the number of its nonzero coordi- nates, i.e., N() = cardfj2f1;:::;ng : j 6= 0g: Then, for any function f dened onN n 0 , we have X jj=k f() = n X l=1 X jj=k1 1 N(e l +) f(e l +) wheree l denotes the multiindex with 1 in thel-th place and zeros elsewhere. This formula states that every multiindex such thatjj =k can be written in N() dierent ways in the form +e l wherejj =k 1 andl2f1;:::;ng. Denoting by (R k f) ^= ( k =ijj) ^ f the k-th Riesz transform, we get X 2jjm 1 ! (i) jj+1 ( p t) k jj 2 ! _ Z y ! kj (y;t)dy = m X l=1 X 1jjm1 1 N(e l +)(e l +)! (i) jj+2 ( p t) l k jj 2 ! _ Z y l y ! kj (y;t)dy = m X l=1 X 1jjm1 (1) jj N(e l +)(e l +)! p t jjn R l R k @ x p t Z y l y ! kj (y;t)dy 6 and we may thus rewrite the relation (2.0.4) as jxj a u j (;t) n X l=1 X 1jjm1 (1) jj+1 N(e l +)(e l +)! p t jjn R l R k @ x p t Z y l y ! kj (y;t)dy ! L p =O t 0 +a=2(n=2)(1=21=p) and the equality holds for all p2 [2;1], m2N, and a2 [0;m +nn=p). Remark 2.0.2 The most natural choice for is () =e jj 2 which is the Fourier Transform of the normalized Gaussian (x) = 1 (4) n=2 e jxj 2 =4 : In this case, we may write the asymptotic expansion in terms of Riesz transforms of the Gaussian kernel G(x;t) = (4t) n=2 exp(jxj 2 =4t) as jxj a u j (;t) n X l=1 X 1jjm1 (1) jj N(e l +)(e l +)! Z y y l ! kj (y;t)dy (R l R k @ G(;t)) ! L p =O t 0 +a=2(n=2)(1=21=p) (2.0.5) 7 which holds for any p2 [2;1], m2N, and a2 [0;m +nn=p). In the case p =1, we obtain u j (;t) n X l=1 X 1jjm1 (1) jj N(e l +)(e l +)! Z y y l ! kj (y;t)dy (R l R k @ G(;t)) C jxj a t 0 a=2+n=4 (2.0.6) for all m2N and a2 [0;m +n). Proof of Theorem 2.0.1. We start by the Biot-Savart law, which is written in Fourier variables as ^ u j (;t) = i k jj 2 ^ ! kj (;t): (2.0.7) From [MS], if follows that ^ ! kj (0;t) = 0 for all t> 0, and by [Ku2] we have @ ^ ! kj (0;t) = Z (ix) ! kj (x;t)dx = 0; jj = 1: (2.0.8) Hence, applying the Taylor formula around the origin to the Fourier transform of the vorticity, we obtain ^ ! kj (;t) = X 2jjm 1 ! @ ^ ! kj (0;t) + (m + 1) Z 1 0 X jj=m+1 1 ! (1s) m @ ^ ! kj (s;t)ds: (2.0.9) 8 With the decomposition ^ ! kj (;t) =( p t)^ ! kj (;t) + 1( p t) ^ ! kj (;t), the identity (2.0.9) can be rewritten as ^ ! kj (;t) = X 2jjm 1 ! ( p t)@ ^ ! kj (0;t) + (m + 1)( p t) X jj=m+1 Z 1 0 1 ! (1s) m @ ^ ! kj (s;t)ds + 1( p t) ^ ! kj (;t): (2.0.10) Substituting (2.0.10) into (2.0.7) we get ^ u j (;t) + X 2jjm 1 ! i k jj 2 ( p t)@ ^ ! kj (0;t) =(m + 1)( p t) X jj=m+1 Z 1 0 i k !jj 2 (1s) m @ ^ ! kj (s;t)ds i k jj 2 1( p t) ^ ! kj (;t): By the Hausdor-Young inequality, we have jxj a 0 @ u j (;t) X 2jjm 1 ! (i) jj+1 ( p t) k jj 2 ! _ Z y ! kj (y;t)dy 1 A L p C a 0 @ ( p t) Z 1 0 X jj=m+1 k !jj 2 (1s) m @ ^ ! kj (s;t)ds 1 A L p 0 +C a 1( p t) k jj 2 ^ ! kj (;t) L p 0 =I 1 +I 2 where p is as in the statement, 1=p + 1=p 0 = 1, and = () 1=2 . In order to estimateI 1 , we need to ndq;r2 (1;1) such that 1=q+1=r = 1=p 0 ,r>n, and q(am)<n. For this, we observe that a<m +n(1 1=p), and thus we can nd M > 0 suciently large so that 11=p> 1=(n+M) anda<m+n(11=p1=(n+M)). 9 Now we dener =n +M andq by the relation 1=q + 1=r = 1=p 0 , which is possible since r > p 0 . Hence, m +n(1 1=p 1=(n +M)) = m +n(1=p 0 1=r) = m +n=q, which implies a<m +n=q, and consequently q(am)<n. By [KPV, Lemma 2.10], we have I 1 C X jj=m+1 a k jj 2 ( p t) L q Z 1 0 (1s) m @ ^ ! kj (s;t)ds L r +C k jj 2 ( p t) L q a Z 1 0 (1s) m @ ^ ! kj (s;t)ds L r : (2.0.11) Now, forjj =m + 1 we have a k jj 2 ( p t) L q =t m=2 a ( p t) ( p t k ) j p tj 2 ( p t) L q and since a (f(b)) =b a ( a f)(b), the latter is equal to t (am)=2 a k jj 2 () p t L q =t (amn=q)=2 a k jj 2 () L q =O t (amn=q)=2 where the last equality holds since q(am) < n implies a ( k ()=jj 2 )2 L q (R n ) ([KT2]). Next, by the Hausdor-Young inequality, we get Z 1 0 (1s) m @ ^ ! kj (s;t)ds L r Z 1 0 (1s) m k@ ^ ! kj (s;t)k L r ds C Z 1 0 (1s) m s n=r jxj m+1 ! kj (x;t) L r 0 ds: Using n=r < 1 andkjxj m+1 ! kj (x;t)k L r 0 =O(t 0 +m=2(n=2)(1=21=r 0 ) ) by (2.0.3), we conclude that Z 1 0 (1s) m @ ^ ! kj (s;t)ds L r =O t 0 +m=2(n=2)(1=21=r 0 ) : 10 Applying the same estimates for the second term in (2.0.11), we get I 1 = O(t 0 +a=2(n=2)(1=21=p) ). For I 2 , we get with q and r as above I 2 C a 1( p t) jj 2 L r k k ^ ! kj k L q +C 1( p t) jj 2 L r k a ( k ^ ! kj (;t))k L q Ct a=2+1n=2r k k ^ ! kj (;t)k L r +Ct 1n=2r k a ( k ^ ! kj (;t))k L r where for the last inequality, we used a (1())=jj 2 2L r (R n ). Ifq< 2, we apply the inequalitykfk L skfk 3=21=s L 2 kjxj n fk 1=s1=2 L 2 fors2 [1; 2] from [GK] withf = k ^ ! kj (;t) and f = a n+1 k ^ ! kj (;t) and get k k ^ ! kj (;t)k L q Ck k ^ ! kj (;t)k 3=21=q L 2 n+1 k ^ ! kj (;t) 1=q1=2 L 2 =Ck (! kj (x;t))k 3=21=q L 2 n+1 (! kj (x;t)) 1=q1=2 L 2 =O t 0 1n=2q+n=4 : If q 2, the estimatek k ^ ! kj (;t)k L q =O(t 0 1n=2q+n=4 ) follows from (2.0.3) and the Hausdor-Young inequality. Similarly, k a ( k ^ ! kj (;t))k L q =O t a=2 0 1n=2q+n=4 : Therefore, I 2 =O t a=2 0 n=2(1=r+1=q1=2) =O t a=2 0 (n=2)(1=21=p) which proves the theorem. 11 Remark 2.0.3 Applying the same proof as in Theorem 2.0.1, we can obtain also the asymptotic development for all the derivatives of u. Namely, we have jxj a @ x u j (;t) X 2jjm 1 ! (1) jj (i) jj+jj+1 ( p t) + k jj 2 ! _ Z y ! kj (y;t)dy ! L p =O t 0 +a=2(n=2)(1=21=p) (2.0.12) withjj =b and a2 [0;m +b +nn=p). Note that the allowed range of exponents of jxj increases as becomes larger. When p =1, the estimate (2.0.12) can be rewritten as @ u j (x;t) X 2jjm 1 ! (1) jj+1 (i) jj+jj+1 ( p t) + k jj 2 ! _ Z y ! kj (y;t)dy C jxj a t 0 a=2+n=4 which is valid for all m2N and 0a<m +b +n. 12 Chapter 3 Lower order asymptotics (jjn + 3) In this chapter, we write the lower order terms of the asymptotics only in terms of the velocity and the initial data, by expressing the moments of the vorticity in terms of quadratic quantities. Of particular interest is the asymptotic development (3.0.10), which shows that the long term behavior of u is determined with scalars of the form R 1 0 R x u j u k dxds. Lemma 3.0.4 Under the conditions of Theorem 2.0.1 we have Z x ! ij (x;t)dx Z x ! ij (x; 0)dx = Z t 0 Z (x )! ij (x;s)dxds + Z t 0 Z @ ik (x )u k u j dxds Z t 0 Z @ jk (x )u k u i dxds (3.0.1) for all 2N n 0 such thatjjn + 3. 13 Proof of Lemma 3.0.4. Let R > 0 and let 2 C 1 0 (R n ) be such that 1 in B 1 (0), supp B 2 (0). Multiplying the equations for the vorticity (2.0.2) by (x=R)x and integrating in both space and time variables, we get Z x ! ij (x;t) x R dx Z x ! ij (x; 0) x R dx = Z t 0 Z x ! ij x R dxds + Z t 0 Z x @ i (u k ! kj ) x R dxds Z t 0 Z x @ j (u k ! ki ) x R dxds: Integrating by parts, we obtain Z x ! ij (x;t) x R dx Z x ! ij (x; 0) x R dx = 1 R 2 Z t 0 Z x x R ! ij dxds + 2 R Z t 0 Z @ k (x )@ k x R ! ij (x;s)dxds + Z t 0 Z (x ) x R ! ij dxds Z t 0 Z @ i (x ) x R u k ! kj dxds 1 R Z t 0 Z x @ i x R u k ! kj dxds + Z t 0 Z @ j (x ) x R u k ! ki dxds + 1 R Z t 0 Z x @ j x R u k ! ki dxds =I 1 +I 2 + +I 7 : (3.0.2) Before sending R!1, we rewrite the terms I 4 and I 6 by using ! ij =@ i u j @ j u i and get I 4 = Z t 0 Z @ i (x ) x R u k (@ k u j @ j u k )dxds: Since divu = 0 and u k @ j u k = (1=2)@ j (u k u k ), we have I 4 = Z t 0 Z @ i (x ) x R @ k (u k u j )dxds + 1 2 Z t 0 Z @ i (x ) x R @ j (u k u k )dxds 14 and thus, integrating by parts, I 4 = Z t 0 Z @ ik (x ) x R u k u j dxds + 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds 1 2 Z t 0 Z @ ij (x ) x R u k u k dxds 1 2R Z t 0 Z @ i (x )@ j x R u k u k dxds: (3.0.3) In a completely analogous way, we get I 6 = Z t 0 Z @ jk (x ) x R u k u i dxds 1 R Z t 0 Z @ j (x )@ k x R u k u i dxds + 1 2 Z t 0 Z @ ij (x ) x R u k u k dxds + 1 2R Z t 0 Z @ j (x )@ i x R u k u k dxds: (3.0.4) We may now verify the assumptions of the Dominated Convergence Theorem for the casejjn + 2. For the sake of illustration, we only show this for the terms 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds and 1 R Z t 0 Z x @ j x R u k ! ki dxds the other terms being treated in a similar way. By [L, Chapter 25] we have sup s2[0;t] sup x2R n (1 +jxj n+1 )ju(x;s)j<1: (3.0.5) Hence, usingjjn + 2, @ i (x )@ k x R u k u j C jxj jj1 1 +jxj 2n+2 C 1 +jxj 2n+3jj 2L 1 ([0;t]R n ): 15 For the second limit, we use a result in [GW] (cf. also [L, KT2]) which states that for any b 0 sup s2[0;t] sup x2R n jxj b j! ij (x;s)j<1 in order to conclude 1 R x @ j x R u k ! ki C jxj jj 1 +jxj b C 1 +jxj bjj 2L 1 ([0;t]R n ) for any xedb>jj +n. Thus, we can take the limit as R!1 to obtain (3.0.1). Note that the third term in (3.0.3) cancels with the third term in (3.0.4) when adding I 4 and I 6 . In the casejj =n+3, we can no longer apply the Dominated Convergence Theorem for the terms 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds or 1 R Z t 0 Z @ j (x )@ k x R u k u i dxds: However, we can still show that as R!1 these two terms tend to zero. Indeed, using (3.0.5) again, we have 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds C R Z t 0 Z Rjxj2R jxj n+2 1 jxj 2n+2 dxds C R (ln(2R) ln(R))! 0: Therefore, (3.0.1) also holds forjj =n + 3. 16 Using the recursive relation (3.0.1), we may express the moments of the vor- ticity in terms of quadratic quantities as follows. For 2 jj 3, we have R t 0 R (x )! ij (x;s)dxds = 0 by ^ !(0;t) = 0 and (2.0.8); hence, in this case Z x ! ij (x;t)dx = Z x ! ij (x; 0)dx + Z t 0 Z @ ik (x )u k u j dxds Z t 0 Z @ jk (x )u k u i dxds: (3.0.6) In the case 4jj 5, we use (3.0.6) to write Z t 0 Z (x )! ij (x;s)dxds = Z t 0 Z (x )! ij (x; 0)dx + Z s 0 Z @ ik (x )u k u j dxd Z s 0 Z @ jk (x )u k u i dxd ds: Applying the Fubini Theorem, we obtain Z t 0 Z (x )! ij (x;s)dxds =t Z (x )! ij (x; 0)dx + Z t 0 Z (ts)@ ik (x )u k u j dxds Z t 0 Z (ts)@ jk (x )u k u i dxds: Hence, for 4jj 5, we have Z x ! ij (x;t)dx = Z x ! ij (x; 0)dx +t Z (x )! ij (x; 0)dx + Z t 0 Z (ts)@ ik (x )u k u j dxds Z t 0 Z (ts)@ jk (x )u k u i dxds + Z t 0 Z @ ik (x )u k u j dxds Z t 0 Z @ jk (x )u k u i dxds: (3.0.7) 17 We may continue in this manner for all such thatjjn + 3. Using induction, we get Z x ! ij (x;t)dx = [jj=2]1 X l=0 t l l! Z l (x )! ij (x; 0)dx + [jj=2]1 X l=0 Z t 0 Z (ts) l l! @ ik l (x )(u k u j )(x;s)dxds [jj=2]1 X l=0 Z t 0 Z (ts) l l! @ jk l (x )(u k u i )(x;s)dxds (3.0.8) for all such thatjj n + 3 and all i;j2f1;:::;ng. Therefore, we have proven the following theorem. Theorem 3.0.5 Let , u 0 , u, and 0 be as in the statement of Theorem 2.0.1. Then jxj a u j (;t) n X l=1 X 1jjm1 (1) jj N(e l +)(e l +)! p t jjn R l R k @ x p t [(jj1)=2] X l=0 t l l! Z l (y l y )! ij (y; 0)dy + [(jj1)=2] X l=0 Z t 0 Z (ts) l l! @ ik l (y l y )(u k u j )(y;s)dyds [(jj1)=2] X l=0 Z t 0 Z (ts) l l! @ jk l (y l y )(u k u i )(y;s)dyds L p =O t 0 +a=2(n=2)(1=21=p) for any p2 [2;1], m2f2; 3;:::;n + 3g, and a2 [0;m +nn=p). As mentioned above, the expression R t 0 R x u j u k dxdt is well dened ifjj n + 3. In this case Z x u j u k dx Ckjxj jj juj 2 k L 1 =Ckjxj jj=2 uk 2 L 2 =O(t 2 0 +jj=2 ): 18 This function is integrable in t around innity if2 0 +jj=2<1, which is the case if jj< 4 0 2. In the case of generic decay 0 = (n + 2)=4, this condition reads asjj<n. We thus obtain Z t 0 Z x u j u k dxdt = Z 1 0 Z x u j u k dxdt +O(t 2 0 +jj=2+1 ): Using the above observation, we may rewrite (3.0.8) as Z x ! ij (x;t)dx = [jj=2]1 X l=0 t l l! Z l (x )! ij (x; 0)dx + [jj=2]1 X l=0 Z 1 0 Z (ts) l l! @ ik l (x )(u k u j )(x;s)dxds [jj=2]1 X l=0 Z 1 0 Z (ts) l l! @ jk l (x )(u k u i )(x;s)dxds +O(t 2 0 +jj=2 ) (3.0.9) and this holds providedjjn + 3 andjj< 2 0 . After a short calculation, we obtain jxj a u j (;t) n X l=1 X 1jjm1 (1) jj+1 N(e l +)(e l +)! p t jjn R l R k @ x p t [(jj1)=2] X l=0 t l l! Z l (y l y )! ij (y; 0)dy + [(jj1)=2] X l=0 Z 1 0 Z (ts) l l! @ ik l (y l y )(u k u j )(y;s)dyds [(jj1)=2] X l=0 Z 1 0 Z (ts) l l! @ jk l (y l y )(u k u i )(y;s)dyds L p =O t 0 +a=2(n=2)(1=21=p) (3.0.10) for any p2 [2;1], m2f2; 3;:::;n + 3g, and a2 [0;m +nn=p) such that m< 4 0 . 19 Chapter 4 Higher order asymptotics (jjn + 4) For the rangejjn + 4, the spatial decay of the velocity is not sucient to even dene the quadratic velocity moments. In order to overcome this diculty, we use the lower order asymptotic formula already obtained in Chapter 3 above. (An alternative way is to use quadratic moments involving the vorticity as in the next chapter.) For simplicity, we address only the case n = 3 andjj = 7. The higher may be treated using the same idea and the recursion. Theorem 4.0.6 Under the conditions of Theorem 2.0.1 the identity Z x ! ij (x;t)dx Z x ! ij (x; 0)dx = Z t 0 Z (x )! ij dxds + Z t 0 Z @ ik (x )(u k f k3 )u j dxds + Z t 0 Z @ ik (x )f k3 (u j f j3 )dxds Z t 0 Z @ jk (x )(u k f k3 )u i dxds Z t 0 Z @ jk (x )f k3 (u i f i3 )dxds + lim R!1 H(R) 20 holds forjj = 7, where H(R) = Z t 0 Z @ ik (x ) x R f k3 f k3 dxds + 1 R Z t 0 Z @ i (x )@ k x R f k3 f j3 dxds 1 2R Z t 0 Z @ i (x )@ j x R f k3 f k3 dxds Z t 0 Z @ jk (x ) x R f k3 f i3 dxds 1 R Z t 0 Z @ j (x )@ k x R f k3 f i3 dxds + 1 2R Z t 0 Z @ j (x )@ i x R f k3 f k3 dxds and f km (x;t) = X 2jjm 1 ! ( p t) i l jj 2 _ (x)@ ^ ! lk (0;t): Proof of Theorem 4.0.6. To illustrate the arguments used in the proof, we choose one of the terms from the equations (3.0.2){(3.0.4). All the other terms can be handled in a completely analogous way. We write 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds = 1 R Z t 0 Z @ i (x )@ k x R (u k f k3 )u j dxds + 1 R Z t 0 Z @ i (x )@ k x R f k3 u j dxds = 1 R Z t 0 Z @ i (x )@ k x R (u k f k3 )u j dxds + 1 R Z t 0 @ i (x )@ k x R f k3 (u j f j3 )dxds + 1 R Z t 0 Z @ i (x )@ k x R f k3 f j3 dxds =I 1 +I 2 +I 3 : 21 Now, applying Theorem 2.0.1 to obtain sup x2R 3 ;0st jxj 5 ju j f j3 j < 1 and using sup x2R 3 ;0st jxj 4 ju j j< +1 we conclude jI 1 j +jI 2 j C R Z t 0 Z @ k x R jxj 6 jxj 9 dxds C R Z t 0 Z 2R R 1 dds C R log 2: Hence, lim R!+1 1 R Z t 0 Z @ i (x )@ k x R u k u j dxds = lim R!+1 1 R Z t 0 Z @ i (x )@ k x R f k3 f j3 dxds: The claim then follows by treating the other terms the same way. 22 Chapter 5 Asymptotics using the vorticity moments As in the previous chapter, our aim is to describe the terms of the asymptotic develop- ment (2.0.4) using quadratic expressions, but here, we use the moments involving the velocity and the vorticity. The advantage of using vorticity is that there is no diculty at the spatial innity. First, we start with the equation (3.0.2). Sending R!1, we obtain the recursive relation Z x ! ij (x;t)dx Z x ! ij (x; 0)dx = Z t 0 Z (x )! ij dxds Z t 0 Z @ i (x )u k ! kj dxds + Z t 0 Z @ j (x )u k ! ki dxds (5.0.1) where we used that ! ij are well-localized to justify passing to the limit. Using the induction, we can easily prove the next statement. Theorem 5.0.7 Let , u 0 , u, and 0 be as in the statement of Theorem 2.0.1. Then jxj a 0 @ u j (;t) X 2jjm 1 ! (i) jj+1 ( p t) k jj 2 ! _ f ;k;j (t) 1 A L p =O t 0 +a=2(n=2)(1=21=p) (5.0.2) 23 for any2 [2;1], m2N, and a2 [0;m +nn=p), where f ;i;j (t) = [jj=2]1 X l=0 t l l! Z l (x )! ij (x; 0)dx [jj=2]1 X l=0 Z t 0 Z (ts) l l! @ i l (x )(u k ! kj )(x;s)dxds + [jj=2]1 X l=0 Z t 0 Z (ts) l l! @ j l (x )(u k ! ki )(x;s)dxds for all 2N n 0 and k;j2f1;:::;ng. 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Abstract (if available)

Abstract We derive an asymptotic expansion for smooth solutions of the Navier-Stokes equations in weighted spaces. This result removes previous restrictions on the number of terms of the asymptotics, as well as on the range of the polynomial weights. We also write the expansion in terms of expressions involving non-linear quantities.

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Creator Reis, Ednei F. (author)

Core Title Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 06/08/2011

Defense Date 05/02/2011

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

Tag asymptotic expansion,Navier-Stokes equation,OAI-PMH Harvest

Language English

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Advisor Kukavica, Igor (committee chair), Newton, Paul K. (committee member), Ziane, Mohammed (committee member)

Creator Email ednei82@gmail.com,edneirei@usc.edu

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