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Global existence, regularity, and asymptotic behavior for nonlinear partial differential equations

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Global existence, regularity, and asymptotic behavior for nonlinear partial differential equations

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Content GLOBAL EXISTENCE, REGULARITY, AND ASYMPTOTIC BEHAVIOR FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS by David Luke Massatt 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 (APPLIED MATHEMATICS) August 2022 Copyright 2022 David Luke Massatt Acknowledgements I would like to thank Igor Kukavica and Nabil Ziane for their assistance in the preparation of this thesis. The author was supported in part by the NSF grant DMS-1907992. ii TableofContents Acknowledgements ii Abstract iv Chapter1: Introduction 1 Chapter2: Kuramoto-SivashinskyEquation 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Notation and the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Properties of the average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Space-time energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Estimate onM 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Two barrier arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter3: Reduced-Kuramoto-SivashinskyEquation 32 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Main Theorem and Supporting Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter4: BoussinesqEquation 52 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Proofs for the global bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.4 Interior bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.5 Uniform Gronwall inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Bibliography 95 iii Abstract We rst address the global existence of solutions for the 2D Kuramoto-Sivashinsky equations in a periodic domain [0;L 1 ] [0;L 2 ] with initial data satisfyingku 0 k L 2C 1 L 2 2 , whereC is a constant. We prove that the global solution exists under the conditionL 2 1=CL 3=5 1 , improving earlier results. The solutions are smooth and decrease in energy until they are dominated by CL 3=2 L 1=2 2 , implying the existence of an absorbing ball in L 2 . Secondly, we prove the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial datau 01 2 L 2 andu 02 2H 1+ for> 0. Lastly, we address the asymptotic properties for the Boussinesq equations with vanishing thermal diusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of theL 2 norm of the velocity and its gradient, convergence of theL 2 norm ofAu, and ano(1)- type exponential growth forkA 3=2 uk L 2. We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time. iv Chapter1 Introduction We consider three partial dierential equation models, the Kuramoto-Sivashinsky equation (KSE), the re- duced Kuramoto-Sivashinsky equation (r-KSE) and the Boussinesq system of equations. We rst address the global existence of the 2D periodic Kuramoto-Sivashinsky equation @ t + + 2 + 1 2 jrj 2 = 0; with the initial data(0) = 0 , in two space dimensions on the domain = [0;L 1 ] [0;L 2 ] under the conditionL 2 L q 1 , whereq > 0 is a certain exponent. As it is more common, we shall also consider the velocity formulationu = (u 1 ;u 2 ) =r, which reads @ t u 1 + 2 u 1 + u 1 +u 1 @ x u 1 +u 2 @ x u 2 = 0 @ t u 2 + 2 u 1 + u 2 +u 1 @ y u 1 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 ; 1 with the initial data u(0) = u 0 = r(0). When u 2 = 0, the system reduces to the well-known 1D Kuramoto-Sivashinsky equationu t +u xxxx +u xx +uu x = 0 which is typically studied on a peri- odic domain [0;L]. The Kuramoto-Sivashinsky equation arises in many important physical contexts [Kur, S, T]. In par- ticular, it has been a model for instabilities of ame fronts and ion plasmas. It also serves as a model for a ow down an inclined plane in a presence of an electric eld. In addition, the KSE also serves as a model for low-dimensional chaos. The global existence, dissipativity, and the existence of the global attractor for the one dimensional KSE has now been well-established, while in two space dimensions, there is a fundamental diculty with global existence due to the lack of suitable energy conservation. In one space dimension, the primary problem for the instability, and thus the global existence, is the backward heat termu xx . The rst result establishing the global existence for the one-dimensional problem is that of Nicolaenko, Scheurer, and Temam [NST] for odd initial data. In [NST] it was also proven that lim sup t!1 ku(t)k L 2CL 5=2 , whereL denotes the size of the periodic domain. The oddness assumption was removed by Ilyashenko [I], and subsequently by Goodman [G] and Collet, Eckmann, Epstein, and Stubbe [CEES]. In particular, in [CEES], the authors proved that lim sup t!1 ku(t)k L 2CL 8=5 for arbitrary initial data. Finally, Giacomelli and Otto proved in [GO], that lim sup t!1 ku(t)k L 2 CL 3=2 , which is currently the most precise upper bound for the size of the absorbing ball for the 1D KSE. More recently, Otto has found in [O] (cf. also [GJO]), estimates for the space time averages of solutions. For other results on regularity of solutions of the 1D model, see also [BG,BiS,GK,M,SS,RK,T,TP,RK,WH] 2 Considering the laminar ame front model, the two-dimensional KSE appears more physically interest- ing. However, the global existence in 2D is a long-standing open problem. The main issue mathematically is that the energy dissipation is not available. In addition, the methods treating the KSE as a perturbation of the Burgers equations [I, G, GJO], do not extend to higher space dimensions. There are however still several results available on the global existence of solutions. In [AM], Ambrose and Mazzucato obtained the global existence of solutions for small initial data forL 1 andL 2 less than 2 along with the decay and analyticity of solutions. Sell and Taboada proved in [ST] that the global solutions still exist if one of the scales, sayL 2 , is suciently small compared to the other and with initial condition smaller than a function ofL 1 . Note that the mechanism for the global existence is dierent here than when both spatial scales are small since the existence is obtained by using oscillations in the vertical direction rather than by damping. The paper [ST] used methods inspired by the work on the 3D Navier-Stokes equations by Raugel and Sell [RS1, RS2, RS3] (cf. also subsequent works [HoS, H1, H2, KZ, KRZ]). Further, Molinet [M1, M2] showed local dissipation of the equation on [0;L 1 ] [0;L 2 ] under the conditionL 2 1=CL 67=35 1 and obtained a global solution provided also that the initial data is of a certain size as a function ofL 1 andL 2 . The size of the domain was extended further in the paper [BKRZ] toL 2 1=CL 22=25 1 . This last paper contains the best results reached at this point on the two dimensional domain size relation to global in time solutions. The main result of this chapter establishes the global existence under the condition L 2 1=CL 3=5 1 , improving earlier results; as in [BKRZ], the initial data needs to satisfy the condition ku 0 k L 2 1 CL 2 2 ; 3 which agrees with the condition in [BKRZ]. In addition, we prove that lim sup t!1 ku(t)k L 2CL 3=2 1 L 1=2 2 : The bound agrees with the one in [GO] if the data is constant in they direction. In the third chapter, we consider the well-posedness of the anisotropically-reduced Kuramoto-Sivashinsky equation (r-KSE) as introduced in [LY] @ t u 1 u 1 +u 1 @ x u 1 +u 2 @ y u 1 =u 1 @ t u 2 + 2 u 2 +u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 with the initial datau(0) =u 0 and,, and> 0 on the torus [0; 2] 2 . This model is constructed from the velocity formulation of the Kuramoto-Sivashinky equation (KSE) @ t u 1 + 2 u 1 + u 1 +u 1 @ x u 1 +u 2 @ y u 1 = 0 @ t u 2 + 2 u 2 + u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 : The anisotropically-reduced equation was devised to address core diculties with the two dimensional problem. In particular, by modifying the model in the rst dimension to resemble Navier-Stokes, the reduced equation acquires something akin to a maximum principle. In [LY], for suciently regular initial data,H 1 , andL 1 in the rst dimension, Larios and Yamazaki proved well-posedness on the torus. They 4 utilized Galerkin approximation to show local existence and uniqueness followed by energy methods and a maximum-like principle to prove the global existence. In this chapter, we reduce the required regularity of the initial data for global well-posedness by utiliz- ing a method involving semi-groups. Namely, we show that for initial data contained inL 2 H 1+ , for 2 (0; 3=2) a unique, strong, global solution exists. This allows for the initial conditions to lie in negative Sobolev spaces. In the last chapter, we address the asymptotic behavior of the Boussinesq equations u t u +uru +rp =e 2 t +ur = 0 ru = 0 with vanishing thermal/density diusivity, in a smooth bounded domain R 2 with the Dirichlet bound- ary condition u @ = 0 and subject to the initial condition (u(0);(0)) = (u 0 ; 0 ). Here,u represents the velocity,p the pressure, and the density or the temperature, depending on the physical context. The 2D Boussinesq system of equations is used in a wide range of physical contexts, from large scale oceanic and atmospheric ows where rotation and stratication are signicant to microuids and biophysics. It also relates closely to fundamental models in uid dynamics. In particular, the vorticity formulation of the incompressible Euler 5 equations away from the singularity can be described by the 2D Boussinesq equations (cf. [DWZZ]). For simplicity of exposition, we shall refer to the variable as the density, although it may also represent a temperature. While global existence results have been well-known in the case of positive viscosity and positive thermal diusivity, i.e., when adding the term in the equation for the density/temperature, we address here the case of vanishing thermal diusivity. In the case when both viscosity and diusion coecients vanish, the global existence and uniqueness remain open questions, although results on the local existence, blow-up criteria, explicit solutions, and nite time singularities have been proven; cf. the blow-up results in [CH, EJ], based on the singularity creation theorem for the Euler equations by Elgindi [E]. The case > 0 and = 0, considered here, was initially considered by Chae [C] and Hou and Li [HL]. In particular, Hou and Li obtained the global existence and persistence of regularity in H s H s1 for integer valueds 3 in the case of periodic boundary conditions. The paper [LLT] by Lai et al extended the result in [HL] to the Dirichlet boundary conditions. The persistence of regularity for the lower value s = 2 in the case of Dirichlet or periodic boundary conditions was addressed in [HKZ1]. Subsequently, Ju obtained in [J] thatCe Ct 2 is an upper bound for theH 1 norm for the density, also for the Dirichlet boundary conditions. The bound was lowered toe Ct in [KW2], where also more precise results were obtained for periodic boundary conditions. In particular, [KW2] contains a uniform in time upper bound for the quantitykD 2 uk L p for allp 2 in the periodic case. In a recent paper by Doering et al [DWZZ], the global existence, uniqueness, and regularity for the Boussinesq for the Lions boundary condition on a Lipschitz domain , was proven along with the dissipation of theL 2 norm of the velocity and its gradient. For other papers on the global existence and the regularity in Sobolev and 6 Besov spaces, see [ACW,ACSetal,BFL,BS,BrS,CD,CG,CN,CW,DP,HK1,HK2,HKR,HKZ2,HS, JMWZ,KTW,KW2,KWZ,LPZ,SW]. Additionally, we prove several results on the asymptotic behavior of solutions of the Boussinesq system (4.1) with the Dirichlet boundary conditions (4.2). In our rst main theorem, Theorem 4.2.1, we show that theH 1 norm of the velocity dissipates. We also establish a balanced convergence ofAu, cf. (4.5) below, whereA is the Stokes operator. Regarding the growth of the density, we prove that the rst Sobolev norm of the density is bounded, up to a constant, bye t for an arbitrarily small > 0, thus improving a result from [KW2] where the bound of the type e Ct was proven. Since the growth of the Sobolev norms of the density is controlled by the time integral ofkruk L 1, it is reasonable to expect that the bound was optimal; however, here we prove that the optimal bound is in facte t . It remains an open problem if one can achieve the estimate of the typee Ct , where2 [0; 1). The theorem holds under the assumption that (u 0 ; 0 ) belongs toH 2 H 1 . The ideas for the proof of Theorems 4.2.1 draw from the approaches in [DWZZ], [HKZ1], [LLT], [HKZ1], [J], [KW1], and [KW2]. Additionally, in Theorem 4.3.1, we show that the theorem and the persistence of regularity also hold under theH 1 H 1 assumption on the initial data. In the second main theorem, Theorem 4.2.2, we address the behavior of the solution in a higher reg- ularity norm. We prove that, under theH 3 H 2 assumption on the initial data, that for every> 0 the norm of (u;) in theH 3 H 2 norm is bounded bye t , up to a constant depending on> 0. This holds under theH 3 H 2 regularity of the initial data (u 0 ; 0 ). We point out that, as in Theorem 4.3.1, the same in fact holds under theH 2 H 2 assumption on the data. 7 In the last main theorem, Theorem 4.2.3, we consider the upper bound for theL p norm of the second derivatives of the velocity. As shown in [HKZ1], one may obtain a uniform bound whenp = 2. When p > 2, this is not known except in the case of periodic boundary condition, which is a result obtained in [KW1]. Here, we prove that we can obtain a polynomial in time bound in the interior of a domain when considering the Dirichlet boundary condition, which is considerably lower than e t type bound that would result from applying the Gagliardo-Sobolev inequality on the conclusions of Theorem 4.2.2. The proof is obtained by the change of variable from [KW1] combined with new localization arguments controlling the nonlocal nature of the transformation in [KW1] (see the double cut-o strategy in the proof of Theorem 4.2.3). 8 Chapter2 Kuramoto-SivashinskyEquation 2.1 Introduction In this chapter, we consider the global existence of the 2D periodic Kuramoto-Sivashinsky equation (KSE) in two space dimensions on the domain = [0;L 1 ] [0;L 2 ] under the conditionL 2 L q 1 , whereq > 0 is a certain exponent in the velocity formulationu = (u 1 ;u 2 ) =r, which reads @ t u 1 + 2 u 1 + u 1 +u 1 @ x u 1 +u 2 @ x u 2 = 0 @ t u 2 + 2 u 1 + u 2 +u 1 @ y u 1 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 ; with the initial data u(0) = u 0 . When u 2 = 0, the system reduces to the well-known 1D Kuramoto- Sivashinsky equationu t +u xxxx +u xx +uu x = 0 which is typically studied on a periodic domain [0;L]. 9 The main result of this chapter establishes the global existence under the conditionL 2 1=CL 3=5 1 ; with the initial data satisfying the condition ku 0 k L 2 1 CL 2 2 : In addition, we prove that the inequality lim sup t!1 ku(t)k L 2CL 3=2 1 L 1=2 2 holds. In Sections 2.2–2.4, we introduce the approach used in analyzing the two dimensional KSE. In particu- lar, we introduce the projection operatorsM andN based on the spatial average in the vertical direction. This method has been used in the work on the Navier-Stokes system [A,KZ,TZ], although in these cases the references considered the average in the thin direction, while we use the average along the long di- rection. This method is expedient here as much of the inherent diculties lie with the spatial average, which parallels the one dimensional problem. In Sections 2.5 and 2.6 we prove energy inequalities. We also provide a technical result, which we use in Section 2.7 to formulate control on the keyL 2 norm of the average. The idea of controlling the space time average of the solution through its space-timeL 4 norm is due to Giacomelli and Otto [GO]. For the zero-average functions, we have dissipation, but to control the average, the dissipation is insucient. To resolve this, we remove time dependence by controlling it with its time-average. This permits us to control the growth of the average by a small factor provided the barrier assumptions are satised. In Section 2.8, we show they are indeed satised by rst providing nite 10 local control and then extending them to global in time through the incremental control of the average. This allows us to acquire the nal result, which in terms of the size of the absorbing ball reduces to bounds achieved for the 1D KSE due Giacomelli and Otto when the initial data are independent ofy. 2.2 Notationandthemaintheorem Deningu =r and observing that xy = yx , and thus alsouru = (1=2)r(juj 2 ), the KSE becomes @ t u + 2 u + u +uru = 0 curlu = 0 (2.1) on a periodic domain = [0;L 1 ] [0;L 2 ], with the initial condition u(; 0) =u 0 : (2.2) Since the average ofu = (u 1 ;u 2 ) over is preserved, we normalize it to zero, i.e., we assume Z u 0 (x;y)dxdy = 0: (2.3) This in turn implies Z u(x;y;t)dxdy = 0; t 0; (2.4) for as long as the solution exists. The following is the main result of the chapter. 11 Theorem2.2.1. There exists 0 2 (0; 1] such that if L 2 0 maxfL 3=5 1 ; 1g (2.5) and ku(0)k L 2 0 L 2 2 ; (2.6) then there exists a solution of (2.1) which is global in time. IfL 1 1, it satises lim sup t!1 ku(t)k L 2CL 3=2 1 L 1=2 2 ; (2.7) whereC is a suciently large universal constant, while ifL 1 1, then lim sup t!1 ku(t)k L 2CL 1 L 1=2 2 ; (2.8) The above theorem, however with a more restrictive assumptionsL 2 0 =L 22=25 1 andL 1 2, was proven in [BKRZ]. Note that whenL 1 1, the above theorem extends [AM] since the initial data do not need to have a smallL 2 norm, but only satisfy (2.6). 12 2.3 Preliminaries Expanding (2.1) out, we obtain the component-wise formulation @ t u 1 + u 1 + 2 u 1 +u 1 @ x u 1 +u 2 @ y u 1 = 0 @ t u 2 + u 1 + 2 u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 ; (2.1) with the initial conditionu(; 0) =u 0 . Consider the average of a functionf2L 1 ( ) in they direction M(f) = 1 L 2 Z L 2 0 f(x;y;t)dy; and the dierence between the function and this average N(f) =f(x;y;t)M(f): Observe that M(u 2 ) = 0; (2.2) for allt 0 such that the solution exists. This property results from applying the operatorM to@ y u 1 = @ x u 2 which shows thatM(u 2 ) is constant inx. The constant then vanishes by (2.4). 13 For simplicity, denote by M 1 (x;t) =M(u 1 ) = 1 L 2 Z L 2 0 u 1 (x;y;t)dy (2.3) the average ofu 1 in they direction, and N 1 (x;y;t) =u 1 (x;y;t)M 1 (x;t) its deviation from the average. Clearly,u 1 =M 1 +N 1 . Analogously, we denote N 2 (x;y;t) =u 2 (x;y;t); since the average ofu 2 in they direction vanishes by (2.2). 2.4 Propertiesoftheaverage First, we recall several projection identities involving the operatorsM andN. Lemma2.4.1. The operatorsM andN satisfy the identities (i)M(M(f)) =M(f), (ii)M(N(f)) = 0, (iii)N(M(f)) = 0, and (iv)N(N(f)) =N(f), for allf2L 1 ( ). 14 The identities above imply the product rules M(fg) =MfMg +M(NfNg) (2.1) and N(fg) =MfNg +NfMg +N(NfNg); (2.2) which hold for all suciently regular functions f and g. Both identities are obtained by writing f = Mf +Nf andg =Mg +Ng and usingM(M(f)N(g)) =M(f)M(N(g)) = 0 andM(N(f)M(g)) = M(g)M(N(g)) = 0, as well asN(M(f)M(g)) = 0. Proof of Lemma 2.4.1. The part (i) is immediate from the denition. For (ii), we have M(N(f)) =M(fM(f)) =M(f)M(M(f)) =M(f)M(f) = 0: Similarly, N(M(f)) =M(f)M(M(f)) =M(f)M(f) = 0 providing (iii), and N(N(f)) =N(fM(f)) =fM(f) (M(f)M(M(f))) =fM(f) =N(f); establishing (iv). 15 Next, we rewrite the rst equation in (2.1) in terms ofM 1 andN 1 . The equation forM 1 is a perturbation of the 1D KSE. Lemma2.4.2. The averageM 1 satises @ t M 1 +@ xxxx M 1 +@ xx M 1 +M 1 @ x M 1 +M(N 1 @ x N 1 ) +M(N 2 @ x N 2 ) = 0: (2.3) Proof of Lemma 2.4.2. Applying an average iny, and recallingu 1 =M 1 +N 1 , we get M(@ t u 1 + u 1 + 2 u 1 +u 1 @ x u 1 +u 2 @ x u 2 ) =@ t M 1 +@ xxxx M 1 +@ xx M 1 +M(u 1 @ x u 1 ) +M(u 2 @ x u 2 ) =@ t M 1 +@ xxxx M 1 +@ xx M 1 +M 1 @ x M 1 +M(N 1 @ x N 1 ) +M(N 2 @ x N 2 ); where in the last inequality we usedM(u 2 ) = 0 and the product rule (2.1). 2.5 EnergyEstimates Now we turn to the energy estimates. Throughout the chapter, we abbreviatekk =kk L 2 ( ) . Lemma2.5.1. The energy inequalities for the quantitiesM 1 ,N 1 , andN 2 read (i) 1 2 d dt kM 1 k 2 k@ x M 1 k 2 +k@ xx M 1 k 2 .L 2 2 (kN 1 k 2 +kN 2 k 2 )kM 1 k, (ii) 1 2 d dt kN 1 k 2 krN 1 k 2 +kN 1 k 2 .L 2 2 (kM 1 k +kN 2 k)kN 1 k 2 , (iii) 1 2 d dt kN 2 k 2 krN 2 k 2 +kN 2 k 2 .L 2 2 (kM 1 k +kN 1 k)kN 2 k 2 . 16 Proof of Lemma 2.5.1. For the rst energy inequality (i), we multiply (2.3) by M 1 and integrate over , obtaining 1 2 d dt kM 1 k 2 k@ x M 1 k 2 +k@ xx M 1 k 2 = Z M(N 1 @ x N 1 ) +M(N 2 @ x N 2 ) M 1 . Z L 1 0 kN 1 k L 2 [0;L 2 ] k@ x N 1 k L 2 [0;L 2 ] jM 1 j + Z L 1 0 kN 2 k L 2 [0;L 2 ] k@ x N 2 k L 2 [0;L 2 ] jM 1 j: (2.1) For the rst term, we have Z L 1 0 kN 1 k L 2 [0;L 2 ] k@ x N 1 k L 2 [0;L 2 ] jM 1 j.kN 1 kk@ x N 1 k L 1 x L 2 y kM 1 k L 2 [0;L 1 ] .kN 1 kk@ x N 1 k L 2 y L 1 x kM 1 k L 2 [0;L 1 ] .kN 1 kk@ x N 1 k 1=2 k@ xx N 1 k 1=2 1 L 1=2 2 kM 1 k .L 2 2 k@ yy N 1 kL 1=2 2 k@ xy N 1 k 1=2 k@ xx N 1 k 1=2 1 L 1=2 2 kM 1 k.L 2 2 kN 1 k 2 kM 1 k; (2.2) where we applied Hölder’s inequality iny in the rst step, Agmon’s inequality inx in the third, and the Poincaré inequalities iny in the fourth. The second term on the far right side of (2.1) is treated analogously. For (ii), we start by applyingN to the rst equation in (2.1), then multiply byN 1 , and integrate over . This yields 1 2 d dt Z N 2 1 Z jrN 1 j 2 + Z (N 1 ) 2 = Z N(u 1 @ x u 1 )N 1 Z N(u 2 @ x u 2 )N 1 : (2.3) 17 In order to complete the proof of (ii), we only need to estimate the nonlinear components. Using the product rule (2.2) and Lemma 2.4.1 (iv), the nonlinear term in (2.3) is rewritten as Z N(u 1 @ x u 1 )N 1 Z N(u 2 @ x u 2 )N 1 = Z N 1 @ x N 1 N 1 Z M 1 @ x N 1 N 1 Z N 1 @ x M 1 N 1 Z N 2 @ x N 2 N 1 = Z N 2 1 @ x N Z M 1 @ x N 1 N 1 + 2 Z M 1 N 1 @ x N 1 + Z N 2 N 1 @ y N 1 = Z M 1 N 1 @ x N 1 + Z N 2 N 1 @ y N 1 .L 2 2 (kM 1 k +kN 2 k)kN 1 k 2 ; (2.4) where in the second equality we used @ x N 2 =@ y N 1 ; by@ y u 1 = @ x u 2 . The nal inequality in (2.4) is obtained by applications of Hölder’s, Agmon’s, and the Poincaré inequalities. For (iii), we observe thatu 2 =N 2 , multiply the second equation in (2.1) byN 2 and integrate over , we nd 1 2 d dt Z N 2 2 Z jrN 2 j 2 + Z N 2 2 = Z N(u 1 @ y u 1 )N 2 Z N(u 2 @ y u 2 )N 2 = 0: (2.5) 18 The nonlinear term in (2.5) is rewritten as Z u 1 @ y u 1 N 2 Z u 2 @ y u 2 N 2 = Z u 1 @ y N 1 N 2 Z N 2 @ y N 2 N 2 = Z u 1 @ y N 1 N 2 = Z u 1 @ x N 2 N 2 = Z (M 1 +N 1 )@ x N 2 N 2 .L 2 2 (kM 1 k +kN 1 k)kN 2 k 2 ; where the nal inequality is obtained by applying Hölder’s inequality iny and Agmon’s inequality iny andx. 2.6 Space-timeenergyestimates We require control of theL 2 norms in space-time of various quantities involvingM 1 ,N 1 , andN 2 , which are then used in the barrier arguments in the next section. Lemma2.6.1. Fors 0, let 1 (s) =L 2 2 (kM 1 (s)k +kN 2 (s)k) (2.1) and 2 (s) =L 2 2 (kM 1 (s)k +kN 1 (s)k): (2.2) 19 LetT2 [0; 1], and assume that 1 (s);(s) 1 C ; s2 [t 0 ;t 0 +T ] (2.3) withC 2 suciently large for somet 0 0. Then the quantity A(s) =kM 1 (s)k 2 +kN 1 (s)k 2 +kN 2 (s)k 2 satises A 0 (s) 2A(s); s2 [t 0 ;t 0 +T ]: (2.4) Also, we have sup t2[t 0 ;t 0 +T ] kM 1 k 2 2 ; sup t2[t 0 ;t 0 +T ] kN 1 k 2 2 ; sup t2[t 0 ;t 0 +T ] kN 2 k 2 2 ; Z t 0 +T t 0 k@ xx M 1 k 2 ; Z t 0 +T t 0 k@ x M 1 k 2 ; Z t 0 +T t 0 kN 1 k 2 ; Z t 0 +T t 0 kN 2 k 2 ; Z t 0 +T t 0 krN 1 k 2 ; Z t 0 +T t 0 krNk 2 .A(t 0 ) (2.5) with Z t 0 +T t 0 Z j@ x M 1 j 3 . 1 L 1=2 2 A(t 0 ) 3=2 : (2.6) 20 Proof of Lemma 2.6.1. Summing the three inequalities in Lemma 2.5.1, we nd 1 2 d dt (kM 1 k 2 +kN 1 k 2 +kN 2 k 2 ) +k@ xx M 1 k 2 + 1CL 2 2 (kM 1 k +kN 1 k) kN 1 k 2 +kN 2 k 2 .k@ x M 1 k 2 +krN 1 k 2 +krN 2 k 2 : (2.7) Integrating by parts and using Young’s inequality on the rst derivatives, we nd k@ x M 1 k 2 1 2 k@ xx M 1 k 2 + 1 2 kM 1 k 2 ; krN 1 k 2 1 2 (1 1 )kN 1 k 2 + 1 2(1 1 ) kN 1 k 2 ; krN 2 k 2 1 2 (1 2 )kN 2 k 2 + 1 2(1 2 ) kN 2 k 2 ; where 1 (t); 2 (t) 1=2 are dened in (2.1) and (2.2). Therefore, the inequality (2.7) becomes d dt (kM 1 k 2 +kN 1 k 2 +kN 2 k 2 ) +k@ xx M 1 k 2 +kN 1 k 2 +kN 2 k 2 2 kM 1 k 2 +kN 1 k 2 +kN 2 k 2 ; (2.8) and the inequality (2.4) follows. In order to obtain (2.5), we then use (2.4) and (2.8). It remains to establish (2.6). By the 1D Gagliardo-Nirenberg inequality we have Z j@ x M 1 j 3 . 1 L 1=2 2 k@ x M 1 k 5=2 k@ xx M 1 k 1=2 . 1 L 1=2 2 kM 1 kk@ x M 1 k 1=2 k@ xx M 1 k 3=2 ; 21 where the factorL 1=2 2 results from the integration in they variable. Therefore, integrating in time and applying Hölder’s inequality, we obtain Z t 0 +T t 0 Z j@ x M 1 j 3 1 L 1=2 2 sup t2(0;T ) kM 1 k Z t 0 +T t 0 k@ x M 1 k 2 1=4 Z t 0 +T t 0 k@ xx M 1 k 2 3=4 ; which by (2.5) implies (2.6). Next, inspired by [GO], we express the space-time integral ofM 4 1 in terms of quantities which we can control. This in turn provides a means to boundkM 1 k 2 . We rst consider thex-integral of (2.3) and thus denote h(x;t) = Z x 0 M 1 : By (2.4) and (2.3), we have Z L 1 0 h = 0: (2.9) The functionh satises the equation @ t h +@ xxxx h +@ xx h + 1 2 (@ x h) 2 + 1 2 M(N 2 1 +N 2 2 ) +g(t) = 0; (2.10) where g(t) = 1 2L 1 L 2 Z (M 2 1 +N 2 1 +N 2 2 ): (2.11) The equation (2.11) follows by integrating (2.10) inx over [0;L 1 ] and then using (2.9) and@ x h =M 1 . 22 Lemma 2.6.2. Under the assumptions of Lemma 2.6.1, i.e., assuming (2.3) for somet 0 0 andT2 [0; 1], the functionh(x;t) satises sup t2[t 0 ;t 0 +T ] sup x jh(x;t)j. L 1 L 2 1=2 A(t 0 ) 1=2 : (2.12) Proof of Lemma 2.6.2. For allt2 [t 0 ;t 0 +T ], we have jhj Z L 1 0 j@ x hj = Z L 1 0 jM 1 jL 1=2 1 kM 1 k L 2 [0;L 1 ] L 1 L 2 1=2 kM 1 k; which implies (2.12) by Lemma 2.6.1. Next, we obtain the following integral identity forM 1 . Lemma2.6.3. For the averageM 1 , we have an integral identity 1 12 Z t 0 +T t 0 Z M 4 1 = Z t 0 +T t 0 Z (@ x M 1 ) 3 + Z t 0 +T t 0 Z (@ xx M 1 ) 2 (@ x M 1 ) 2 h + 1 2 Z M 1 (t 0 +T ) 2 h(t 0 +T )M 1 (t 0 ) 2 h(t 0 ) + 1 2 Z t 0 +T t 0 Z M 2 1 g(t 0 ) + 1 4 Z t 0 +T t 0 Z (N 2 1 +N 2 2 )M 2 1 + Z t 0 +T t 0 Z (N 1 @ x N 1 +N 2 @ x N 2 )M 1 h: (2.13) 23 Proof of Lemma 2.6.3. Multiplying (2.3) byM 1 h and integrating over , we obtain Z M 1 h@ t M 1 + Z M 1 h@ xxxx M 1 + Z M 1 h@ xx M 1 + Z M 2 1 h@ x M 1 + Z M(N 1 @ x N 1 +N 2 @ x N 2 )M 1 h = 0: (2.14) For the rst term in (2.14), we have Z M 1 h@ t M 1 = 1 2 Z @ t (M 2 1 h) 1 2 Z M 2 1 @ t h = 1 2 Z @ t (M 2 1 h) + 1 2 Z M 2 1 (@ xxx M 1 +@ x M 1 ) + 1 4 Z (M 4 1 +M 2 1 M(N 2 1 +N 2 2 )) + 1 2 Z M 2 1 g(t) = 1 2 Z @ t (M 2 1 h) Z M 1 @ x M 1 @ xx M 1 + 1 4 Z M 4 1 + 1 4 Z M 2 1 (N 2 1 +N 2 2 ) + 1 2 Z M 2 1 g(t); (2.15) where we used (2.10), periodicity ofM 1 , and the fact that the integral over absorbs the average overy ofN 2 1 +N 2 2 . For the second term in (2.14), we have Z M 1 h@ xxxx M 1 = Z @ x M 1 @ xxx M 1 h Z M 2 1 @ xxx M 1 = Z (@ xx M 1 ) 2 h + Z M 1 @ x M 1 @ xx M 1 + 2 Z M 1 @ x M 1 @ xx M 1 = Z (@ xx M 1 ) 2 h 3 2 Z (@ x M 1 ) 2 ; 24 while for the third term in (2.14) we write Z M 1 h@ xx M 1 = Z (@ x M 1 ) 2 h Z M 2 1 @ x M 1 = Z (@ x M 1 ) 2 h: For the fourth term in (2.14), we integrate by parts obtaining Z M 2 1 @ x M 1 h = 1 3 Z M 4 1 : (2.16) Using (2.15)–(2.16), combined with R M 1 @ x M 1 @ xx M 1 =(1=2) R (@ x M 1 ) 3 for the second term in (2.15), yields 1 2 Z @ t (M 2 1 h) + 1 2 3 2 Z (@ x M 1 ) 3 + 1 4 1 3 Z M 4 1 + 1 4 Z M 2 1 (N 2 1 +N 2 2 ) + 1 2 Z M 2 1 g(t) + Z (@ xx M 1 ) 2 h Z (@ x M 1 ) 2 h + Z M(N 1 @ x N 1 +N 2 @ x N 2 )M 1 h = 0: (2.17) SinceM(M 1 h) =M 1 h, the last term equals R (N 1 @ x N 1 +N 2 @ x N 2 )M 1 h. The argument is then completed by integrating (2.17) int and combining fractions. 25 2.7 EstimateonM 1 Here we apply the energy inequalities in Lemma 2.6.1 and the integral identity (2.13) to estimate the space- time integral ofM 1 . Suppose that (2.3) holds for somet 0 0 andT2 [0; 1]. The nonlinear components of the integral identity (2.13) are bounded as Z t 0 +T t 0 Z (N 1 @ x N 1 +N 2 @ x N 2 )M 1 h. sup t2[t 0 ;t 0 +T ] sup x jhj Z t 0 +T t 0 Z jN 1 @ x N 1 +N 2 @ x N 2 jjM 1 j .L 2 2 sup t2[t 0 ;t 0 +T ] sup x jhj sup t2[t 0 ;t 0 +T ] kM 1 k Z t 0 +T t 0 (kN 1 k 2 +kN 2 k 2 ) .L 1=2 1 L 3=2 2 A(t 0 ) 1=2 sup t2[t 0 ;t 0 +T ] kM 1 k Z t 0 +T t 0 (kN 1 k 2 +kN 2 k 2 ); where the rst inequality is obtained as in (2.2), while the last follows from (2.12). For the fth term on the right-hand side of (2.13), we use the Cauchy-Schwarz inequality 1 4 Z t 0 +T t 0 Z M 2 1 N 2 1 1 48 Z t 0 +T t 0 Z M 4 1 + 3 Z t 0 +T t 0 Z N 4 1 ; and for theL 4 norm ofN 1 , we have Z t 0 +T t 0 Z N 4 1 . Z t 0 +T t 0 kN 1 k 2 krN 1 k 2 sup t kN 1 k 2 Z t 0 +T t 0 krN 1 k 2 .L 2 2 A(t 0 ) Z t 0 +T t 0 kN 1 k 2 .L 2 2 A(t 0 ) 2 : In the rst inequality, we used kN(f)k L p ( ) Cp 1=2 kN(f)k 2=p krN(f)k 12=p ; 2p<1 26 similarly to [KZ]. (This is obtained as in [TZ] by the stacking principle and the 2D Gagliardo-Nirenberg inequality.) The argument forN 2 is the same. Thus, for the nonlinear terms in (2.13) we have Z t 0 +T t 0 Z (N 1 @ x N 1 +N 2 @ x N 2 )M 1 h.L 1=2 1 L 3=2 2 A(t 0 ) 2 (2.1) and 1 4 Z t 0 +T t 0 Z M 2 1 (N 2 1 +N 2 2 )CL 2 2 A(t 0 ) 2 + 1 24 Z t 0 +T t 0 Z M 4 1 : (2.2) Using that R M 2 1 g(t) 0 in (2.13) along with (2.1)–(2.2), we get 1 24 Z t 0 +T t 0 Z M 4 1 . Z t 0 +T t 0 Z j@ x M 1 j 3 + sup x;t jhj Z t 0 +T t 0 Z (@ xx M 1 ) 2 + (@ x M 1 ) 2 + Z (M 2 1 (T ) +M 2 1 (0)) +L 1=2 1 L 3=2 2 A(t 0 ) 2 +L 2 2 A(t 0 ) 2 . 1 L 1=2 2 A(t 0 ) 3=2 + L 1 L 2 1=2 A(t 0 ) 3=2 +L 1=2 1 L 3=2 2 A(t 0 ) 2 +L 2 2 A(t 0 ) 2 . maxf1;L 1 g L 2 1=2 A(t 0 ) 3=2 + (L 1=2 1 L 3=2 2 +L 2 2 )A(t 0 ) 2 ; where we used (2.12) and Lemma 2.6.1 in the second inequality. Thus, by the Cauchy-Schwarz inequality, we obtain an estimate for the time integral ofkM 1 k 2 , which reads Z t 0 +T t 0 Z M 2 1 L 1=2 1 L 1=2 2 Z t 0 +T t 0 Z M 4 1 1=2 .L 1=2 1 L 1=2 2 maxf1;L 1 g L 2 1=2 A(t 0 ) 3=2 + (L 1=2 1 L 3=2 2 +L 2 2 )A(t 0 ) 2 ! 1=2 . maxfL 1=2 1 ;L 3=4 1 gL 1=4 2 A(t 0 ) 3=4 + (L 3=4 1 L 5=4 2 +L 1=2 1 L 3=2 2 )A(t 0 ); (2.3) 27 recalling thatT 1. Now, consider a solution which is dened on a time interval [0;T 0 + 1], whereT 0 0, and satises 1 (t); 2 (t) 1 C ; t2 [0;T 0 + 1]: By Lemma 2.6.1, we have Z t+T t Z (N 2 1 +N 2 2 ).L 4 2 Z t+T t (kN 1 k 2 +kN 2 k 2 ).L 4 2 A(t); (2.4) fort2 [0;T 0 ] andT2 [0; 1] . Summing (2.3) witht 0 =t and (2.4), translated int, we get Z t+T t A(s)ds. maxfL 1=2 1 ;L 3=4 1 gL 1=4 2 A(t) 3=4 + (L 3=4 1 L 5=4 2 +L 1=2 1 L 3=2 2 +L 4 2 )A(t) (2.5) fort2 [0;T 0 ] andT2 [0; 1]. In addition, ift 1, Lemma 2.6.1 implies A(t). Z t t1 A(s)ds (2.6) sinceA(t).A(ts) fors2 (0; 1). The inequalities (2.5) and (2.6) then imply fort 1 Z t+1 t A(s)dsC 0 maxfL 1=2 1 ;L 3=4 1 gL 1=4 2 Z t t1 A(s)ds 3=4 +C 0 (L 3=4 1 L 5=4 2 +L 1=2 1 L 3=2 2 +L 4 2 ) Z t t1 A(s)ds; (2.7) 28 where C 0 1 is a xed constant. Using (2.7) and the assumption (2.5), we then obtain the following lemma. Lemma2.7.1. For every2 (0; 1=2], there exists a suciently large constantC 1 and 0 2 (0; 1] with the following property: IfA(s) 1=CL 4 2 fors2 [t 1;t + 1], wheret 1, and if we have (2.5), then Z t+1 t A(s)dsC maxfL 2 1 ;L 3 1 gL 2 + Z t t1 A(s)ds (2.8) holds. Proof of Lemma 2.7.1. Applying Young’s inequality on the rst term in (2.7), we get C 0 maxfL 1=2 1 ;L 3=4 1 gL 1=4 2 Z t t1 A(s)ds 3=4 2C 4 0 3 maxfL 2 1 ;L 3 1 gL 2 + 2 Z t t1 A(s)ds: (2.9) Also, by (2.5), we have for the second term in (2.7) C 0 (L 3=4 1 L 5=4 2 +L 1=2 1 L 3=2 2 +L 4 2 ) Z t t1 A(s)dsC 0 ( 5=4 0 + 3=2 0 + 4 0 ) Z t t1 A(s)ds: (2.10) Now, we restrict 0 so that C 0 ( 5=4 0 + 3=2 0 + 4 0 ) 2 : (2.11) Replacing the inequalities (2.9) and (2.10) in (2.7), with the help of (2.11), then gives (2.8). 29 2.8 Twobarrierarguments Recalling that 1 =L 2 2 (kM 1 k +kN 2 k) and 2 =L 2 2 (kM 1 k +kN 1 k) from (2.1) and (2.2), we need to prove that both quantities are bounded by 1=C, whereC is an in Lemma 2.6.3 on an initial interval of time. In order to establish this, it suces to prove the following statement. Lemma2.8.1. Under the conditions (2.5) and (2.6) with 0 > 0 suciently small, we have A(t) = (kM 1 k 2 +kN 1 k 2 +kN 2 k 2 )(t) C 0 L 4 2 ; fort2 [0; 2]. Proof. Assume thatA(0) 0 =L 4 2 , and suppose thatT > 0 is the rst time such thatA(T ) = e 6 0 =L 4 2 . If 0 > 0 is suciently small, then Lemma 2.6.1 applies on [0;T ] and thus, in particular, (2.4) holds for t2 [0;T ]. Therefore fort2 [0;T ], we haveA(t)A(0)e 2T < 0 e 5 A(0) ifT 2, which contradicts the minimality ofT ifT 2. Therefore,T > 2 if it exists, and the claim is established. Now, we proceed to the second barrier argument, leading to the conclusion of the main theorem. Proof of Theorem 2.2.1. Assume that A(t 0 ) 1 C 0 L 4 2 ; (2.1) whereC 0 is a suciently large xed constant, andt 0 1. Note that Lemma 2.8.1 shown that (2.1) holds fort 0 = 1 provided 0 is suciently small. Lemma 2.6.1 then implies A(t) C C 0 L 4 2 ; 30 fortt 0 + 1. Then, by Lemma 2.7.1, Z t 0 +1 t 0 A(s)dsC maxfL 2 1 ;L 3 1 gL 2 + C C 0 L 4 2 and then applying Lemma 2.6.1 again, A(t 0 + 1)C maxfL 2 1 ;L 3 1 gL 2 + C C 0 L 4 2 ; (2.2) with a possibly largerC. Note that if 0 and are suciently small, then the right hand-side of (2.2) is smaller than the right-hand side of (2.1) and we obtain (2.1) wheret 0 is replaced byt 0 + 1. This provides the global existence. Finally, applying (2.8) iteratively, we also get lim sup t!1 Z t+1 t A(s)dsC maxfL 2 1 ;L 3 1 gL 2 ; from where, using (2.4), the assertions (2.7) and (2.8) follow. 31 Chapter3 Reduced-Kuramoto-SivashinskyEquation 3.1 Introduction Here, we consider the well-posedness of the anisotropically-reduced Kuramoto-Sivashinsky equation (r- KSE) as introduced in [LY] @ t u 1 u 1 +u 1 @ x u 1 +u 2 @ y u 1 =u 1 @ t u 2 + 2 u 2 +u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 (3.1) 32 with the initial datau(0) =u 0 and,, and> 0 on the torus [0; 2] 2 . This model is constructed from the velocity formulation of the Kuramoto-Sivashinky equation (KSE) @ t u 1 + 2 u 1 + u 1 +u 1 @ x u 1 +u 2 @ y u 1 = 0 @ t u 2 + 2 u 2 + u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 : In this chapter, we reduce the required regularity of the initial data for global well-posedness by uti- lizing a method involving semi-groups. Namely, we show that for initial data contained inL 2 H 1+ , for 2 (0; 3=2) a unique, strong, global solution exists. This allows for the initial conditions to lie in negative Sobolev spaces. We rst introduce an operator based on semi-groups and the r-KSE. This semi- group based approach provides a convenient approach to the negative Sobolev spaces, and additionally yields instantaneous smoothing properties. We then prepare a norm to prove our chosen operator is a closed, bounded, contraction mapping with respect to it, which ensures local existence of a unique solu- tion. The norm needs to be selected such that it balances the relative regularity foru 1 andu 2 . Once the correct choice is made, we ensure the operator is bounded with respect to the norm. The uniform bound and subsequently continuity of the operator requires control of the norm, which we achieve through local time, smallness of the norm of the semi-group operator, and by utilizing a large constant to balance the borderline case. We apply bootstrapping through a similar argument to prove the solution is smooth. With smooth solutions, we can then utilize the proof for the global existence of a solution as provided in [LY] to show existence of unique, global solutions past an initial time. Overlapping the two unique solutions ensures global existence and uniqueness of the solution for all time. At the end, we also provide a summary 33 of a simpler, alternative proof to the main result of [LY] by utilizing semi-groups to prove local existence instead of Galerkin approximation for the same initial regularity assumptions. 3.2 MainTheoremandSupportingResults The r-KSE equation @ t u 1 u 1 +u 1 @ x u 1 +u 2 @ y u 1 =u 1 @ t u 2 + 2 u 2 +u 2 +u 1 @ x u 2 +u 2 @ y u 2 = 0 @ y u 1 =@ x u 2 on the torusT = [0; 2] 2 has been shown in [LY] to have a strong, unique, global solutionu = (u 1 ;u 2 ) for initial datau 0 = (u 01 ;u 02 )2 (H 1 ) 2 andu 01 2L 1 . We dene a strong solutionu as satisfying Z T @ t u 1 1 dx + Z T ru 1 r 1 dx + Z T (ur)u 1 1 dx = Z T u 1 1 dx Z T @ t u 2 2 dx + Z T u 2 2 dx + Z T ru 2 r 2 dx + Z T (ur)u 2 2 dx = 0 for = ( 1 ; 2 )2C 1 (T) 2 on [0;T ]. In addition, we require u2C b ((0;T ];H k (T)) 2 \C b ([0;T ];L 2 (T))C b ([0;T ];H 1+ (T)); for allk > 0 and2 (0; 3=2) where we dene the negative Sobolev spacesH 1+ (T) as the dual space ofH 1 (T) when < 1. Then, we show that for weaker initial data than in [LY], we still have a strong, global solution. 34 Theorem 3.2.1. Foru 01 2 L 2 andu 02 2 H 1+ where2 (0; 3=2), there exists a unique, strong, global solution. For 3=2, an argument using boostrapping shows that the same result holds as well. We next recall known properties of Sobolev spaces that are used in the proof. Lemma 3.2.1. Let 0 < s 1 ;s 2 s 3 be such that 1 +s 3 s 1 +s 2 with a strict inequality ifs 3 = s 1 or s 3 =s 2 . Then, we have kuvk H s 3 .kuk H s 1kvk H s 2; foru2H s 1 andv2H s 2 . The Sobolev multiplicative inequality for two dimensions follows directly from Hölder and embedding inequalities. Furthermore, we have the following well known lemma. Lemma3.2.2. Ifs ~ s,t> 0, and2N, then we have ke t() fk H s .t s~ s kfk H ~ s; forf2H ~ s . This allows us to interchange between derivatives in space and powers in time. From [W], we recall the following property. 35 Lemma3.2.3. Forf(t) =e t f 0 andf 0 2H ~ s where ~ s2R, we have lim T!0 sup 0 ~ s. The proof follows from a density argument as in [W]. This helps us to show continuity in borderline cases. We suspect that the main theorem continues to hold if we change the domain from the torusT 2 to the whole spaceR 2 or to a bounded domain withu subject to appropriate boundary conditions. In both of these cases, what qualities the r-KSE displays comparatively with the KSE remains an open problem. 3.3 ProofofMainTheorem The proof of the main theorem utilizes a contraction mapping argument to show existence, uniqueness and regularity of the solution for local time. Then, due to the smoothness, we can extend by arguments made in [LY] our local solution to a global solution. Proof of Theorem 3.2.1. We begin with proving the local existence of a solution and its uniqueness. Thus, we dene a sequenceu (m) = (u (m) 1 ;u (m) 2 ) such that form2N 0 we have u (m+1) =e tA u 0 Z t 0 e (ts)A (u (m) ru (m) )(s)ds + Z t 0 e (ts)A L(u (m) )(s)ds (3.1) 36 where A = (; 2 ) T , and L((u 1 ;u 2 )) = (u 1 ;u 2 ) T , with the starting term u (0) = e tA u 0 . Furthermore, dene the norm k(u 1 ;u 2 )k X = sup tT ku 1 k L 2 +K sup tT t 2 ku 1 k H + sup tT ku 2 k H + +K sup tT t 1 4 (2) ku 2 k H 2 where2 [1=2; 1),2 (0; 1], andK 1 is a large constant to be determined. Denote = 1 + so that + =1 +. Here,X =C b ((0;T ];H k (T)) 2 \ C b ([0;T ];L 2 (T))C b ([0;T ];H 1+ (T)) . We claim thatu (m) is uniformly bounded inX byM whereM = 8 maxf1;ku 01 k L 2;ku 02 k H 1+g. To show this, we begin by considering the base case in an induction argument to nd that ku (0) (t)k X sup tT ke t u 01 k L 2 +K sup tT t 2 ke t u 01 k H + sup tT ke t 2 u 02 k H + +K sup tT t 1 4 (2) ke t 2 u 02 k H 2: Using Lemma 3.2.3 and choosingT > 0 to be suciently small, we determine sup tT t 2 ke t u 01 k H M 8K and sup tT t 1 4 (2) ke t 2 u 02 k H 2 M 8K : 37 Then, by the prior observations,ku (0) k X M=2. By the induction hypothesis, assume thatku (m) k X M. Then, observe that ku (m+1) k X ke tA u 0 k X + Z t 0 e (ts)A (u (m) ru (m) )ds X + Z t 0 e (ts)A L(u (m) )ds X : (3.2) Note that the bound onu (m+1) is determined by the bound on the preceding term in the sequence. Thus, for notational simplicity, we denoteu =u (m) . Examining the middle term, we observe that it expands as sup tT Z t 0 ke (ts) (u 1 @ x u 1 )k L 2ds + sup tT Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k L 2ds + sup tT Z t 0 ke (ts) 2 (u 1 @ x u 2 )k H +ds + sup tT Z t 0 ke (ts) 2 (u 2 @ y u 2 )k H +ds +K sup tT t 2 Z t 0 ke (ts) (u 1 @ x u 1 )k H ds + Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k H ds +K sup tT t 1 4 (2) Z t 0 ke (ts) 2 (u 1 @ x u 2 )k H 2ds + Z t 0 ke (ts) 2 (u 2 @ y u 2 )k H 2ds : (3.3) Neglecting the supremum in time for now, for the rst term in (3.3), using Lemmas 3.2.1 and 3.2.2, we have Z t 0 ke (ts) (u 1 @ x u 1 )k L 2ds. Z t 0 ke (ts) @ x (u 2 1 ) k L 2ds. Z t 0 (ts) 1 2 (1(21)) ku 2 1 k H 21ds . Z t 0 (ts) (1) ku 1 k 2 H ds. Z t 0 (ts) (1) s (s 2 ku 1 k H ) 2 ds . sup tT t 2 ku 1 k H 2 Z t 0 (ts) (1) s ds; (3.4) 38 noting that2 [1=2; 1), which ensures 0 2 1<. Thus, we determine that Z t 0 ke (ts) (u 1 @ x u 1 )k L 2dsC T sup tT t 2 ku 1 k H 2 C T M 2 K 2 (3.5) where C T is a constant depending only on T . The second term in (3.3), containing both u 1 and u 2 , is bounded by Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k L 2ds. Z t 0 ke (ts) @ y (u 1 u 2 )k L 2 +ke (ts) (u 1 @ y u 2 )k L 2 ds . Z t 0 (ts) 1 2 (1) ku 1 u 2 k H +ku 1 @ y u 2 k L 2 ds . Z t 0 (ts) 1 2 (1) ku 1 k H ku 2 k H 2 +ku 1 k H k@ y u 2 k H 1 ds . sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 Z t 0 (ts) 1 2 (1) + 1 s 2 s 1 4 (2) ds; (3.6) which implies that we have Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k L 2dsC T sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 C T M 2 K 2 : (3.7) The third term in (3.3) reads Z t 0 ke (ts) 2 (u 1 @ x u 2 )k H +ds. Z t 0 ku 1 @ x u 2 k L 2ds. Z t 0 ku 1 k H k@ x u 2 k H 1ds . sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 Z s 2 s 1 4 (2) ds: (3.8) 39 Thus, we nd that Z t 0 ke (ts) 2 u 1 @ x u 2 k H +ds C T M 2 K 2 : (3.9) The fourth term in (3.3) is estimated as Z t 0 ke (ts) 2 @ y (u 2 2 )k H +ds. Z t 0 ku 2 2 k H 1+ds. Z t 0 ku 2 k 2 H 2 ds . sup tT t 1 4 (2) ku 2 k H 2 2 Z t 0 s 1 2 (2) ds; (3.10) where we again used Lemmas 3.2.1, 3.2.2 and 1. Therefore, we can bound the term as Z t 0 ke (ts) 2 @ y (u 2 2 )k H +dsC T sup tT t 1 4 (2) ku 2 k H 2 2 C T M 2 K 2 : (3.11) To bound the remaining four terms in (3.3) we note that t 2 . (ts) 2 +s 2 : (3.12) 40 We use arguments in (3.4) to nd that Kt 2 Z t 0 ke (ts) (u 1 @ x u 1 )k H ds CK Z t 0 (ts) 2 +s 2 (ts) 2 (ts) (1) ku 2 1 k H 21ds CK Z t 0 (ts) 2 +s 2 (ts) (1 2 ) s ku 1 k 2 H ds CK sup tT t 2 ku 1 k H 2 Z t 0 (ts) (1 2 ) s 2 + (ts) (1) s ds: (3.13) As in (3.6), we observe the sixth term is bounded as Kt 2 Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k H ds CK sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 Z t 0 s 2 + (ts) 2 (ts) 1 2 (1) + 1 s 1 4 (2) ds: (3.14) Recalling (3.8) for the seventh term, we nd that Kt 1 4 (2) Z t 0 ke (ts) 2 u 1 @ x u 2 k H 2ds CK sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 Z t 0 (ts) 1 4 (2) +s 1 4 (2) s 2 s 1 4 (2) (ts) 1 4 (2) ds CK sup tT t 2 ku 1 k H sup tT t 1 4 (2) ku 2 k H 2 Z t 0 (ts) 1 4 () s 1 4 (2) + (ts) 1 4 (2) s 2 ds: (3.15) 41 For the eighth term, combining Lemmas 3.2.1 and 3.2.2 with (3.10), we observe that 1 2 Kt 1 4 (2) Z t 0 ke (ts) 2 @ y (u 2 2 )k H 2dsCK Z t 0 (ts) 1 4 (2) +s 1 4 (2) (ts) 1 4 ku 2 2 k H 2ds CK Z t 0 (ts) 1 4 (2) +s 1 4 (2) (ts) 1 4 ku 2 k 2 H 2 ds CK sup tT t 1 4 (2) ku 2 k H 2 2 Z t 0 (ts) 1 4 (2) +s 1 4 (2) (ts) 1 4 s 1 2 (2) ds CK sup tT t 1 4 (2) ku 2 k H 2 2 Z t 0 (ts) 1 4 (1) s 1 2 (2) + (ts) 1 4 s 1 4 (2) ds (3.16) noting that 2 > 1 ensures that Lemma 3.2.1 applies. In each case, the sum of the exponents under the integral is greater than or equal to1. Therefore, these four terms can be bounded by C T M 2 =K using (3.13)–(3.16). Lastly we consider the linear term in (3.2) where we have the inequality Z t 0 ke (ts)A L(u)k X ds. Z t 0 ke (ts) u 1 k L 2ds + Z t 0 ke (ts) 2 u 2 k H +ds +t 2 Z t 0 ke (ts) u 1 k H ds +t 1 4 (2) Z t 0 ke (ts) 2 u 2 k H 2ds . sup tT t 2 ku 1 k H Z t 0 s 2 + (ts) 2 s 2 + 1 ds + sup tT t 1 4 (2) ku 2 k H 2 Z t 0 (ts) 2 s 1 4 (2) + (ts) 1 2 ds; (3.17) 42 which we can bound byM=12 by the induction hypothesis and choosingT appropriately small. So, com- bining (3.4)–(3.11) with our bounds on (3.13)–(3.17) and choosingK 12C T maxf1;Mg we determine, with the bounds on the initial data and re-introducing supremums fortT , that the equation ku (m+1) k X ke tA u 0 k X + Z t 0 ke (ts)A (uru)k X ds + Z t 0 ke (ts)A L(u)k X ds M 2 + 5M 12 + M 12 =M holds. Therefore, we conclude by induction thatku (m) k X M for allm2N 0 . Regarding continuity, we claim that if u (m) 2C b ([0;T ];L 2 )C b ([0;T ];H + ) and t 2 u (m) 1 ;t 2 4 u (m) 2 2C b ([0;T ];H )C([0;T ];H 2 ); thenu (m+1) also resides in these spaces. We assume the second term to equal zero att = 0. We omit the proof for continuity forT > 0 since the argument is analogous to the one atT = 0. To prove continuity atT = 0, we examine the inequalities ku (m+1) 1 (t)u (m+1) 1 (0)k L 2ke t u 01 u 01 k L 2 + Z t 0 ke (ts) (u 1 @ x u 1 +u 2 @ y u 1 +u 1 )k L 2ds (3.18) 43 and ku (m+1) 1 (t)u (m+1) 1 (0)k H + ke t 2 u 02 u 02 k H + + Z t 0 ke (ts) 2 (u 1 @ x u 2 +u 2 @ y u 2 +u 2 )k H +ds (3.19) with t 2 ku (m+1) 1 (t)u (m+1) 1 (0)k H t 2 ke t u 01 k H +t 2 Z t 0 ke (ts) (u 1 @ x u 1 +u 2 @ y u 1 +u 1 )k H ds (3.20) and t 1 4 (2) ku (m+1) 2 (t)u (m+1) 2 (0)k H 2 t 1 4 (2) ke t 2 u 02 k H 2 +t 1 4 (2) Z t 0 ke (ts) 2 (u 1 @ x u 2 +u 2 @ y u 2 +u 2 )k H 2ds: (3.21) Considering the bounds in (3.4) and (3.13), we observe that for terms involving onlyu 1 , the sum of the exponents under the integrals in the bounds is equal to1, and thus the integrals are constants in time. By Lemma 3.2.3, these upper bounds decrease to zero whenT converges. In (3.6)–(3.10) and (3.14)–(3.17) the sum of the time exponents is strictly greater than1, so these integrals integrate tot for some> 0, which also converges to zero asT converges to zero. Combined with initial conditions being bounded in L 2 H + , we conclude that the right hand sides of (3.18)–(3.21) each decrease to 0 asT decreases to 0. 44 Finally, we claim that the sequenceu (m) is contracting. Considering forn2N 0 , we observe that ku (m+1) u (m) k X Z t 0 e (ts)A (u (m) ru (m) u (m1) ru (m1) )ds X + Z t 0 e (ts)A (L(u (m) )L(u (m1) ))ds X : (3.22) Expanding out the norms, we nd another eight terms analogous to (3.3). Thus, by using similar arguments to (3.4) and neglecting the supremum in time, we have for the rst term Z t 0 ke (ts) (u (m) 1 @ x u (m) 1 u (m1) 1 @ x u (m1) 1 )k L 2ds . Z t 0 (ts) 1 k(u (m) 1 u (m1) 1 )(u (m) 1 +u (m1) 1 )k H 21ds . Z t 0 (ts) 1 ku (m) 1 u (m1) k H ku (m) 1 +u (m1) 1 k H ds . sup tT t 2 ku (m) 1 u (m1) 1 k H sup tT t 2 ku (m) 1 +u (m1) 1 k H Z t 0 (ts) 1 s ds; (3.23) which we may bound byC T ku (m) u (m1) k X =K 2 allowing the implicit constants to depend on M. Re- ferring to (3.6), we nd for the second term that Z t 0 ke (ts) (u (m) 2 @ y u (m) 1 u (m1) 2 @ y u (m1) 1 )k L 2ds = Z t 0 ke (ts) @ y (u (m) 1 u (m) 2 u (m1) 1 u (m1) 2 ) (u (m) 1 @ y u (m) 2 u (m1) 1 @ y u (m1) 2 ) k L 2ds . Z t 0 (ts) 1 2 (1) k(u (m) 1 u (m) 2 u (m1) 1 u (m1) 2 )k H ds + Z t 0 ku (m) 1 @ y u (m) 2 u (m1) 1 @ y u (m1) 2 k L 2ds; 45 and by adding and subtracting terms and factoring, we conclude that the last expression equals Z t 0 (ts) 1 2 (1) ku (m) 2 (u (m) 1 u (m1) 1 )u (m1) 1 (u (m) 2 u (m1) 2 )k H ds + Z t 0 k(@ y u (m) 2 (u (m) 1 u (m1) 1 ) +u (m1) 1 @ y (u (m) 2 u (m1) 2 ))k L 2ds . Z t 0 (ts) 1 2 (1) ku (m) 2 k H 2ku (m) 1 u (m1) 1 k H +ku (m1) 1 k H k(u (m) 2 u (m1) 2 )k H 2 ds + Z t 0 k(@ y u (m) 2 k H 1ku (m) 1 u (m1) 1 k H +ku (m1) 1 k H k@ y (u (m) 2 u (m1) 2 ))k H 1ds where we applied Lemma 3.2.1. We can then bound this by the equation sup tT t 2 ku (m) 1 u (m1) 1 k H Z t 0 (ts) 1 2 (1) s 2 s 1 4 (2) ds + sup tT t 1 4 (2) ku (m) 2 u (m1) 2 k H 2 Z t 0 s 2 s 1 4 (2) ds: (3.24) The second term may then be bounded byC T ku (m) u (m1) k X =K 2 . We refer to (3.10) for the fourth term to deduce that Z t 0 ke (ts) 2 @ y ((u (m) 2 ) 2 (u (m1) 2 ) 2 )k H +ds. Z t 0 k(u (m) 2 u (m1) 2 )(u (m) 2 +u (m1) 2 )k H 1+ds . Z t 0 ku (m) 2 u (m1) 2 k H 2k(u (m) 2 +u (m1) 2 )k H 2ds . sup tT t 1 4 (2) ku (m) 2 u (m1) 2 k H 2 sup tT t 1 4 (2) ku (m) 2 +u (m1) 2 k H 2 Z t 0 s 1 2 (2) ds; (3.25) which is also bounded byC T ku (m) u (m1) k X =K 2 . A similar result holds for the third term using (3.8). By using arguments found in (3.13)–(3.17), we can also bound the last four terms byC T ku (m) u (m1) k X =K. 46 Finally, the linear term in (3.22) is also bounded by 1=12 for smallT by arguments analogous to (3.17). By choosingK 12C T , the constants can be made small. These new choices ofK andT are made such that they still satisfy the choices made for uniform boundedness. Therefore, using (3.22)–(3.25) with the analogous results for the remaining terms, and combining with our choices ofK andT , we have ku (m+1) u (m) k X 1 2 ku (m) 1 u (m1) k X : Therefore, the sequence is contracting, and u (m) converges by the Contraction Mapping Theorem to a unique, local solution to (3.1), which we denote asu. Since the solution is a xed point, we furthermore have the solution satisfying u2C b ([0;T ];L 2 )C b ([0;T ];H + ) and t 2 u 1 ;t 2 4 u 2 2C b ([0;T ];H )C([0;T ];H 2 ): By using bootstrapping, we can repeat the above arguments to show thatu2C b ((0;T ];H k (T)) 2 for k> 0. Thus, the local solution is strong. Up to this point, we have shown the existence of a strong, unique solution on [0;T ]. To prove the solution is global, we chooset 0 = T for some2 (0;T ). We have shown thatu(t 0 )2 H k (T) 2 for k> 0, which satises the initial condition regularity requirements of [LY]. Thus, we have a unique, global 47 solution ~ u with initial condition ~ u(t 0 ) =u(t 0 ). Sinceu = ~ u on [t 0 ;t 0 +] by uniqueness the solutionu is global. For completeness, we also provide a summary of an alternative proof to the main theorem of [LY] using semi-groups instead of Galerkin approximation. In [LY], it was found using approximations that whenu 01 2 L 1 andu 0 2 (H 1 ) 2 , a strong, unique, local solution exists. This was then extended to a global solution using a maximum principle. Under the same assumptions, we dene the sequence as in (3.1), but utilize a new space-time norm kuk X = sup tT kuk H 1 + sup tT t 1 2 ku 1 k H 2 + sup tT t 1 4 ku 2 k H 2 to prove the sequence is uniformly bounded with respect to theX-norm. The techniques to prove this have to be adapted however, as while we have stronger regularity, we also have lower exponents under the integral. Attempting to proceed as before results in the sum of the exponents under the integrals being less than1. The primary alteration to our argument is to use the stronger assumptions on our initial conditions to put less weight onto the higher regularity norms. We expandu (m+1) in a similar manner to (3.2) to nd that ku (m+1) k X ke tA u 0 k X + Z t 0 e (ts)A (u (m) ru (m) )ds X + Z t 0 e (ts)A L(u (m) )ds X : 48 Therefore, denotingu =u (m) , we nd that the center term expands as sup tT Z t 0 ke (ts) (u 1 @ x u 1 )k H 1ds + sup tT Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k H 1ds + sup tT Z t 0 ke (ts) 2 (u 1 @ x u 2 )k H 1ds + sup tT Z t 0 ke (ts) 2 (u 2 @ y u 2 )k H 1ds + sup tT t 1 2 Z t 0 ke (ts) (u 1 @ x u 1 )k H 2ds + Z t 0 ke (ts) (@ y (u 1 u 2 )u 1 @ y u 2 )k H 2ds + sup tT t 1 4 Z t 0 ke (ts) 2 (u 1 @ x u 2 )k H 2ds + Z t 0 ke (ts) 2 (u 2 @ y u 2 )k H 2ds : (3.26) We then, for brevity, consider the second term, which is of highest order. Using Hölder and Agmon’s inequalities, the second term satises Z t 0 ke (ts) u 2 @ y u 1 k H 1ds. Z t 0 (ts) 1 2 ku 2 @ y u 1 k L 2ds . Z t 0 (ts) 1 2 ku 2 k 1 3 L 2 kru 2 k 2 3 L 2 kru 1 k 2 3 L 2 ku 1 k 1 3 L 2 ds . sup tT ku 2 k H 1 sup tT ku 1 k 2 3 H 1 sup tT t 1 2 ku 1 k H 2 1 3 Z t 0 (ts) 1 2 s 1 6 ds: Thus, we have Z t 0 ke (ts) u 2 @ y u 1 k H 1dsC T kuk 2 X ; (3.27) 49 whereC T decreases to 0 withT . We observe that the base case is bounded byM=2 using Lemma 3.2.2, for someM > 0. Since the remaining terms from (3.26) all proceed similarly, using (3.27), and assuming by induction, the prior step is bounded byM, we nd that ku (m+1) k X C T M 2 : Thus,ku (m) k X M forT suciently small such thatC T 1=M. Continuity follows using arguments as in (3.18)–(3.21). To prove that the sequence is contracting, we observe that Z t 0 ke (ts) (u (m) 2 @ y u (m) 1 u (m1) 2 @ y u (m1) 1 )k H 1ds . Z t 0 (ts) 1 2 k@ y u (m) 1 (u (m) 2 u (m1) 2 )k L 2 +ku (m1) 2 @ y (u (m) 1 u (m1) 1 ))k L 2 ds . Z t 0 (ts) 1 2 k@ y u (m) 1 k L 3ku (m) 2 u (m1) 2 k L 6 +ku (m1) 2 k L 6k@ y (u (m) 1 u (m1) 1 )k L 3 ds . Z t 0 (ts) 1 2 kru (m) 1 k 2 3 L 2 ku (m) 1 k 1 3 L 2 ku (m) 2 u (m1) 2 k 1 3 L 2 kr(u (m) 2 u (m1) 2 )k 2 3 L 2 +ku (m1) 2 k 1 3 L 2 kru (m1) 2 k 2 3 L 2 kr(u (m) 1 u (m1) 1 )k 2 3 L 2 k(u (m) 1 u (m1) 1 )k 1 3 L 2 ds; which follows from Lemma 3.2.2, Hölder and Agmon’s inequalities. This can be bounded by sup tT ku (m) 1 k H 1 2 3 sup tT t 1 2 ku (m) 1 k H 2 1 3 sup tT ku (m) 2 u (m1) 2 k H 1 + sup tT ku (m1) 2 k H 1 sup tT ku (m) 1 u (m1) 1 k 2 3 H 1 sup tT t 1 2 ku (m) 1 u (m1) 1 k H 2 1 3 Z t 0 (ts) 1 2 s 1 6 ds 50 from which we conclude that the term is bounded by (T )Mku (m) u (m1) k X ; where(T ) may be made small forT close to 0. Thus, using analogous results for the remaining terms from the expansion (3.26), we conclude the sequence is contracting for T suciently small. Therefore using the Contraction Mapping Principle, we conclude the existence of a unique, strong solution on the torus for local time. Using bootstrapping, the denition of strong solution is satised locally. This provides an alternative to the Galerkin approximations for existence of local, strong solutions. In order to obtain the global solution we then proceed as in [LY] by showing that the boundedness ofku 1 k L 1 achieved through a maximum principle implies that the solution does not blow up in theH 1 norm. 51 Chapter4 BoussinesqEquation 4.1 Introduction In this chapter, we address the asymptotic behavior of the Boussinesq equations u t u +uru +rp =e 2 t +ur = 0 ru = 0 (4.1) with vanishing thermal/density diusivity, in a smooth bounded domain R 2 with the Dirichlet bound- ary condition u @ = 0 (4.2) 52 and subject to the initial condition (u(0);(0)) = (u 0 ; 0 ). Here,u represents the velocity,p the pressure, and the density or the temperature, depending on the physical context. In particular, we consider the case for > 0 and = 0. In a recent paper by Doering et al [DWZZ], the global existence, uniqueness, and regularity for the Boussinesq for the Lions boundary condition on a Lipschitz domain , was proven along with the dissi- pation of theL 2 norm of the velocity and its gradient. In this chapter, we prove several results on the asymptotic behavior of solutions of the Boussinesq system (4.1) with the Dirichlet boundary conditions (4.2). In Theorem 4.2.1, we show that theH 1 norm of the velocity dissipates. We also establish a balanced convergence ofAu, cf. (4.5) below, whereA is the Stokes operator. Regarding the growth of the density, we prove that the rst Sobolev norm of the density is bounded, up to a constant, bye t for an arbitrarily small> 0, thus improving a result from [KW2] where the bound of the typee Ct was proven. The ideas for the proof of Theorems 4.2.1 draw from the approaches in [DWZZ], [HKZ1], [LLT], [HKZ1], [J], [KW1], and [KW2]. Additionally, in Theorem 4.3.1, we show that the theorem and the persistence of regularity also hold under theH 1 H 1 assumption on the initial data. In Theorem 4.2.2, we address the behavior of the solution in a higher regularity norm. We prove that, under theH 3 H 2 assumption on the initial data, that for every> 0 the norm of (u;) in theH 3 H 2 norm is bounded bye t , up to a constant depending on> 0. This holds under theH 3 H 2 regularity of the initial data (u 0 ; 0 ). We point out that, as in Theorem 4.3.1, the same in fact holds under theH 2 H 2 assumption on the data. 53 Lastly, in Theorem 4.2.3, we consider the upper bound for the L p norm of the second derivatives of the velocity. We prove that we can obtain a polynomial in time bound in the interior of a domain when considering the Dirichlet boundary condition, which is considerably lower than e t type bound that would result from applying the Gagliardo-Sobolev inequality on the conclusions of Theorem 4.2.2. The proof is obtained by the change of variable from [KW1] combined with new localization arguments controlling the nonlocal nature of the transformation in [KW1] (see the double cut-o strategy in the proof of Theorem 4.2.3 below). We emphasize that all our results extend also in the often-studied problem of the channel with Dirichlet boundary conditions on top and the bottom and periodic boundary conditions on the sides. Also, our proofs are completely self-contained. 4.2 Maintheorems We consider the asymptotic behavior of the Boussinesq equations u t u +uru +rp =e 2 t +ur = 0 ru = 0 (4.1) and u @ = 0; (4.2) 54 coupling the Navier-Stokes equations [CF,DG,K1,K2,R,T1,T2,T3] for the velocityu = (u 1 ;u 2 ) and the pressurep with the equation for the density. The system is set on a smooth, bounded, and connected domain R 2 and supplemented with the initial condition (u;)(0) = (u 0 ; 0 ) in : Here,u denotes the velocity,p the pressure, and the density. Note that we set = 1 for simplicity of exposition; all the results extend to other values of with constants depending additionally on. From [CF,T1], we recall the classical spaces H =fu2L 2 ( ) :ru = 0 in ;un = 0 on@ g; wheren denotes the outward unit normal, and V =fu2H 1 0 ( ) :ru = 0 in g; utilized in the study of the Navier-Stokes equations. WithP: L 2 !H the Leray projector, denote by A =P; the Stokes operator with the domainD(A) =H 2 ( )\V . 55 It is known that for a suciently regular initial condition there exists a unique, global in time solution for (4.1)–(4.2) (cf. [C,HL]). In the rst theorem, we obtain the asymptotic properties ofA 1=2 u andAu in the energy norm. Theorem4.2.1. Let (u 0 ; 0 )2 (H 2 ( )\V )H 1 ( ). Then the solution (u;)2 (C([0;1);H)\L 2 loc ([0;1);D(A)))L 1 loc ([0;1);H 1 ( )) of (4.1)–(4.2) satises kAuk L 2C; (4.3) whereC depends on the size of the initial data, i.e., on the normskAu 0 k L 2 andk 0 k H 1. Moreover, kA 1=2 uk L 2 =kruk L 2! 0 ast!1; (4.4) and kAuP(e 2 )k L 2! 0 ast!1; (4.5) and for every> 0 we have k(t)k H 1C e t ; t 0; (4.6) whereC is a constant depending on and the size of initial data. 56 Above and in the sequel, we allow all constants to depend on . We note that in Theorem 4.2.1 the assumption ofH 2 regularity on the initial velocity can be relaxed tou 0 2V , as shown in Theorem 4.3.1 below. In the next statement, we obtain the asymptotic behavior of theH 3 H 2 norm of the solution (u;). From [LLT,T5], the local existence requires the initial data to satisfy the compatibility condition (u 0 rp 0 0 e 2 )j @ = 0; (4.7) wherep 0 denotes the initial pressure, which solves the Neumann boundary problem p 0 =r ( 0 e 2 u 0 ru 0 ) in@ rp 0 n @ = (u 0 + 0 e 2 )n @ withn denoting the outward unit normal. Theorem4.2.2. Assume that (u 0 ; 0 )2 (H 3 ( )\V )H 2 ( ) satises the compatibility condition (4.7), and let (u;) be the corresponding solution of (4.1)–(4.2). Then for every> 0, we have ku(t)k H 3C e t ; t 0 and k(t)k H 2C e t ; t 0; (4.8) whereC is a constant depending on. 57 Using the ideas in the proof of Theorem 4.3.1, the same long time behavior can be obtained with initial data (u 0 ; 0 )2D(A)H 2 ( ), now without the compatibility condition (4.7). In the next theorem, we obtain the interior bounds for theL p norm of the HessianD 2 u of the velocity in the interior, for anyp 2. Theorem4.2.3. Let (u 0 ; 0 )2 (H 2 ( )\V )H 1 ( ) andp2 [2;1), and suppose that 0 is open and relatively compact. Then for the corresponding solution (u;) of (4.1)–(4.2) and allt 0 > 0 we have a space-time bound kD 2 uk L p ([t 0 ;T ];L p ( 0 )) C(T 1=p + 1); (4.9) forTt 0 > 0, while in addition we have a pointwise in time bound kD 2 u(t)k L p ( 0 ) Ct (p+4)=4 ; tt 0 ; (4.10) where the constants in (4.9) and (4.10) depend ont 0 ,p, and dist( 0 ;@ ). 58 4.3 Proofsfortheglobalbounds First, we recall prior results on theL 2 norms corresponding to Theorem 4.2.1. Let (u 0 ; 0 )2 (H 2 ( )\ V )H 1 ( ). Then there exists a unique global solution (u;) such that u 2 L 1 ((0;1);H 2 ( ))\ L 2 loc ((0;1);H 3 ( )) and2L 1 ((0;1);H 1 ( )) of (4.1)–(4.2). Furthermore, the solution (u;) satises ku(t)k L 2 +k(t)k L 2. 1; t 0: (4.1) Here and below, the notationa.b meansaCb, whereC is a constant, which is allowed to depend on the size of the initial data in the pertinent norms. We denote by B(u;v) =P(urv) u;v2V the bilinear term corresponding to the Navier-Stokes equations. This allows us to rewrite (4.1) as u t +Au +B(u;u) =P(e 2 ) t +ur = 0: (4.2) We now turn to the proof of the rst theorem. Proof of Theorem 4.2.1. We begin by proving thatkuk L 2 dissipates. Inspired by [DWZZ], we shift the density byx 2 , i.e., introduce (x 1 ;x 2 ;t) =(x 1 ;x 2 ;t)x 2 ; (4.3) 59 and compensate withP =p(x 1 ;x 2 ;t)x 2 2 =2 to derive an equivalent system of equations u t u +uru +rP =e 2 t +ur =ue 2 ru = 0; (4.4) withu @ = 0. Multiplying the rst equation of (4.4) withu and the second by, integrating, and applying the Dirichlet boundary conditions and incompressibility, we obtain 1 2 d dt (kuk 2 L 2 +kk 2 L 2 ) +kruk 2 L 2 = 0: (4.5) Observe that the normkk L 2 may increase, thus no direct conclusion on decay rates can be reached from (4.5). The identity (4.5) implieskuk 2 L 2 andkk 2 L 2 are uniformly bounded in time and Z 1 0 kruk 2 L 2 . 1; where we allow all constants to depend onku 0 k H 2 andk 0 k H 1. Utilizing the Poincaré inequality, we also get Z 1 0 kuk 2 L 2 . 1: (4.6) 60 To prove the uniform continuity from above of theL 2 norm ofu, we multiply the rst equation in (4.4) withu and integrate by parts to nd that 1 2 d dt kuk 2 L 2 +kruk 2 L 2 = Z ue 2 kuk L 2kk L 2.kuk L 2; which, by Poincaré and Young’s inequalities, implies d dt kuk 2 L 2 +kruk 2 L 2 . 1: (4.7) It is elementary to show that if a dierentiable functionf : [0;1)! [0;1) satises R 1 0 f(s)ds <1 andf 0 (t). 1, then lim t!1 f(t) = 0. Applying the statement withf(t) =kuk 2 L 2 , the inequalities (4.6) and (4.7) imply kuk L 2! 0 ast!1: (4.8) Next, we aim to prove thatkruk 2 L 2 ! 0. We take theL 2 inner product of (4.2) 1 withAu to nd that 1 2 d dt kA 1=2 uk 2 L 2 +kAuk 2 L 2 =hB(u;u);Aui L 2 +hP(e 2 );Aui L 2 kB(u;u)k L 2kAuk L 2 +kk L 2kAuk L 2.kuk 1=2 L 2 kA 1=2 uk L 2kAuk 3=2 L 2 +kAuk L 2; (4.9) where we used kB(u;u)k L 2.kuk L 4kruk L 4.kuk 1=2 L 2 kuk H 1kuk 1=2 H 2 .kuk 1=2 L 2 kA 1=2 uk L 2kAuk 1=2 L 2 : (4.10) 61 In (4.9), we apply Young’s inequality and absorb the factorskAuk L 2 into the second term on the left side, obtaining d dt kA 1=2 uk 2 L 2 +kAuk 2 L 2 .kuk 2 L 2 kA 1=2 uk 4 L 2 + 1.kA 1=2 uk 4 L 2 + 1: Utilizing Lemma 4.5.1 in the Appendix, we obtain kA 1=2 u(t)k L 2. 1; t 0 (4.11) and kA 1=2 u(t)k L 2! 0 ast!1; giving (4.4). In addition, by the same lemma, lim sup t!1 Z t+t 0 t kAuk 2 L 2 .t 0 ; t 0 0: (4.12) We note in passing, and since it is needed in the proof of Theorem 4.2.3, that the inequality of type (4.12) also holds withAu replaced withu t . To show thatu t dissipates in theL 2 norm, we take the time derivative of (4.4) 1 , multiply byu t , and integrate by parts, to get the equation 1 2 d dt ku t k 2 L 2 +kru t k 2 L 2 =h t e 2 ;u t i L 2hu t ru;u t i L 2: (4.13) 62 For the rst term on the right, we apply (4.4) 2 to obtain h t e 2 ;u t i L 2 = Z (ur)(@ t u 2 ) Z u 2 @ t u 2 = Z ur@ t u 2 Z u 2 @ t u 2 .kk L 4kuk 1=2 L 2 kA 1=2 uk 1=2 L 2 kru t k L 2 +kuk L 2ku t k L 2 .kA 1=2 uk 1=2 L 2 kru t k L 2 +kuk L 2kru t k L 2; (4.14) where we usedkk L 4 . 1 andku t k L 2 .kru t k L 2 in the last inequality. For the second term on the right-hand side of (4.13), we write hu t ru;u t i L 2.ku t k 2 L 4 kruk L 2.ku t k L 2kru t k L 2kA 1=2 uk L 2: (4.15) Using (4.14) and (4.15) in (4.13) and then absorbing the factorskru t k L 2 by Young’s inequality, we get d dt ku t k 2 L 2 +kru t k 2 L 2 .kA 1=2 uk L 2 +kuk 2 L 2 +ku t k 2 L 2 kA 1=2 uk 2 L 2 .(t)(1 +ku t k 2 L 2 ); where: [0;1)! [0;1) is a bounded function, which satises lim t!1 (t) = 0. By Lemma 4.5.2, we get ku t k L 2. 1; t2 [0;1) (4.16) and ku t (t)k L 2! 0 ast!1 (4.17) 63 as well as lim sup t!1 Z t+t 0 t kru t k 2 L 2 = 0; t 0 0: (4.18) Next, from (4.2) 1 , we obtain kAuk L 2.ku t k L 2 +kB(u;u)k L 2 +kk L 2.ku t k L 2 +kuk 1=2 L 2 kA 1=2 uk L 2kAuk 1=2 L 2 + 1: Absorbing the factorkAuk 1=2 L 2 in the left-hand side by using Young’s inequality, we get kAuk L 2.ku t k L 2 +kuk L 2kA 1=2 uk 2 L 2 + 1; from where, by (4.4) and (4.17), we get (4.3). Note, in passing, that (4.3) and (4.8) imply ku(t)k L 1! 0 ast!1; (4.19) by Agmon’s inequality. From (4.2) 1 , we get kAuP(e 2 )k L 2.ku t k L 2 +kB(u;u)k L 2.ku t k L 2 +kuk 1=2 L 2 kA 1=2 uk L 2kAuk 1=2 L 2 : (4.20) By (4.3), (4.4), (4.8), and (4.17), the right-hand side of (4.20) converges to 0 ast!1, and we obtain (4.5). 64 We lastly proceed to prove the o(1)-type exponential estimate on the growth ofkrk L 2. For this, we rst need to prove the local in time boundedness ofkk H 1, which in turn requires us to rst bound R T 0 kruk L 1 for someT > 0. As above, we have Z T 0 kru t k 2 L 2 . 1; for allT > 0, where the constant depends onT . Now, consider the Stokes problem u t u +rp =uru +e 2 ru = 0 uj @ = 0: By [SvW] (see also [GS]) applied withs =p = 3, we obtain that for any ~ > 0 Z T 0 kuk 3 W 2;3 .kA 2=3+~ 3 u 0 k 3 L 3 + Z T 0 kurue 2 k 3 L 3 ; (4.21) for allT > 0, where the constant depends onT and ~ . In (4.21),A 3 denotes theL 3 version of the Stokes operator (cf. [SvW]). For the rst term on the right-hand side in (4.21), we use kA 2=3+~ 3 u 0 k L 3.kAu 0 k L 2. 1 (4.22) 65 with ~ = 1=6 from the embedding property on [SvW], while for the second term we estimate kurue 2 k 3 L 3 .kuk 3 L 6 kruk 3 L 6 +kk 3 L 3 .kuk L 2kA 1=2 uk 3 L 2 kAuk 2 L 2 + 1. 1: (4.23) Applying (4.22) and (4.23) in (4.21), we get Z T 0 kD 2 uk 3 L 3 . 1; (4.24) where the constant depends onT and consequently Z T 0 kruk L 1 . 1 (4.25) for allT > 0, where the constant depends onT , due to the Gagliardo-Nirenberg type inequality kvk L 1 .kvk 1=4 L 2 krvk 3=4 L 3 +kvk L 2: (4.26) By applying the gradient to (4.4) 2 and taking the inner product withr, we nd that 1 2 d dt krk 2 L 2 =hr(ur);ri L 2hr(ue 2 );ri L 2: (4.27) 66 The second term is estimated byCkruk L 2krk L 2, using the Cauchy-Schwarz inequality. The rst term is likewise bounded as hr(ur);ri L 2 = Z @ j (u i @ i )@ j = Z @ j u i @ i @ j 1 2 Z u i @ i jrj 2 .kruk L 1krk 2 L 2 ; by (4.4) 3 andu @ = 0. Thus, estimating the two terms in (4.27) as indicated, we conclude that d dt krk L 2.kruk L 1krk L 2 +kruk L 2.kruk L 1(krk L 2 + 1); (4.28) which implies that the exponential growth ofkrk L 2 is determined by the time integral ofkruk L 1. In particular, applying (4.25) to (4.28) yields kk H 1. 1; t2 [0;T ]; (4.29) for allT > 0, where the constant depends onT . Next, we x2 (0; 1] and claim that k(t)k H 1.e t ; t 0; (4.30) where we allow all constants to depend on. Note that (4.30) directly implies (4.6) by the denition (4.3). To prove (4.30), we need to estimate the time integral ofkruk L 1. Let 0<t 0 t 1 , wheret 0 2 is a large 67 time to be determined based on. By the Gagliardo-Nirenberg in space and Hölder’s inequalities in time, we have, using (4.26) Z t 1 +1 t 1 kruk L 1 Z t 1 +1 t 1 kruk 1=4 L 2 kuk 3=4 L 3 +kruk L 2 C Z t 1 +1 t 1 kruk 1=3 L 2 3=4 Z t 1 +1 t 1 kuk 3 L 3 1=4 + 1 2 ; (4.31) providedt 0 is suciently large. To bound theL 3 L 3 norm of u, we introduce a smooth cut-o function : [0;1)! [0; 1], where(t) = 0 on [0;t 1 1] and(t) = 1 on [t 1 ;1] withj 0 j. 1. Now we consider the equation (u) t (u) +r(p) = 0 uur(u) +e 2 which follows from (4.1) 1 ; note thatr (u) = 0 since is a function of time only. Using the W 2;3 estimate due to Sohr and Von Wahl [SvW] we have, similarly to (4.21)–(4.24), Z t 1 +1 t 1 kD 2 uk 3 L 3 . Z t 1 +1 t 1 1 kur(u)k 3 L 3 + Z t 1 +1 t 1 1 k 0 uk 3 L 3 + Z t 1 +1 t 1 1 kk 3 L 3 . Z t 1 +1 t 1 1 kuk 3 L 6 kruk 3 L 6 + Z t 1 +1 t 1 1 kuk 3 L 3 + 1 . Z t 1 +1 t 1 1 kuk L 2kruk 3 L 2 kAuk 2 L 2 + Z t 1 +1 t 1 1 kuk 4 L 4 + 1. 1; t 0 (4.32) where we used (4.3), (4.1), and (4.11). Also, for the rst factor of the rst term in (4.31), we use (4.4) to obtain that for any 0 > 0 there existst 0 1 suciently large so that Z t 1 +1 t 1 1 kruk 1=3 L 2 dt 3=4 0 : (4.33) 68 Thus, using (4.32) and (4.33) in (4.31), we obtain Z t 1 +1 t 1 kruk L 1dtC 0 + 1 2 ; tt 0 ; fort 0 1 suciently large, which in turn implies Z t t 0 kruk L 1dt(tt 0 ); tt 0 (4.34) if we choose 0 a suciently small constant. Note that (4.34) is obtained by adding the integrals of unit length. Returning to (4.28), we nd that Gronwall’s inequality implies kr(t)k L 2 (kr(t 0 )k L 2 + 1)e (tt 0 ) : (4.35) Finally, we use (4.29) implying k(t 0 )k H 1. 1; (4.36) where the constant depends ont 0 , which in turn only depends on. Combining (4.35) and (4.36) leads to the claimed inequality (4.30). We noted that the initial assumptions of Theorem 4.2.1 can be relaxed, implying the conclusions of Theorem 4.2.1 for less restrictive initial conditions than those required for (4.1). 69 Theorem 4.3.1. Let (u 0 ; 0 ) 2 V H 1 ( ). Then there exists a unique solution (u;) such that u 2 L 2 loc ([0;1);D(A))\C([0;1);H) and2L 1 loc ([0;1);H 1 ( )), which moreover satises kAuk L 2C ; t; (4.37) where> 0 is arbitrary. Proof of Theorem 4.3.1. Let (u;) be a solution to (4.4) on [0;T ] whereT2 (0; 1]. Integrating (4.5) in time, we obtain ku(t)k 2 L 2 +k(t)k 2 L 2 + Z t 0 kruk 2 L 2 .ku 0 k 2 L 2 +k 0 k 2 L 2 . 1; t2 [0;T ]: (4.38) We note that all constants are allowed to depend onku 0 k V andk 0 k H 1. We use this inequality in (4.9) obtaining d dt kA 1=2 uk 2 L 2 +kAuk 2 L 2 .kA 1=2 uk 4 L 2 + 1; (4.39) which implies, along with (4.38), that upon suitably reducing T > 0, we have u 2 L 1 ([0;T ];V )\ L 2 ([0;T ];D(A)) and kA 1=2 uk 2 L 2 . 1; t2 [0;T ]; (4.40) and then Z T 0 kAuk 2 L 2 . 1; t2 [0;T ]; (4.41) 70 upon returning to (4.39). Note that ku t k 2 L 2 .kAuk 2 L 2 +kB(u;u)k 2 L 2 +kk 2 L 2 .kAuk 2 L 2 +kuk L 2kA 1=2 uk 2 L 2 kAuk L 2 + 1 .kAuk 2 L 2 +kuk 2 L 2 kA 1=2 uk 4 L 2 + 1.kAuk 2 L 2 + 1; (4.42) by (4.1), (4.10), and (4.11). From (4.41) and (4.42), we obtainu t 2 L 2 ([0;T ];H). Thus, we may modifyu on a measure zero subset of [0;T ] so thatu2C([0;T ];H). In order to prove (4.37), we rst need to show uniqueness in the classVH 1 ( ). Thus, let (u (1) ; (1) ) and (u (2) ; (2) ) be solutions to the Boussinesq equation, and deneu =u (1) u (2) and = (1) (2) with both solutions satisfying the bounds (4.40) and (4.41) on [0;T ]. Then subtracting the evolution equations (4.1) 1 foru (1) andu (2) and testing the equation for the dierence withAu, we acquire 1 2 d dt kA 1=2 uk 2 L 2 +kAuk 2 L 2 .ku (1) k 1=2 L 2 kA 1=2 u (1) k 1=2 L 2 kA 1=2 uk 1=2 L 2 kAuk 3=2 L 2 +kuk 1=2 L 2 kA 1=2 uk 1=2 L 2 kA 1=2 u (2) k 1=2 L 2 kAu (2) k 1=2 L 2 kAuk L 2 +kk L 2kAuk L 2; (4.43) whence, using the bounds onu (1) andu (2) and absorbing factors ofkAuk L 2, we get d dt kA 1=2 uk 2 L 2 +kAuk 2 L 2 .kA 1=2 uk 2 L 2 +kuk L 2kA 1=2 uk L 2kAu (2) k L 2 +kk 2 L 2 : 71 On the other hand, from the density equations for (1) and (2) , we get d dt kk 2 L 2 .kuk L 1kr (2) k L 2kk L 2.kuk 1=2 L 2 kAuk 1=2 L 2 kr (2) k L 2kk L 2 . 0 kAuk 2 L 2 +kuk 2=3 L 2 kr (2) k 4=3 L 2 kk 4=3 L 2 . 0 kAuk 2 L 2 + (kA 1=2 uk 2 L 2 +kk 2 L 2 )kr (2) k 4=3 L 2 ; (4.44) where 0 is a suciently small constant to be determined. Adding (4.43) and (4.44), choosing 0 suciently small and absorbing factors ofkAuk 2 L 2 , we obtain d dt (kA 1=2 uk 2 L 2 +kk 2 L 2 ).kA 1=2 uk 2 L 2 +kAu (2) k L 2kA 1=2 uk 2 L 2 +kuk 2 L 2 +kk 2 L 2 ; (4.45) where we usedkr (2) k L 2 . 1 fort2 [0;T ] on the last term in (4.44), subject to reducingT . Applying a Gronwall argument to (4.45) and using (4.41) foru (2) , we conclude thatu = 0, whenceu (1) = u (2) on [0;T ]. In order to obtain (4.37), we observe thatu(t)2D(A) for a.e.t2 [0;T ] by (4.41). We chooset 0 2 (0;) such thatu(t 0 )2D(A). Sinceu is unique on [t 0 ;1), we may apply Theorem 4.2.1 and obtain (4.37) for tt 0 , concluding the proof. We remark that similar arguments show analogous reduced required regularity foru in Theorems 4.2.2 and 4.2.3. Next, we address a higher regularity norm. 72 Proof of Theorem 4.2.2. We start with a priori estimates and at the end of the proof we provide a sketch of the justication. Taking a time derivative of (4.1) 1 , we obtain u tt u t +u t ru +uru t +rp t = t e 2 ; which, after testing withu tt gives 1 2 d dt kru t k 2 L 2 +ku tt k 2 L 2 = Z u t ru j @ tt u j Z ur@ t u j @ tt u j + Z ( t e 2 )u tt .ku t k 1=2 L 2 kru t k 1=2 L 2 kruk 1=2 L 2 kAuk 1=2 L 2 ku tt k L 2 +kuk L 1kru t k L 2ku tt k L 2 +k t k L 2ku tt k L 2: Now we applykuk L 1 .kuk 1=2 L 2 kAuk 1=2 L 2 for the second term and k t k L 2 =kurk L 2.kuk L 1krk L 2; by (4.1) 2 , on the last. Absorbing the factors ofku tt k L 2, we obtain 1 2 d dt kru t k 2 L 2 +ku tt k 2 L 2 .ku t k L 2kru t k L 2kruk L 2kAuk L 2 +kuk 2 L 1kru t k 2 L 2 +kuk 2 L 1krk 2 L 2 . 1 +kru t k 2 L 2 +C e 2t ; (4.46) 73 where> 0 is arbitrarily small. In (4.46), we also used (4.3). Combining (4.18) and (4.46) with a uniform Gronwall argument, we get kru t k L 2.e t ; t 0 (4.47) and Z t 0 ku tt k 2 L 2 .e t ; t 0; where we allow constants to depend on . Now, consider the stationary, i.e., pointwise in time, Stokes problem u +rp =uruu t +e 2 uj @ = 0: (4.48) Note that kuru +u t e 2 k H 1.kD(uruu t )k L 2 +kk H 1 .kDuk 2 L 4 +kuk L 1kD 2 uk L 2 +kru t k L 2 +e t .kA 1=2 uk L 2kAuk L 2 +kuk L 1kAuk L 2 +kru t k L 2 +e t .e t ; using (4.47) in the last step. Applying theH 3 regularity for the Stokes problem (4.48), cf. [T4], leads to kuk H 3 +krpk H 1.e t : (4.49) 74 In order to obtain (4.8), we apply@ ij , fori;j = 1; 2, to (4.1) 2 , test with@ ij , and sum which leads to 1 2 d dt k@ ij k 2 L 2 =h@ ij (ur);@ ij i L 2 = Z @ ij u k @ k @ ij + 2 Z @ i u k @ jk @ ij + Z u k @ ijk @ ij ; (4.50) which holds for allt 0. The last term vanishes due to the incompressibility, while the second is bounded byCkruk L 1kD 2 k 2 L 2 . For the rst term on the far right side of (4.50), we write Z @ ij u k @ k @ ij .kuk L 4krk L 4kD 2 k L 2 . (kuk 1=2 L 2 kD 3 uk 1=2 L 2 +kuk L 2)(krk 1=2 L 2 kD 2 k 1=2 L 2 +krk L 2)kD 2 k L 2; (4.51) where we utilized the Gagliardo-Nirenberg inequalities. Now, we use (4.6) and (4.49) in (4.51), sum ini andj, and cancel a factor ofkD 2 k L 2 on both sides to obtain 1 2 d dt kD 2 k L 2.e 3t=2 +C e t kD 2 k 1=2 L 2 +kruk L 1kD 2 k L 2; whence, applying Young’s inequality 1 2 d dt kD 2 k L 2.e 2t + ( +kruk L 1)kD 2 k L 2; 75 for allt 0. Applying a Gronwall argument and using (4.34), which holds fort 0 > 0 suciently large depending on, we get kD 2 (t)k L 2C e Ct (kD 2 k L 2(t 0 ) + 1); tt 0 : (4.52) On the other hand, using Gronwall’s argument on [0;t 0 ] with (4.25) forT =t 0 , we get kD 2 (t)k L 2.kD 2 k L 2(0) + 1; t2 [0;t 0 ]; (4.53) where the constant depends ont 0 and thus on. Combining (4.52) and (4.53), we nally obtain (4.8) with C replacing. To justify the a priori bounds above, we consider the sequence of solutions u (n+1) t u (n+1) +u (n) ru (n+1) +rP (n+1) = (n+1) e 2 (n+1) t +u (n) r (n+1) =u (n+1) e 2 ru (n+1) = 0; (4.54) with the boundary conditionu (n+1) j @ = 0 and with the initial data (u (n+1) (0); (n+1) (0)) = (u 0 ; 0 x 2 ); 76 forn2N 0 . Forn = 0, we dene u (0) t u (0) +rP (0) = (0) e 2 (0) t =u (0) e 2 ru (0) = 0; with the boundary conditionu (0) j @ = 0 and with the initial data (u (0) (0); (0) (0)) = (u 0 ; 0 x 2 ): Since the system (4.54) is linear in (u (n+1) ; (n+1) ), it is easy to construct a local solution (u (n+1) ; (n+1) ). Also, our a priori estimates apply to the sequence and one may pass uniform bounds to the limit. Since the arguments are standard, we omit further details. 4.4 Interiorbounds In this section, we establish the nal result on the interior regularity of the second order derivatives. Proof of Theorem 4.2.3. In the proof, we work in the interior of the domain and thus localize the vorticity equation using a smooth cut-o function. With 0 as in the statement, consider a smooth function :R 2 77 [0;1)! [0; 1] such that supp [t 0 =2;1) with = 1 on 00 [3t 0 =4;1), where 00 is an open set such that 0 b 00 b . In order to prove (4.9), we rst claim that the vorticity! = curlu satises kr!k L p ([t 0 ;T ]:L p ( 0 )) .T 1=p + 1; (4.1) where the constant depends ont 0 , and dist( 0 ;@ ). Since (4.9) and (4.10) forp = 2 follow from (4.3), we xp> 2. We allow all constants to depend onp andt 0 , wheret 0 > 0 should be considered small. As in [KW2], we introduce the operator R =@ 1 (I ) 1 and a change of variable =!R(): (4.2) We shall applyR to functions which are compactly supported in , and we consider such functions ex- tended toR 2 by setting them identically to zero on c . Recalling the vorticity formulation for (4.1), ! t ! +ur! =@ 1 ; 78 we have, as in [KW2], that t +ur = [R;ur]()N()@ 1 2@ j (!@ j ) +!( t + +ur)R((ur)); (4.3) where N = ((I ) 1 I)@ 1 ; which has the property thatrN is in the Calderón-Zygmund class. The equation (4.3) is obtained by a direct computation from (!) t (!) +ur(!) =! t +! 2@ j (!@ j ) +!ur +@ 1 ()@ 1 and () t +ur() =ur and then using the identity@ 1 +R =N. Note that both operatorsR andN commute with translations (and hence derivatives) and they are smoothing of order one, i.e., they satisfy kRfk W 1;p;kNfk W 1;p.kfk L p; f2L p (R 2 ); (4.4) 79 for p2 (1;1), where the constant depends on p; the property (4.4) can be veried by computing the Fourier multiplier symbols corresponding toR andN (or cf. [KW2]). Sinceu is divergence free, we may rewrite [R;u j @ j ]() =R(u j @ j ())u j @ j R() =@ j R(u j )u j @ j R(): To acquireL p space-time estimates, we rewrite our solution as = (1) + (2) , where (1) satises (1) t (1) =f (1) t=0 = 0 with f =!( t + +ur)R((ur))N()urR()@ 1 ; while for (2) we have (2) t (2) =rg (2) t=0 = 0; where g =u 2!r +R(u): 80 Using theL p W 2;p regularity for the nonhomogeneous heat equation and the Gagliardo-Nirenberg inequal- ity, we have kD (1) k L p L p (R 2 (0;1)) .kD 2 (1) k L p L 2p=(p+2) (R 2 (0;1)) .kfk L p L 2p=(p+2) (R 2 (0;1)) ; (4.5) observe that 2p=(p + 2)> 1 sincep> 2. Similarly, using theL p W 1;p regularity for the nonhomogeneous heat equation in divergence form, we have kD (2) k L p L p (R 2 (0;1)) .kgk L p L p (R 2 (0;1)) : (4.6) For the right-hand side of (4.5), we use (4.4) to obtain kfk L 2p=(p+2) .k!k L 2(k t k L p +kk L p +kuk L 1krk L p) +kk L 2kuk L 1krk L p +kk L 2kk L p +kuk L 1kk L 2kk L p +kk L 2k@ 1 k L p .k!k L 2 + 1. 1; (4.7) for everyt 0, where the domains are understood to beR 2 . For the right-hand side in (4.6), we determine that kgk L p .kuk L 2pkk L 2p +k!k L 2pkrk L 2p +kuk L 1kk L pkk L 1 .kuk L 2pkk L 2p +k!k L 2p + 1 (4.8) 81 for everyt 0, by (4.4). To bound the right-hand side of (4.8), we write kk L q .k!k L q +kR()k L q .k!k L q +kk L 2. 1; q2 [2;1): (4.9) Therefore, we obtainkgk L p . 1 for allt 0. This fact and (4.7) imply by integration that the left-hand sides of (4.5) and (4.6) are bounded byT 1=p forTt 0 , from where kDk L p L p (R 2 (0;1)) .T 1=p (4.10) and thus kr(!)k L p L p (R 2 (0;1)) .krk L p L p (R 2 (0;1)) +kRr()k L p L p (R 2 (0;1)) .T 1=p + 1; which proves (4.1). The bound (4.9) then follows by a simple application of the interior elliptic estimate connectingu and!. The pointwise in time bound in (4.10) follows once we obtain kr!(t)k L p ( 0 ) .t 1=4+2=p+1=p 2 ; tt 0 ; (4.11) where the constant depends ont 0 ,p, and dist( 0 ;@ ). To prove (4.11), we begin by introducing a second smooth cut-o function:R 2 [0;1)! [0; 1] for which suppf = 1g =f(x;t)2R 2 [0;1) :(x;t) = 1g 82 and is such that = 1 on 0 [t 0 ;1). Denote ~ =: Using (4.3), we nd that ~ t ~ +ur ~ = [R;ur]()N() R((ur)) 2rr +( t +ur); (4.12) note that the terms in (4.3) containing derivatives of vanish after multiplication with, except for the term involving R, which is a non-local operator. The main reason for introducing the second cut-o function is that does not vanish on the boundary@ due to nonlocality ofR; cf. the denition (4.2). In order to estimater ~ , we apply@ k to (4.12) fork = 1, 2, multiply byj@ k ~ j 2p2 @ k ~ , integrate, and sum ink to acquire 1 2p d dt X k k@ k ~ k 2p L 2p X k Z @ k ~ j@ k ~ j 2p2 @ k ~ = X k Z @ k (u j @ j ~ )j@ k ~ j 2p2 @ k ~ + X k Z @ k ([R;ur]())j@ k ~ j 2p2 @ k ~ X k Z @ k (N())j@ k ~ j 2p2 @ k ~ X k Z @ k (R((ur)))j@ k ~ j 2p2 @ k ~ 2 X k Z @ k (@ j @ j )j@ k ~ j 2p2 @ k ~ + X k Z @ k (( t +ur))j@ k ~ j 2p2 @ k ~ : (4.13) 83 The second term on the left-hand side of (4.13) is estimated as X k Z @ k ~ j@ k ~ j 2p2 @ k ~ = 2p 1 p 2 X k Z @ j (j@ k ~ j p )@ j (j@ k ~ j p ) 1 p X k kr(j@ k ~ j p )k 2 L 2 = 1 p D; (4.14) where we denoted D = P k kr(j@ k ~ j p )k 2 L 2 . For the rst term on the right-hand side of (4.13), we use the incompressibility ofu to determine that X k Z @ k (u j @ j ~ )j@ k ~ j 2p2 @ k ~ = X k Z @ k u j @ j ~ j@ k ~ j 2p2 @ k ~ .kruk L 2kr ~ k L 4p X k kj@ k ~ j 2p1 k L 4p=(2p1) .o(1) X k k@ k ~ k 2p L 4p ; (4.15) whereo(1) denotes a function which is bounded on [0;1) and converges to 0 ast!1. Applying the estimate k@ k ~ k 2p L 4p =kj@ k ~ j p k 2 L 4 .kj@ k ~ j p k L 2kr(j@ k ~ j p )k L 2. D 1=2 k@ k ~ k p L 2p in (4.15), we obtain X k Z @ k (u j @ j ~ )j@ k ~ j 2p2 @ k ~ o(1) D 1=2 X k k@ k ~ k p L 2p D 8 +o(1) X k k@ k ~ k 2p L 2p : (4.16) 84 For the second term on the right-hand side of (4.13), we use integration by parts and write X k Z @ k ([R;ur]())j@ k ~ j 2p2 @ k ~ =(2p 1) X k Z [R;ur]()j@ k ~ j 2p2 @ kk ~ = 2p 1 p X k Z [R;ur]()j@ k ~ j p2 @ k ~ @ k (j@ k ~ j p ) .k[R;ur]()k L 2p X k kj@ k ~ j p1 k L 2p=(p1)kr(j@ k ~ j p )k L 2 . D 1=2 k[R;ur]()k L 2p X k k@ k ~ k p1 L 2p : (4.17) For the second factor in the last expression, we have k[R;ur]()k L 2p.kk L 1kR(u j @ j ())u j @ j R()k L 2p .k@ j R(u j ())k L 2p +kuk L 1k@ j R()k L 2p.kk L 2p. 1; (4.18) using the incompressibility ofu and k(t)k L 2p. 1; (4.19) which follows fromk 0 k L 2p .k 0 k H 1 . 1 and the L p conservation for . (Recall that all constants depend onp.) Thus, by (4.17)–(4.18), we have X k Z @ k ([R;ur]())j@ k ~ j 2p2 @ k ~ C D 1=2 X k k@ k ~ k p1 L 2p D 8 +C X k k@ k ~ k 2p2 L 2p : (4.20) 85 For the third term on the right-hand side of (4.13), we obtain X k Z @ k (N())j@ k ~ j 2p2 @ k ~ . X k k@ k (N())k L 2pkj@ k ~ j 2p1 k L 2p=(2p1) . (krk L 1kN()k L 2p +kk L 1krN()k L 2p) X k k@ k ~ k 2p1 L 2p .kk L 2p X k k@ k ~ k 2p1 L 2p . X k k@ k ~ k 2p1 L 2p : For the fourth term on the right-hand side of (4.13), we observe that X k Z @ k (R((ur)))j@ k ~ j 2p2 @ k ~ . X k k@ k (R((ur)))k L 2pkj@ k ~ j 2p1 k L 2p=(2p1) . (kk L 1krR((ur))k L 2p +krk L 1kR((ur))k L 2p) X k k@ k ~ k 2p1 L 2p .kk L 2pkuk L 1krk L 1 X k k@ k ~ k 2p1 L 2p . X k k@ k ~ k 2p1 L 2p ; where we used (4.19). For the fth term on the right-hand side of (4.13), we determine that 2 X k Z @ k (@ j @ j )j@ k ~ j 2p2 @ k ~ = 2p 1 p X k Z @ j @ j j@ k ~ j p2 @ k ~ @ k (j@ k ~ j p ) .k@ j @ j k L 4 X k kj@ k ~ j p1 k L 4kr(j@ k ~ j p )k L 2. D 1=2 krk L 4 X k kj@ k ~ j p k (p1)=p L 4(p1)=p : By the Gagliardo-Nirenberg inequality, we have for the last factor kj@ k ~ j p k (p1)=p L 4(p1)=p . kj@ k ~ j p k p=(2p2) L 2 kr(j@ k ~ j p )k (p2)=(2p2) L 2 (p1)=p .kj@ k ~ j p k 1=2 L 2 kr(j@ k ~ j p )k (p2)=2p L 2 . D (p2)=4p k@ k ~ k p=2 L 2p ; 86 fork = 1; 2. Therefore, by Young’s inequality, we conclude that 2 X k Z @ k (@ j @ j )j@ k ~ j 2p2 @ k ~ . D (3p2)=4p jrj L 4 X k j@ k ~ j p=2 L 2p D 8 +Cjrj 4p=(p+2) L 4 X k j@ k ~ j 2p 2 =(p+2) L 2p : For the nal term of (4.13), we integrate by parts and obtain X k Z @ k (( t +ur))j@ k ~ j 2p2 @ k ~ = 2p 1 p X k Z ( t +ur)j@ k ~ j p2 @ k ~ @ k (j@ k ~ j p ) .kk L 2pk t +urk L 1 X k kj@ k ~ j p1 k L 2p=(p1)kr(j@ k ~ j p )k L 2 . D 1=2 X k k@ k ~ k p1 L 2p ; using (4.19) and (4.9). Therefore, we have X k Z @ k (( t +ur))j@ k ~ j 2p2 @ k ~ D 8 +C X k k@ k ~ k 2p2 L 2p : (4.21) Introducing (t) = X k Z j@ k ~ j 2p ; 87 we may rewrite (4.13) by applying (4.14), (4.16), (4.20)–(4.21) as (1 + ) 0 + D 2 .o(1)(1 + ) + (1 + ) (p1)=p + (1 + ) (2p1)=2p +krk 4p=(p+2) L 4 (1 + ) p=(p+2) : (4.22) It may seem that the rst term in (4.22) causes an exponential increase of , but importantly we have the property Z t 0 (1 + ).t +krk 2p L 2p ([0;t];L 2p ) .t; t 0; (4.23) where we used (4.10) in the second step. Now we show that the inequality (4.23) implies that the growth is algebraic. We divide the inequality (4.22) by (1 + ) p=(p+2) , obtaining ((1 + ) 2=(p+2) ) 0 .o(1)(1 + ) 2=(p+2) + (1 + ) (p 2 2)=(p 2 +2p) + (1 + ) (3p2)=(2p(p+2)) +krk 4p=(p+2) L 4 ; 88 which upon integration and applying Jensen’s (or Hölder’s) inequality yields fort 0, (1 + ) 2=(p+2) . 1 +o(1) Z t 0 (1 + ) 2=(p+2) + Z t 0 (1 + ) (p 2 2)=(p(p+2)) + Z t 0 (1 + ) (3p2)=(2p(p+2)) + Z t 0 krk 4p=(p+2) L 4 . 1 +o(1)t 12=(p+2) Z t 0 (1 + ) 2=(p+2) +t 1(p 2 2)=(p(p+2)) Z t 0 (1 + ) (p 2 2)=(p(p+2)) +t 1(3p2)=(2p(p+2)) Z t 0 (1 + ) (3p2)=(2p(p+2)) + Z t 0 krk 4 L 4 p=(p+2) t 1p=(p+2) ; (4.24) where we also used (0) = 0 since vanishes in a neighborhood offt = 0g. Therefore, recalling (4.23), we have fortt 0 the inequality (1 + ) 2=(p+2) .t; where we used R t krk 4 L 4 . t for > 0 on the last term in (4.24). Raising the resulting inequality to (p + 2)=2, we obtain 1 + C p t (p+2)=2 : 89 By the support properties of and, we get forp 1 kr!k L 2p ( 0 ) .krk L 2p ( 0 ) +p 3=2 .kr ~ k L 2p + 1. 1=2p + 1.t (p+2)=2 ; concluding the proof. 4.5 UniformGronwallinequalities In the appendix, we state and prove two Gronwall inequalities needed in the proof of Theorem 4.2.1. The following lemma is used to show (4.8). Lemma 4.5.1. Assume thatx;y : [0;1)! [0;1) are measurable functions withx dierentiable, which satisfy _ x +yC(x 2 + 1) (4.1) and xCy; (4.2) for some positive constantC. If Z 1 0 x(s)ds<1; (4.3) thenx(t)C fort 0 and lim t!1 x(t) = 0: (4.4) 90 Moreover, lim sup t!1 Z t+a t y(s)dsCa (4.5) for everya> 0, where the constant in (4.5) depends on the constants in (4.1) and (4.2). Proof of Lemma 4.5.1. Let2 (0; 1], and denoteb = p . Based on (4.3), there existst 0 > 0 such that Z t+2b t x(s)ds; tt 0 : (4.6) Integrating the inequality _ x.x 2 + 1 and using (4.6), we obtain x(t 2 )e C (x(t 1 ) +Cb).x(t 1 ) +b (4.7) for allt 1 andt 2 such thatt 1 t 2 t 1 + 2b. By (4.6), for everytt 0 , there exists ~ t2 [t;t +b] such that x( ~ t). b ; and thus applying (4.7) witht 1 = ~ t leads to x(t 2 ). b +b. p ; ~ tt 2 t 1 + 2b; (4.8) where we usedb = p in the last step. The inequality (4.8) holds for allt 2 t 0 +a, and since > 0 is arbitrarily small, (4.4) follows. The inequality (4.5) is obtained by integratingy.x 2 +1 and using (4.4). 91 The next Gronwall-type lemma is needed to establish (4.16) and (4.17), which are necessary for the proofs of (4.3) and (4.5). Lemma 4.5.2. Assume thatx;y : [0;1)! [0;1) are measurable functions withx dierentiable, which satisfy _ x +y(t)(x + 1) (4.9) and xCy; (4.10) where: [0;1)! [0;1)issuchthat(t)C fort2 [0;1)and(t)! 0,ast!1. Ifalsox(0)C, then x(t). 1; t2 [0;1) and lim t!1 x(t) = 0 (4.11) as well as lim sup t!1 Z t+a t y(s)ds = 0; (4.12) for everya> 0. Proof of Lemma 4.5.2. First, by the boundedness of, we have x(t). 1; t2 [0;T ]; 92 for everyT > 0, where the constant depends onT . Next, there existst 0 > 0 such that _ x + 1 2 y(t); tt 0 ; which is obtained by choosing t 0 so large that the term containing x on the right-hand side of (4.9) is absorbed in the half of the second term on the left-hand side, cf. (4.10). Let> 0. Then there existst 1 t 0 such that _ x + 1 C x 2 ; tt 1 ; which shows that as long as x , we have _ x + (1=C)x 0, implying an exponential decay of x. Therefore, by increasingt 1 , we can assume that x(t); tt 1 : (4.13) Since> 0 was arbitrary, we obtain (4.11). To prove (4.12), note that we may assume _ x + 1 2 y 2 ; tt 1 ; (4.14) by increasingt 1 if necessary. Integrating (4.14) betweent andt +a, we get Z t+a t y(s)ds.x(t) +a.(1 +a); tt 1 ; 93 where we used (4.13) in the last step. 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Abstract (if available)

Abstract We first address the global existence of solutions for the 2D~Kuramoto-Sivashinsky equations in a periodic domain [0,L₁] × [0,L₂] with initial data satisfying ∥u₀∥_{L²} ≤ C⁻¹L₂⁻², where C is a constant. We prove that the global solution exists under the condition L₂ ≤ 1/C L₁³/⁵, improving earlier results. The solutions are smooth and decrease in energy until they are dominated by C L³/²L₂¹/², implying the existence of an absorbing ball in L². Secondly, we prove the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data u₀₁ ∈ L² and u₀₂ ∈ H⁻¹⁺^η for η > 0. Lastly, we address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the L² norm of the velocity and its gradient, convergence of the L² norm of Au, and an o(1)-type exponential growth for ∥A³/²u∥_L². We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

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Creator Massatt, David (author)

Core Title Global existence, regularity, and asymptotic behavior for nonlinear partial differential equations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Degree Conferral Date 2022-08

Publication Date 07/20/2022

Defense Date 05/04/2022

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

Tag asymptotics,Boussinesq equation,global existence,Kuramoto-Sivashinsky equation,OAI-PMH Harvest,partial differential equations,reduced Kuramoto-Sivashinsky equation,uniqueness

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kukavica, Igor (committee chair), Ghanem, Roger (committee member), Jang, Juhi (committee member), Ziane, Nabil (committee member)

Creator Email davidmassatt@me.com,dmassatt@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111373257

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Source 20220720-usctheses-batch-957 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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