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On some nonlinearly damped Navier-Stokes and Boussinesq equations

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On some nonlinearly damped Navier-Stokes and Boussinesq equations

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Content On some nonlinearly damped Navier-Stokes and Boussinesq equations by Zhanerke Temirgaliyeva A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Mathematics) August 2020 Copyright 2020 Zhanerke Temirgaliyeva Acknowledgements First of all, I would like to thank my advisor Nabil Ziane for his constant support, patient guidance, and warm encouragement over the course of my degree. I am also greatly indebted to Professor Igor Kukavica not only for his time and patience, but for his intellectual contributions to my development — I learned so much from him. A very special thank you to Professor Sergey Lototsky for his extensive knowledge and for all of the invaluable help and advice. I would also like to extend my deepest gratitude to Professor Susan Montgomery. No one could have asked for a more supportive and understanding Graduate Chair. Her constant enthusiasm and encouragement motivated and inspired me in various ways. As a teaching assistant, I have been lucky to learn from many brilliant teachers, and I have used them as role models in developing my own teaching philosophy. Thank you to Guillermo Reyes Souto, Nathaniel Emerson, and Richard Arratia for teaching me how to teach mathematics. I am grateful to the staff in the Department of Mathematics — Amy Yung, Arnold Deal, and Chaunte Lavene Williams — not only for all their help with the formalities of the graduate program, but also for their friendly and caring presence. My time at USC was made enjoyable in large part due to the many friends that became and remain a big part of my life. I thank my KAP 411 officemates and friends Irmak Balçık, Lernik Asserian, and Maria Allayioti for all the fun times we have shared during these years. Thank you, girls! Last but certainly not least I would like to thank my family for all their love and encouragement. Thank you to my parents Nurlan Temirgaliyev and Gulbarshin Jumakhayeva who raised me with a love ii of mathematics and supported me in all my pursuits. To my brothers Satzhan and Magzhan and sister- in-law Dinara who always kept my spirits high. To my nephews Zhangir, Amirzhan, Madiyar, and nieces Adiya and Alima who have been a constant source of joy and inspiration. In the end, I would like to dedicate this thesis to the loving memory of my dear grandmother, Aqkün äzhe, who didn’t get to share this moment with us, but who, I know, would have been proud. iii Table of Contents Acknowledgements ii Abstract v Chapter 1: Navier-Stokes and Euler equations with damping 1 1.1 Introduction and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Local-in-time solutions in the whole space . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Local-in-time solutions to the Euler equations in a bounded domain . . . . . . . . . . . . 16 1.4 Global-in-time solutions in the whole space . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 2: Boussinesq equations with damping 41 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Global estimates in L 2 L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3 Global estimates in H 1 L 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Global regularity in H 2 H 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.5 Global regularity in H 1 H 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.6 Global regularity in H 2 H 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References 70 iv Abstract This thesis is devoted to the study of three-dimensional incompressible Navier-Stokes, Euler and Boussinesq equations with damping. It consists of two chapters. In Chapter 1, we study the Euler and Navier-Stokes equations with nonlinear damping. Namely, we consider the following modification of the incompressible 3D Navier-Stokes equations: u t u + (ur)u +juj 1 u +r = 0; div u = 0; u(x; 0) =u 0 (x): In the damping term juj 1 u above, > 0 and 1 are real parameters. We address the questions of existence and uniqueness of local-in-time and global-in-time solutions inR 3 and in a bounded set R 3 . Chapter 2 is devoted to the study of the long time behavior of weak solutions to the damped incom- pressible 3D Boussinesq equations without diffusivity: u t u + (ur)u +juj 1 u +r =e 3 ; t + (ur) = 0; div u = 0; u(x; 0) =u 0 (x); (x; 0) = 0 (x) v in a bounded domain R 3 and inR 3 . The analysis of the long time behavior of weak solutions is carried out in the Sobolev space H 2 H 2 . vi Chapter 1 Navier-Stokes and Euler equations with damping 1.1 Introduction and notations The main purpose of this chapter is to study the well-posedness and long time behavior of solutions to the three-dimensional incompressible Navier-Stokes and Euler equations with nonlinear damping u t u + (ur)u +juj 1 u +r = 0; (x;t)2 (0;T ); div u = 0; (x;t)2 [0;T ); u(x; 0) =u 0 (x); x2 : (1.1.1) (1.1.2) (1.1.3) Here is eitherR 3 or an open bounded set with smooth boundary inR 3 ,> 0 and 1 are real param- eters, the unknown functions are the velocity u: R + 7!R 3 , u =u(x;t) = (u 1 (x;t);u 2 (x;t);u 3 (x;t)) with x = (x 1 ;x 2 ;x 3 ), and the pressure : R + 7!R, = (x;t). The given function u 0 = u 0 (x) is the initial velocity and the constant 0 represents the viscosity coefficient of the flow. Coordinatewise (1.1.1) can be written as follows: @ t u i + 3 X j=1 (u j r)u i +ju i j 1 u i +@ i = 0; i = 1; 2; 3: 1 The damping comes from the resistance to the motion of the flow and describes various physical phe- nomena such as porous media flow, drag or friction effects, and some dissipative mechanisms (see [CJ] and the references therein). When = 0, the problem (1.1.1)-(1.1.3) reduces to the classical Navier-Stokes equations. In 1934 Leray constructed a global-in-time weak solution and a local strong solution (see [L]), and in 1951 Hopf proved the existence of a global-in-time weak solution to the initial boundary value problem (see [H]). Since then, many mathematicians studied the uniqueness and regularity of Leray-Hopf solutions, however, inR 3 the uniqueness and regularity of Leray-Hopf solutions still remain open problems. Up to now, the global existence of strong solutions has been proved only when the initial data is sufficiently small. This chapter is structured as follows. In the Section 1.2 we prove the existence of local-in-time solutions inR 3 and in Section 1.3 we prove the existence of local-in-time solutions in a bounded domain R 3 . In Section 1.4 we study the global well-posedness of the system (1.1.1)-(1.1.3) inR 3 . Throughout this thesis, the symbol C will denote a generic positive constant, which is allowed to depend on the domain as well as on indicated parameters and to change from line to line. The notation A.B means that there exists a constant C, independent of the effective parameters of A and B, such that ACB. For R n , the function space L p ( ), 1 p 1 represents the Lebesgue space of scalar or vector-valued functions with the norm kfk p = 8 > > > < > > > : R jf(x)j p dx 1=p if 1p<1; ess sup x2 jf(x)j if p =1: We write (;) to denote the inner product in L 2 ( ): (f;g) = Z fgdx: 2 The notation W k;p ( ), 1p1, k2N denotes the Sobolev space with the norm kfk W k;p = 8 > > > < > > > : P j jk k@ fk p p if 1p<1; P j jk k@ fk 1 if p =1; where @ is the distributional derivative @ =@ 1 1 @ n n . When k2N, W k;2 ( ) can also be denoted H k ( ). The Sobolev space H k generalizes to the case k =s2R with the norm kuk H s = Z R n (1 +jj 2 ) s jb u()j 2 d 1=2 ; whereb u denotes the Fourier transform of u. We write ((;)) H m to denote the inner product in H m ( ): ((f;g)) H m = X j jm (@ f;@ g): Given a Banach space X with normkk X , we denote by L p (0;T ;X), 1p<1 the set of functions f(t) defined on (0;T ) with values in X such that R T 0 kf(t)k p X dt <1, and by L 1 (0;T ;X) the set of functions f(t) defined on (0;T ) with values in X such that ess sup t2(0;T) kf(t)k X <1. 1.2 Local-in-time solutions in the whole space Our goal is to prove the existence of the local-in-time solutions to the Navier-Stokes and Euler equa- tions in the whole spaceR 3 . Theorem 1.2.1. Assume 0, 4, m = 4; 5;:::; [], u 0 2 H m (R 3 ), and div u 0 = 0. Then there exists a time T > 0 such that there exists a unique solution u2C(0;T ;H m (R 3 ))\C 1 (0;T ;C(R 3 ))of the problem (1.1.1)-(1.1.3). 3 The proof of the Theorem requires several steps. First, we prove existence of solutions to the regular- ized equations. Then we show that there exists time T > 0 and a subsequence (u " ) convergent to a limit function u that solves Navier-Stokes or Euler equations. We need to treat the damping termjuj 1 u by posing a question “If u2 W m;p , what can be said aboutjuj 1 u?” We start with the following basic result: Proposition 1.2.1 ([E]). For all s>n=2, H s (R n ) is a Banach algebra. That is, for all u;v2H s (R n ), uv2H s (R n ), and kuvk H s.kuk H skvk H s: (1.2.1) We derive an obvious corollary: Corollary 1.2.2. Assume s>n=2, = 2; 3;:::. Then for any u2H s (R n ),juj 1 u2H s (R n ), and kjuj 1 uk H s.kuk H s: (1.2.2) The general answer to the question posed above was given in [B]: Theorem 1.2.3. Assume F (0) = 0, either mp > n or m 2, or m = n 2 and p = 1, and u2W m;p (R n ). Then the condition F (m) 2L p loc (R) is necessary and sufficient for F (u)2W m;p (R n ). In particular, the general case has been discussed in [Si]: Theorem 1.2.4. Assume m = 1; 2;:::, > 1 are such that m2 (n=p;). Then for all u2W m;p (R n ), juj 1 u2W m;p (R n ), and kjuj 1 uk W m;p.kuk W m;p: (1.2.3) Lemma 1.2.5. Assume > 1, m2Z + . Then for u2L 1 \H m (R n ),juj 1 u2H m (R n ) and kjuj 1 uk H m.kuk 1 1 kuk H m: (1.2.4) 4 To proof this lemma, we will need Theorem 1.2.6 (Hölder’s inequality, [F]). Assume f m : ! R are measurable with f m 2 L pm ( ) (m = 1;:::;k). Assume that 0<r1 and 0<p m 1 (m = 1;:::;k) are such that k X m=1 1 p m = 1 r : Then Q k m=1 f m 2L r ( ), and k Y m=1 f m r k Y m=1 kf m k pm : Theorem 1.2.7 (Gagliardo-Nirenberg interpolation inequality, [G, N]). Assume u:R n !R. Assume 1q;r1, m2N. Suppose 2R and j2N satisfy 1 p = j n + 1 r m n + 1 q and j m 1: Then k@ j uk p Ck@ m uk r kuk 1 q : Proof of Lemma 1.2.5. We have kjuj 1 uk H m = X jjm k@ juj 1 uk 2 2 : Whenjj =k, the Leibniz rule implies @ (juj 1 u) = X j1j++j k j=k C(k;j 1 j;:::;j k j)juj k1 u @ 1 u@ k u; 5 so by Hölder’s inequality k@ (juj 1 u)k 2 .kuk k 1 X j1j++j k j=k k@ 1 u@ k uk 2 kuk k 1 k@ 1 uk 2k j 1 j k@ k uk 2k j k j : Now Gagliardo-Nirenberg interpolation inequality implies for each i = 1;:::;k k@ i uk 2k j i j .k@ uk j i j k 2 kuk 1 j i j k 1 : Therefore, k@ (juj 1 u)k 2 .kuk k 1 X j1j++j k j=k 2 4 X j j=jj k@ uk 2 3 5 j 1 j jj kuk 1 j 1 j jj 1 2 4 X j j=jj k@ uk 2 3 5 j k j jj kuk 1 k jj 1 .kuk k+k1 1 k@ uk 2 =kuk 1 1 k@ uk 2 : Now kjuj 1 uk H m = X jjm kD (juj 1 u)k 2 2 X jjm (kuk 1 1 kD uk 2 ) 2 .kuk 1 1 X jjm kD uk 2 2 =kuk 1 1 kuk H m: Regularized equations Recall the definition of the mollifier operator: 6 Definition 1.2.8. Given a radial function (jxj)2C 1 0 (R 3 ), 0, R R 3 dx = 1, the mollification K " v ("> 0) of a function v2L p (R 3 ), 1p1 is defined as (K " v)(x) =" 3 Z R 3 xy " v(y)dy: (1.2.5) We consider the following regularization of the problem (1.1.1)-(1.1.3): 8 > > > > > > > > < > > > > > > > > : @ t u " +K " [(K " u " )r(K " u " )] +K " [jK " u " j 1 K " u " ] =r " +K " (K " u " ); div u " = 0; u " (0) =u 0 : (1.2.6) To eliminate the pressure, we project these equations onto space V m =fv2H m (R 3 ): div u = 0g; to get @ t u " +PK " [(K " u " )r(K " u " )] +PK " [jK " u " j 1 K " u " ] =K 2 " u " ; (1.2.7) whereP is the Leray projector. Now the system (1.2.6) can be viewed as a system of ordinary differential equations (ODEs) in V s : 8 > > > < > > > : du " dt =F " (u " ); u " j t=0 =u 0 ; (1.2.8) where the operator F " (u " ) =K 2 " u " PK " [jK " u " j 1 K " u " ]PK " [(K " u " )r(K " u " )]F 1 " (u " )F 2 " (u " )F 3 " (u " ): (1.2.9) 7 Our first goal is to show the existence and uniqueness of solutions to the regularized system (1.2.8). We will be using the following standard theorems: Theorem 1.2.9 (Picard Theorem on a Banach Space, [BM]). Let B be a Banach space, U B be an open set and let F : U!B be a mapping such that F is locally Lipschitz continuous, i.e., for any X2U there exists L> 0 and an open neighborhood U X U of X such that kF (y)F (z)kLkyzk 8y;z2U X : Then for any X 0 2U, there exists a time T > 0 such that the ODE dX dt =F (X); Xj t=0 =X 0 2U; has a unique (local) solution X2C 1 (T;T ;U). and Theorem 1.2.10 (Continuation of an Autonomous ODE on a Banach Space, [BM]). Let OB be an open subset of a Banach space B, and let F : O7!B be a locally Lipschitz continuous operator. Then the unique solution X2C 1 (0;T ;O) to the autonomous ODE, dX dt =F (X); X t=0 =X 0 2O; either exists globally in time, or T <1 and X(t) leaves the open set O as t!T. Theorem 1.2.11. Assume u 0 2H m (R 3 ), 2 and m = 2; 3;:::; [], div u 0 = 0. Then (1) for any "> 0 there exists the unique solution u " 2C 1 (0;T " ;V m ) to the system (1.2.8) for some T " =T (ku 0 k H m;"). (2) for any"> 0 there exists a global-in-time unique solutionu " 2C 1 (0;1;V m ) to the system (1.2.8) 8 Proof. First we prove the existence of regularized solutions u " locally in time using Picard’s Theorem. To apply Picard’s Theorem, we need to show that the function F " in (1.2.9) maps V m into H m and is locally Lipschitz continuous. First noteF " : V m 7!H m because of the Theorem 1.2.4 andK " commuting with derivatives. To estimate F 2 , we use the following standard properties of the mollifiers and Leray projector: (1)P commutes with the mollifierK " , K " (Pv) =P(K " v); "> 0; v2H m ; (1.2.10) (2) For all v2H m (R n );k = 0; 1; 2;:::, and "> 0 kK " vk H m+k c mk " k kvk H m; (1.2.11) kPvk 2 H mkvk 2 H m: (1.2.12) We have kF 2 " (u 1 )F 2 " (u 2 )k 2 =kPK " [jK " u 1 j 1 K " u 1 ]PK " [jK " u 2 j 1 K " u 2 ]k 2 kK " (jK " u 1 j 1 K " u 1 jK " u 2 j 1 K " u 2 )k 2 =" 3 Z R 3 xy " (jK " u 1 j 1 K " u 1 jK " u 2 j 1 K " u 2 )(y)dy 2 C(ku 1 k 2 ;ku 2 k 2 ;;")kK " u 1 K " u 1 k 2 C(ku 1 k 2 ;ku 2 k 2 ;;")ku 1 u 2 k 2 : (1.2.13) We have proved the statement of the Theorem for m = 0; the result for m> 0 follows automatically. For F 1 and F 3 , using standard results (see [BM]), we obtain kF 1 " (u 1 )F 1 " (u 2 )k H m c " 2 ku 1 u 2 k H m 9 and kF 3 " (u 1 )F 3 " (u 2 )k H m c " 5=2+m (ku 1 k 2 +ku 2 k 2 )ku 1 u 2 k H m: The final result is kF " (u 1 )F " (u 2 )k H mc(ku 1 k H m;ku 2 k H m;")ku 1 u 2 k H m; (1.2.14) so that F is locally Lipschitz continuous on any open set O M =fv2V m :kvk H m <Mg: Thus the Picard theorem implies that, given any initial conditionu 0 2H m , there exists a unique solution u " 2C 1 (0;T " ;H m \O M ) for some T " > 0. We now prove that if u " 2C 1 (0;T ;L 2 ), then sup 0tT ku " k 2 ku 0 k 2 : (1.2.15) Taking the L 2 inner product of (1.2.8) with u " we obtain 1 2 d dt Z R 3 ju " j 2 dx = Z R 3 u " K 2 " u " dx Z R 3 PK " [jK " u " j 1 K " u " )u " dx Z R 3 u " PK " [(K " u " )r(K " u " )] dx: We have, using (Pu;v) = (u;Pv),Pu " =u " , and the definition (1.2.5), Z R 3 PK " (jK " u " j 1 K " u " )u " dx = Z R 3 K " (jK " u " j 1 K " u " )u " dx =" 3 Z R 3 Z R 3 xy " jK " u " j 1 K " u " (y)dy u " (x)dx =" 3 Z R 3 jK " u " j +1 dx 0: 10 Since Z R 3 u " K 2 " u " dx =krK " u " k 2 2 and Z R 3 u " PK " [(K " u " )r(K " u " )] dx = 0; we come to d dt ku " k 2 2 + 2krK " u " k 2 2 0 implying (1.2.15). Now, the solution can be continued for all time, provided that we can show an a priori bound on ku " (;t)k H m. Note that relation (1.2.14) with u 2 (x;t) = 0 gives the bound d dt ku " (;t)k H mC(ku " k 2 ;")ku " k H m: Energy bound (1.2.15) gives d dt ku " (;t)k H mC(ku 0 k 2 ;")ku " k H m: which, by Grönwall’s lemma implies that the a priori boundku " (;t)k H me CT . The Theorem 1.2.11 is proven. Proof of the Theorem 1.2.1 In order to prove local-in-time existence of solutions to the Navier-Stokes equation, we will need several lemmas. Lemma 1.2.12. Assume u 0 2 V m (R 3 ), 2 and m = 2; 3;:::; []. Then the unique regularized solution u " 2C 1 (0;1;H m ) to (1.2.8) satisfies 1 2 d dt ku " k 2 H m +krK " u " k 2 H mku " k +1 H m +c m kK " ru " k 1 ku " k 2 H m: (1.2.16) 11 Proof. Let u " be a smooth solution to (1.2.8): u " t =K " u " PK " [jK " u " j 1 K " u " ]PK " [(K " u " )r(K " u " )]: We take the derivative @ ,jjm of this equation and then the L 2 inner product with @ u " to obtain: (@ u " t ;@ u " ) =(@ K 2 " u " ;@ u " )(@ PK " [jK " u " j 1 K " u " ];@ u " ) (@ PK " [(K " u " )r(K " u " )];@ u " ) =kK " @ ru " k 2 2 (@ PK " [jK " u " j 1 K " u " ];@ u " ) (PK " [(K " u " )r(@ K " u " )];@ u " ) (f@ PK " [(K " u " )r(K " u " )]PK " [(K " u " )r(@ K " u " )]g;@ u " ): Summing over jj m, we have using @ Pv = P@ v, (Pu;v) = (u;Pv), Pu " = u " , kK " vk H m C(m)kvk H m, and Theorem 1.2.4, X jjm (@ PK " [jK " u " j 1 K " u " ];@ u " ) = X jjm (@ K " [jK " u " j 1 K " u " ];@ u " ) X jjm k@ K " [jK " u " j 1 K " u " ]k 2 k@ u " k 2 ku " k H m X jjm k@ K " [jK " u " j 1 K " u " ]k 2 =ku " k H mkK " [ju " j 1 u " ]k H m.ku " k H mk[ju " j 1 u " ]k H mku " k +1 H m: By standard methods we obtain (PK " [(K " u " )r(@ K " u " )];@ u " ) = 0 and (f@ PK " [(K " u " )r(K " u " )]PK " [(K " u " )r(@ K " u " )]g;@ u " )c m kK " ru " k 1 kK " u " k 2 H m; 12 proving the lemma. Lemma 1.2.13. Assume 4 and m = 4; 5;:::; []. The family (u " ) of regularized solutions is uniformly bounded in H m on some (0;T ). Proof. Energy estimate (1.2.16) and Sobolev inequality for m> 5=2 kK " ru " k 1 .kK " ru " k H m1kru " k H m1ku " k H m imply that d dt ku " k H mku " k H m +c m jK " ru " j 1 ku " k H m.ku " k H m +ku " k 2 H mCku " k H m (1.2.17) for some C > 0 and > 1, hence, or all "> 0, 1 + 1 1 ku " k 1 H m 1 ku 0 k 1 H m ! CT; or, sup 0tT ku " k H m 1 1 ku 0 k 1 H m +C(1 )T : (1.2.18) Thus the family u " is uniformly bounded in C(0;T ;H m ), provided that T < 1 C( 1)ku 0 k 1 H m : Lemma 1.2.14. The family u " forms a Cauchy sequence in C(0;T ;L 2 (R 3 )). 13 Proof. Using (1.2.8), we have that 1 2 d dt ku " u " 0 k 2 2 =(K 2 " u " K 2 " 0u " 0 ;u " u " 0 )(PK " [jK " u " j 1 K " u " ]PK " 0[jK " 0u " 0 j 1 K " 0u " 0 ];u " u " 0 ) (PK " [(K 2 " u " )r(K " u " )PK " 0[(K 2 " 0u " 0 )r(K " 0u " 0 );u " u " 0 )T 1 T 2 T 3 : For T 2 we have, usingkK " vvk 2 C"kvk H 1 and (1.2.13), (PK " [jK " u " j 1 K " u " ]PK " 0[ju " 0 j 1 u " 0 ];u " u " 0 ) =(K " [jK " u " j 1 K " u " ]K " 0[jK " 0u " 0 j 1 K " 0u " 0 ];u " u " 0 ) =(K " [jK " u " j 1 K " u " ]jK " u " j 1 K " u " (K " 0[jK " 0u " 0 j 1 K " 0u " 0 ]jK " 0u " 0 j 1 K " 0u " 0 ) + (jK " u " j 1 K " u " jK " 0u " 0 j 1 K " 0u " 0 );u " u " 0 ) =(K " [jK " u " j 1 K " u " ]jK " u " j 1 K " u " ;u " u " 0 ) + (K " 0[jK " 0u " 0 j 1 K " 0u " 0 ]jK " 0u " 0 j 1 K " 0u " 0 );u " u " 0 ) + (jK " u " j 1 K " u " jK " 0u " 0 j 1 K " 0u " 0 ;u " u " 0 ) kK " [jK " u " j 1 K " u " ]jK " u " j 1 K " u " k 2 ku " u " 0 k 2 +kK " 0[ju " 0 j 1 K " 0u " 0 ]ju " 0 j 1 K " 0u " 0 k 2 ku " u " 0 k 2 +kjK " u " j 1 K " u " jK " 0u " 0 j 1 K " 0u " 0 k 2 ku " u " 0 k 2 (c 1 "kju " j 1 u " k H 1 +c 2 " 0 kju " 0 j 1 u " 0 k H 1 +c 3 ku " u " 0 k 2 )ku " u " 0 k 2 (C(ku " k H 1;ku " 0 k H 1) +ku " u " 0 k 2 )ku " u " 0 k 2 : For T 1 and T 3 we have T 1 C max(";" 0 )ku " k H 3ku " u " 0 k 2 and T 3 sup 0<t<T ku " (t)k H m max(";" 0 ): Putting this all together gives d dt ku " u " 0 k 2 C(M)[max(";" 0 ) +ku " u " 0 k 2 ]; 14 where M = sup 0<t<T ku " (t)k H m. Integrating this yields sup 0<t<T ku " u " 0 k 2 e C(M)T [max(";" 0 ) +ku " 0 u " 0 0 k 2 ] max(";" 0 ) C(M;T;";" 0 ) (1.2.19) since u " 0 =u " 0 0 . So, there exists a constant C that depends on onlyku 0 k H m and the time T so that, for all " and " 0 , sup 0<t<T ku " u " 0 k 2 C(";" 0 ): Now we are ready to apply the compactness theorem and finish the proof of the main theorem. Proof of Theorem 1.2.1. We showed thatu " is a Cauchy sequence inC(0;T ;L 2 (R 3 )) so that it converges strongly to a value u2C([0;T ];L 2 (R 3 )): We have just proved the existence of a u such that sup 0tT ku " uk 2 C": (1.2.20) We now use the fact that the u " are uniformly bounded in a high norm to show that we have strong convergence in all the intermediate norms. To do this, we use the following interpolation lemma for the Sobolev spaces: given s> 0, there exists a constant C s so that for all v2H s (R n ) and 0<s 0 <s, kvk s 0C s kvk 1s 0 =s 2 kvk s 0 =s s : (1.2.21) We now apply the interpolation lemma to the differenceu " u. Takings =m and using relations (1.2.18) and (1.2.20) gives sup 0tT ku " uk m 0C(kv 0 k;T )" 1m 0 =m : 15 Hence for all m 0 < m we have strong convergence in C(0;T ;H m 0 (R 3 )). With 0 < 7=2 < m 0 < m, this implies strong convergence in C(0;T ;C 2 (R 3 )). Also, from the equation u " t =K 2 " u " PK " [jK " u " j 1 K " u " ]PK " [(K " u " )r(K " u " )]; so thatu " t converges inC(0;T;C(R 3 )) toujuj 1 uP(uru). Becauseu " !u, the distribution limit of u " t must be u t so, in particular, u is a classical solution of the Navier-Stokes equations. 1.3 Local-in-time solutions to the Euler equations in a bounded domain Let be an open cube inR 3 , i.e. = (0;L) 3 for L> 0, =@ . Consider the problem u t + (ur)u +juj 1 u +r =f in (0;T ); div u = 0 in (0;T ); un = 0 on (0;T ); u(x; 0) =u 0 (x) in : (1.3.1) (1.3.2) (1.3.3) (1.3.4) Here the force f :R 3 R + 7!R 3 . In the case when = 0, the system reduces to the standard Euler system governing the motion of an incompressible perfect fluid. Note that since is a cube, un = 0 impliesjuj 1 un = 0 on (0;T ). The main goal of the current section is to prove the following theorem. Theorem 1.3.1. Assume that is an open cube inR 3 , i.e. = (0;L) 3 for L> 0, =@ , let 2N and m = 3; 4;:::, or > 3, 2RnN and m = 3; 4;:::; [] be fixed. Then for each u 0 and f such that u 0 2H m ( ); div u 0 = 0; u 0 n = 0 on @ ; (1.3.5) 16 f2L 1 (0;T ;H m ( )); (1.3.6) there exist unique functions u and , defined on some (0;T ), such that u2L 1 (0;T ;H m ( ))\L +1 (0;T ;L +1 ( )); (1.3.7) 2L 1 (0;T ;H m+1 ( )); (1.3.8) satisfying (1.3.1)-(1.3.4) on (0;T ). In order to prove the Theorem, we first obtain some apriori estimates. Then we apply Galerkin’s method to prove the existence of the unique solution. Lemma 1.3.2. If u and satisfy (1.3.1)-(1.3.4) then = div f X i;j @ j u i @ i u j div(juj 1 u) in ; (1.3.9) @ @n =fn on ; (1.3.10) where n =fn 1 ;n 2 ;n 3 g. Proof. We get (1.3.9) by applying the divergence operator to (1.3.1): div u t = 0; divr = ; div (ur)u = X i;j @ j u i @ i u j : 17 Now take the scalar product of each side of (1.3.1) with n, we get on : @ @n =fnjuj 1 un X i;j v i (D i v j )n j =fn: (1.3.11) Lemma 1.3.3. Let 2N and m = 3; 4;:::, or > 3, 2RnN and m = 3; 4;:::; []. If u and satisfy (1.3.1)-(1.3.3) then for each t> 0 kr(t)k H m.kf(t)k H m +ku(t)k H m +ku(t)k 2 H m: (1.3.12) Proof. From (1.3.9), (1.3.10) and [?], we obtain kr(t)k H m. div f X i;j @ j u i @ i u j div(juj 1 u) H m1 +kfnk H m1=2 () : Since m> 5=2, H m1 ( ) is an algebra and k@ j u i @ i u j k H m1.k@ j u i k H m1k@ i u j k H m1kuk H mkuk H m: Clearly kdiv fk H m1kfk H m and for 2N and m = 3; 4;:::, or > 3, 2RnN and m = 3; 4;:::; [] divjuj 1 u H m1 kjuj 1 uk H m.kuk H m: 18 Lemma 1.3.4. If u and satisfy (1.3.1)-(1.3.4), then ku(t)k H my(t); 0t<T 0 ; (1.3.13) where y is the solution of the differential equation 8 > > < > > : dy(t) dt =c 1 y(t) 2 +c 2 y(t) +c 3 kf(t)k H m; y(0) =ku 0 k H m; (1.3.14) and (0;T 0 ), 0 < T 0 1 is the interval of existence of y, T 0 depends only on c 1 , c 2 , c 3 , and the H m norms of f and u 0 . Proof. Let be a multi-index,j j m. We apply the operator @ to (1.3.1), then multiply by @ u, integrate over , and add these equalities forj jm: u t + (ur)u +juj 1 u +r =f; @ u t +@ (ur)u +@ juj 1 u +@ r =@ f; (@ u t ;@ u) + (@ (ur)u;@ u) +(@ juj 1 u;@ u) + (@ r;@ u) = (@ f;@ u); X j jm (@ u t ;@ u) + X j jm (@ (ur)u;@ u) + X j jm (@ juj 1 u;@ u) + X j jm (@ r;@ u) = X j jm (@ f;@ u); or 1 2 d dt kuk 2 H m = 3 X j=1 u j @u @x j ;u H m ((r;u)) H m((juj 1 u;u)) H m + ((f;u)) H m 19 The first term can be majorized using [K]: 3 X j=1 u j @u @x j ;u H m .kuk 3 H m; for the other terms ((f;u)) H mkfk H mkuk H m; ((juj 1 u;u)) H mkjuj 1 uk H mkuk H m.kuk +1 H m: ((r;u)) H mkrk H mkuk H m. n kfk H m +ku(t)k H m +kuk 2 H m o kuk H m by Lemma 1.3.3. Thus d dt kuk H m.kuk 2 H m +kuk H m +kfk H m; (1.3.15) implying the statement of the Lemma. Now we are ready to prove Theorem 1.3.1 Proof of Theorem 1.3.1. We recall Theorem 1.3.5 (Lax-Milgram Theorem, [E]). Assume B : HH7!R is a bilinear mapping for which there exist constants a, > 0 such that jB[u;v]jakukkvk (u;v2H) and kuk 2 B[u;u] (u2H): Finally, let f : H!R be a bounded linear functional on H. Then there exists a unique element u2H such that B[u;v] =hf;vi 20 for all v2H. Denote X m =fv2H m ( ); div v = 0; vn = 0 on g: For m = 0, X 0 is a closed subspace of L 2 ( ). Applying the Lax-Milgram theorem with H = X m , B[u;v] = ((u;v)) H m, we have that for each g2X 0 X H m, there exists a unique w2X m such that ((w(g);v)) H m = (g;v) 8v2X m : (1.3.16) The linear mapping g7!w(g) is a compact self-adjoint operator in X 0 and it possesses an orthonormal complete family of eigenvectors w k : 8 > > < > > : w k 2X H m; ((w k ;v)) H m = k (w k ;v) for any v2X m : (1.3.17) Let us use the Galerkin method with this basis. For N > 0 fixed we look for u N = N X j=1 g jN (t)w j (1.3.18) satisfying d dt (u N ;w k ) + ((u N r)u N ;w k ) +(ju N j 1 u N ;w k ) = (f;w k ); 1kN; (1.3.19) u N (0) =u 0N =P N u 0 ; (1.3.20) whereP N is the orthogonal projection ofX 0 (or as well inX m ) on the space spanned byw 1 ;:::;w k . The equations (1.3.19), (1.3.20) are equivalent to a system of ordinary differential equations for the g jN , and the existence of solution on some interval (0;T N ) follows from the Picard theorem (Theorem 1.2.9). The 21 following a priori estimates on u N show that T N =T is independent of N. The first a priori estimate is obtained by multiplying (1.3.19) by g kN (t) and adding in k: N X k=1 d dt (u N ;g kN (t)w k ) + N X k=1 ((u N r)u N ;g kN (t)w k ) + N X k=1 (ju N j 1 u N ;g kN (t)w k ) = N X k=1 (f;g kN (t)w k ); or d dt (u N ;u N ) + ((u N r)u N ;u N ) +(ju N j 1 u N ;u N ) = (f;u N ): Since ((u N r)u N ;u N ) = 0; we have 1 2 d dt ku N k 2 2 +ku N k +1 +1 = (f;u N )kfk 2 ku N k 2 : This shows that T N =T and that u N remains bounded in L 1 (0;T ;L 2 ( )) as N!1: (1.3.21) We can also write (1.3.19) as du N dt ;w k + (P[(u N r)u N ];w k ) +(Pju N j 1 u N ;w k ) = (Pf;w k ): (1.3.22) NowP[(u N (t)r)u N (t)]2X m ,Pf(t)2X m , and we can use (1.3.17). We multiply (1.3.22) by k g kN and add in k: N X k=1 k du N dt ;w k g kN + N X k=1 k (P[(u N r)u N ];w k )g kN + N X k=1 k (Pju N j 1 u N ;w k )g kN = N X k=1 k (Pf;w k )g kN ; 22 and since ((v;w k )) H m = k (v;w k ), N X k=1 du N dt ;w k H m g kN + N X k=1 ((P[(u N r)u N ];w k )) H mg kN + N X k=1 ((Pjuj 1 u;w k )) H mg kN = N X k=1 ((Pf;w k )) H mg kN ; or d dt ((u N ;u N )) H m + ((P[(u N r)u N ];u N )) H m +((Pju N j 1 u N ;u N )) H m = ((Pf;u N )) H m: We obtain 1 2 d dt jju N jj 2 H m = ((P(f (u N r)u N ju N j 1 u N );u N )) H m; (1.3.23) or, P[f (u N r)u N ju N j 1 u N ] =f (u N r)u N ju N j 1 u N r N : The relation similar to (1.3.12) is satisfied and we get exactly the same relation as (1.3.15), d dt ku N k 2 H m.ku N k 2 H m +ku N k H m +kfk H m: We recall that ku N (0)k H m =ku 0N k H mku 0 k H m; therefore, ku N (t)ky(t); 8t< inf(T;T 0 ); and as N!1; u N remains bounded in L 1 (0;T ;H m ( )); 8T < min(T;T 0 ): (1.3.24) 23 In order to pass to the limit in the nonlinear term using a compactness theorem, we need an estimate on du N =dt. Since the w k are orthogonal in X 0 , we deduce from (1.3.22) that du N dt =P n P(f (u N r)u N ju N j 1 u N ): Hence du N dt (t) 2 kf(t) (u N (t)r)u N (t)ju N j 1 u N k 2 and with (1.3.24) it is easily found that As N!1; du N =dt remains bounded in L 1 (0;T ;L 2 ( )): (1.3.25) We pass to the limit using (1.3.24), (1.3.25), and the compactness theorem. We obtain at the limit the existence of u2L 1 (0;T ;X m ) such that d dt (u(t);v) + ((u(t)r)u(t);v) +(ju N j 1 u N ;v) = (f(t);v); 8v2X 0 ; 0<t<T ; (1.3.26) u(0) =u 0 : (1.3.27) Now, u satisfies all the properties announced. 24 1.4 Global-in-time solutions in the whole space Linear damping Euler equations with the linear damping ( = 1, = 0) 8 > > > > > > > < > > > > > > > : @ t u + (ur)u +u +r = 0; (x;t)2R n (0;T ); div u = 0; (x;t)2R n [0;T ); u(x; 0) =u 0 (x); x2R n ; (1.4.1) have been studied by several authors. Theorem1.4.1 (Wu-Xu-Ye, 2015, [WXY]). Consider the problem (1.4.1) withn 2 and> 0. Suppose that initial vorticity ! 0 2B s p;r with 1<p<1 and s = n p if r = 1 and s> n p if r2 (1;1] satisfies the smallness condition k! 0 k B s p;r (R n ) < 2C for some constant C independent of . Then there exists a unique global solution of the problem (1.4.1) satisfying u2L 1 (0;1;L 2 (R n ));!2L 1 (0;1;B s p;r (R n )): Moreover, for any T > 0 k!(;T )k B s p;r (R n ) < 2C : Here B s p;r are usual Besov spaces. The proof is based on energy estimates and Littlewood-Paley decom- positions. Theorem 1.4.2 (Zhou-Zhu, 2018, [ZZ]). Consider the problem (1.4.1) with n = 3 and > 0. Suppose that div u 0 = 0, and the initial vorticity satisfies k! 0 k H 2 (R 3 ) <C 25 for some constant C. Then there exists a constant " ="(C;) such that the system (1.4.1) has a unique global solution provided that k! 0 u 0 k H 2 (R 3 ) ": To prove this theorem, the authors introduce a new quantity = !u, and rewrite the problem (1.4.1) as follows: 8 > > > > > > > < > > > > > > > : ! t +! r! +!r = 0; t + ru +ur ur( + 1 2 juj 2 ) = 0; r! = 0; r = 0: The existence of global solutions follows from the following a priori estimates d dt k!k 2 H 2 (R 3 ) + 2k!k 2 H 2 (R 3 ) .k k H 2 (R 3 ) k!k 2 H 2 (R 3 ) ; d dt k k 2 H 2 (R 3 ) + 2k k 2 H 2 (R 3 ) .k!k H 2 (R 3 ) k k 2 H 2 (R 3 ) : Weak solutions for nonlinear damping In this section, C 1 0; ( ) denotes the space of all C 1 ( ) vector-valued functions u = (u 1 ;u 2 ;u 3 ) with compact support inR 3 such that div u = 0, L p ( ), 1 < p <1 is closure of C 1 0; ( ) in L p ( ) with respect tokk p , W k;p 0; ( ), 1<p<1 is closure of C 1 0; ( ) in W k;p ( ) with respect tokk k;p . Navier-Stokes equations (> 0) with nonlinear damping have been first studied in [CJ]. In order to prove the existence of weak solutions, we need to add the following condition to the system (1.1.1)-(1.1.3): lim jxj!1 ju(x;t)j = 0: (1.4.2) 26 Definition 1.4.3. The pair (u;) is called a weak solution of the problem (1.1.1)-(1.1.3), (1.4.2) if for any T > 0: (1) u2L 1 (0;T ;L 2 (R 3 ))\L 2 (0;T ;W 1;2 0; (R 3 ))\L +1 (0;T ;L +1 (R 3 )); (2) for any '2C 1 0; ([0;T ]R 3 ) with '(;T ) = 0; Z T 0 (u;' t )dt Z T 0 Z R 3 (ur)u'dxdt + Z T 0 Z R 3 juj 1 u'dxdt = (u 0 ;' 0 ): (1.4.3) Due to its importance, we present the proof of existence of weak solutions for completeness. Theorem 1.4.4 (Cai-Jiu, 2008, [CJ]). Suppose that 1 and u 0 2L 2 (R 3 ). Then for any T > 0 there exists a weak solution (u;) to the problem (1.1.1)-(1.1.3), (1.4.2). Proof. We employ the Galerkin approximations and follow the proof for the classical Navier-Stokes equa- tions ([T]). Since W 1;2 0; is separable and C 1 0; is dense in W 1;2 0; , there exists a sequence ! 1 ;:::;! m of elements of C 1 0; which is free and total in W 1;2 0; . For each m we define an approximate solution u m as follows: u m = m X i=1 g im (t)! i (x); and (u 0 m (t);! j ) +(ru m (t);r! j ) + (u m (t)ru m (t);! j ) +(ju m (t)j 1 u m (t);! j ) = 0; (1.4.4) for t2 [0;T ], j = 1;:::;m, and u m (0) =u 0m . Multiplying (1.4.4) by g jm (t) and summing over j = 1;:::;m, we obtain 1 2 d dt ku m k 2 +kru m k 2 +ku m k +1 +1 0: 27 Therefore, approximate solutions (u m ), given u 0 2L 2 (R 3 ), for any T > 0 satisfy sup 0tT ku m (t)k 2 2 + 2ku m k L 2 (0;T;H 1 ) + 2ku m k L +1 (0;T;L +1 ) dtku 0 k 2 2 ; implying the global existence of the approximate solutions u2L 1 (0;T ;L 2 (R 3 ))\L 2 (0;T ;H 1 0; (R 3 ))\L +1 (0;T ;L +1 (R 3 )): Next, we use the following theorem to prove the strong convergence of u m (or its subsequence) in L 2 \L ([0;T ] ) for any R 3 : Lemma 1.4.5 (Compactness theorem). Let X 0 and X be Hilbert spaces satisfying X 0 ,! X. Let 0 < 1 andfv j g 1 j=1 be a sequence in L 2 (R;X 0 ) satisfying sup j Z 1 1 kv j k 2 X0 dt <1; sup j Z 1 1 jj 2 kb v j k 2 X d <1; where b v() = Z 1 1 v(t)e 2it dt is the Fourier transformation of v(t). Then there exists a subsequence offv j g 1 j=1 with converges strongly to some v2L 2 (R;X) in L 2 (R;X). Denotebye u m thefunctionfromRintoH 1 0; , whichisequaltou m on [0;T ]andto0onthecomplement of this interval. Similarly, we prolong g im (t) toR by defining e g i;m = 0 for t2Rn [0;T ]. The Fourier transforms on time variable ofe u m ande g i;m are denoted by b e u m and b e g i;m respectively. 28 Note that the approximate solutionse u m satisfy d dt (e u m (t);! j ) =(re u m (t);r! j ) + (e u m (t)re u m (t);! j ) +(je u m (t)j 1 e u m (t);! j ) + (u 0m ;! j ) 0 + (u m (T );! j ) T ( e f;! j ) +(je u m (t)j 1 e u m (t);! j ) + (u 0m ;! j ) 0 + (u m (T );! j ) T (1.4.5) where 0 ; T are Dirac distributions at 0 and T. Taking the Fourier transform about the time variable, (1.4.5) gives 2i( b e u m ;! j ) =( ^ e f m ;! j ) +( \ je u m (t)j 1 e u m ;! j ) + (u 0m ;! j ) (u m (T );! j )e 2iT : (1.4.6) Multiplying (1.4.6) by b e g jm () and summing up the resulting equations for j = 1;:::;m, we get 2ik b e u m k 2 2 = ( ^ e f m (); b e u m j) +( \ je u m (t)j 1 e u m ; b e u m ) + (u 0m ; b e u m ) (u m (T ); b e u m )e 2iT : For any v2L 2 (0;T ;H 1 0 )\L +1 (0;T ;L +1 ), we have (ru m ;rv) + (u m ru m ;v)C(ku m k 2 2 +kru m k 2 2 +kru m k 2 )kvk H 1 : It follows that for any given T > 0 Z T 0 ku m ru m (t)k H 1dt. Z T 0 (kru m k 2 2 +kru m k 2 )dtC; and thus sup 2R k ^ e f m ()k H 1 Z T 0 kf m (t)k H 1dtC: 29 Also sup 2R k \ je u m j 1 e u m ()k+1 C: Sinceku m (0)k 2 C andku m (T )k 2 C, if follows that jjk b e u m ()k 2 2 C(k b e u m ()k H 1 +k b e u m ()k +1 ): For fixed 0< < 1=4 jj 2 C 1 +jj 1 +jj 12 ; 2R; thus Z +1 1 jj 2 k b e u m ()k 2 2 dC Z +1 1 1 +jj 1 +jj 12 k b e u m ()k 2 2 d C Z +1 1 k b e u m ()k 2 2 d +C Z +1 1 k b e u m ()k H 1 1 +jj 12 d +C Z +1 1 k b e u m ()k +1 1 +jj 12 d: The first to integrals are bounded for 0< < 1=4; and for 0< < 1 2(+1) Z +1 1 k b e u m ()k +1 1 +jj 12 d Z +1 1 d (1 +jj 12 ) +1 ! +1 Z +1 1 k b e u m ()k +1 +1 d 1 +1 C Z +1 1 ke u m ()k +1 +1 d +1 CT 1 +1 Z T 0 ku m ()k +1 +1 d ! 1 : Therefore, Z +1 1 jj 2 k b e u m ()k 2 2 dC: Thus there exists a subsequence of (u m ) such that u m !u weakly-* in L 1 (0;T ;L 2 (R 3 )) and weakly in L 2 (0;T ;W 1;2 0; (R 3 ))\L +1 (0;T ;L +1 (R 3 )). Then we choose 1 2 with smooth boundary such that[ 1 i=1 i =R 3 . For any fixed i = 1; 2;::: we take X 0 = W 1;2 0 ( i ), X = L 2 ( i ) in the compactness 30 theorem. Thenthereexistsasubsequencesuchthatu m !ustronglyinL 2 (0;T ;L 2 ( i )). Bythediagonal principle, there exists a subsequence (u mj ) of (u m ), such that u mj ! u strongly in L 2 (0;T ;L 2 ( i )) for any i = 1; 2;:::, and hence in L 2 (0;T ;L 2 loc (R 3 )). The existence of weak solution is proved. Remarkably, the weak solutions to the problem (1.1.1)-(1.1.3), (1.4.2) are unique for 4. Theorem 1.4.6. Suppose that 4 and u 0 2 L 2 (R 3 ). Then the weak solution (u;) to the problem (1.1.1)-(1.1.3), (1.4.2) is unique. Proof. Let u 1 and u 2 be two weak solutions to the problem (1.1.1)-(1.1.3), (1.4.2) with the same initial u 0 . Then Z R 3 ((u 1 ) t u 1 + (u 1 r)u 1 +ju 1 j 1 u 1 )dx = 0; (1.4.7) as well as Z R 3 ((u 2 ) t u 2 + (u 2 r)u 2 +ju 2 j 1 u 2 )dx = 0; (1.4.8) 31 for any 2 C 1 c ([0;T )R 3 ). Subtracting (1.4.8) from (1.4.7), taking = u 1 u 2 in the resulting equation, using Hölder, Young, and Gagliardo-Nirenberg inequalities we obtain for any > 2 1 2 d dt ku 1 u 2 k 2 2 +kr(u 1 u 2 )k 2 2 + Z R 3 (ju 1 j 1 uju 2 j 1 u 2 ) (u 1 u 2 )dx = Z R 3 (u 1 u 2 )ru 1 (u 1 u 2 )dx Z R 3 u 2 r(u 1 u 2 ) (u 1 u 2 )dx = Z R 3 (u 1 u 2 )ru 1 (u 1 u 2 )dx = 2 Z R 3 u 1 r(u 1 u 2 ) (u 1 u 2 )dx 2kr(u 1 u 2 )k 2 ku 1 k +1 ku 1 u 2 k2(+1) 1 4 kr(u 1 u 2 )k 2 2 +Cku 1 k 2 +1 kuu 2 k 2 2(+1) 1 4 kr(u 1 u 2 )k 2 2 +Cku 1 k 2 +1 kr(u 1 u 2 )k 3 +1 2 ku 1 u 2 k 2 +1 2 2 = 4 kr(u 1 u 2 )k 2 2 +Ckr(u 1 u 2 )k 6 +1 2 ku 1 k 2 +1 ku 1 u 2 k 2(2) +1 2 4 kr(u 1 u 2 )k 2 2 + 4 kr(u 1 u 2 )k 6 +1 2 +1 3 +C ku 1 k 2 +1 ku 1 u 2 k 2(2) +1 2 +1 2 = 2 kr(u 1 u 2 )k 2 2 +Cku 1 k 2(+1) 2 +1 ku 1 u 2 k 2 2 : Clearly Z R 3 (ju 1 j 1 u 1 ju 2 j 1 u 2 ) (u 1 u 2 )dx = Z R 3 ju 1 j 1 ju 1 j 2 dx + Z R 3 ju 2 j 1 ju 2 j 2 dx Z R 3 ju 1 j 1 u 1 u 2 dx Z R 3 ju 2 j 1 u 2 u 1 dx ku 1 k +1 +1 +ku 2 k +1 +1 ku 2 k +1 kju 1 j 1 u 1 k+1 ku 1 k +1 kju 2 j 1 u 2 k+1 =ku 1 k +1 +1 +ku 2 k +1 +1 ku 2 k +1 ku 1 k +1 ku 1 k +1 ku 2 k +1 = (ku 1 k +1 ku 2 k +1 )(ku 1 k +1 ku 2 k +1 ) 0: Therefore, 1 2 d dt ku 1 u 2 k 2 2 Cku 1 k 2(+1) 2 +1 ku 1 u 2 k 2 2 : (1.4.9) 32 Now, taking =u 1 in (1.4.7) we obtain 1 2 d dt ku 1 k 2 2 +kru 1 k 2 2 +ku 1 k +1 +1 = 0: (1.4.10) Integrating (1.4.10) for any T > 0 on [0;T ] implies Z T 0 ku 1 k +1 +1 dt ku 0 k 2 2 2 : Note that 2( + 1) 2 + 1 whenever 4, therefore, for such ku 1 k 2(+1) 2 +1 ku 1 k +1 +1 + 1; so for any T > 0 Z T 0 ku 1 k 2(+1) 2 +1 dt Z T 0 (ku 1 k +1 +1 + 1)dtT + ku 0 k 2 2 2 <1: Applying Grönwall inequality in (1.4.9), we obtain for any T > 0 k(u 1 u 2 )(T )k 2 2 k(u 1 u 2 )(0)k 0 2 exp Z T 0 ku 1 k 2(+1) 2 +1 dt ! = 0: Therefore u 1 u 2 for a.e. (x;t)2R 3 [0;T ], implying the uniqueness of weak solutions to the problem (1.1.1)-(1.4.2). Theorem 1.4.6 is proven. 33 Strong solutions for nonlinear damping Definition 1.4.7. The pair (u;) is called a strong solution of the problem (1.1.1)-(1.1.3), (1.4.2) if it is a weak solution of (1.1.1)-(1.1.3), (1.4.2) satisfying u2L 1 (0;T ;H 1 0; (R 3 ))\L 1 (0;T ;L +1 (R 3 ))\L 2 (0;T ;H 2 (R 3 )): Multiplying (1.1.1) by u t ,u and adding, we get + 1 2 d dt Z R 3 jruj 2 dx + + 1 d dt Z R 3 juj +1 dx + 3 4 Z R 3 juj 2 dx + 1 2 Z R 3 ju t j 2 dx + Z R 3 juj 1 jruj 2 dx + ( 1) 4 Z R 3 jjuj 3 rjuj 2 j 2 dxC Z R 3 juruj 2 dxJ: (1.4.11) Cai and Jiu in [CJ] have shown that the strong solutions exist for any 7=2. Theorem 1.4.8 (Cai-Jiu, 2008). Suppose that 7=2 and u 0 2H 1 0; \L +1 (R 3 ). Then for any T > 0 there exists a strong solution (u;) to the problem (1.1.1)-(1.1.3), (1.4.2). They applied Gagliardo-Nirenberg inequality to estimate kruk2(+1) 1 Ckuk 11 +7 2 kuk 24 +7 +1 ; 2 5; and then Hölder and Young inequalities to get JCkuk 2 +1 kruk 2 2(+1) 1 Ckuk 2 +1 kuk 2(11) +7 2 kuk 24 +7 +1 Ckuk 2(11) +7 2 kuk 6(+1) +7 +1 4 kuk 2 2 +Ckuk 3(+1) 2 +1 ; 7 2 5: 34 Using similar estimates, they showed that for all 7=2 sup 0tT kru(t)k 2 2 +ku(t)k +1 +1 +ku t k 2 2;2;T +kuk 2 2;2;T +kjruju 1 2 k 2 2;2;T + ( 1) 2 Z T 0 Z R 3 jjuj 3 rjuj 2 j 2 dxdtC(T ): The result of Cai and Jiu was improved in 2011. Theorem 1.4.9 (Zhang-Wu-Lu, 2011, [ZWL]). Suppose that > 3,u 0 2H 1 \L +1 (R 3 ) with div u 0 = 0. Then there exists a strong solution (u;) to the problem (1.1.1)-(1.1.3), (1.4.2) satisfying juj 1 2 ru; rjuj +1 2 ; u t 2L 2 (0;T ;L 2 (R 3 )): The proof is based on a more careful application of the Hölder, Young and Gagliardo-Nirenberg inequal- ities: for 3< 7=2, Zhang, Wu, and Lu estimated in (1.4.11) J = Z R 3 juj 2 jruj 2 dx = Z R 3 (juj 1 2 jruj) 2(5) 1 (jruj 4(3) 1 juj 3 )dx 2 Z R 3 juj 1 jruj 2 dx + 4 Z R 3 juj 2 dx +Cjuj +1 dx: Therefore, (1.4.11) turns into + 1 2 d dt Z R 3 jruj 2 dx + + 1 d dt Z R 3 juj +1 dxC Z R 3 juj +1 dx; and the result follows from the Grönwall lemma. In [CJ] and [ZWL], the existence of the strong solutions of the problem (1.1.1)-(1.1.3), (1.4.2) follows from the a priori estimate d dt kruk 2 2 +C 1 d dt kuk +1 +1 C 2 kruk 2 L 2 +C 3 kuk +1 +1 ; 35 where C 1 ;C 2 ;C 3 > 0, whereas we can prove the theorem with C 1 =C 3 = 0. Theorem 1.4.10. Suppose that u 0 2 H 1 (R 3 ) with div u 0 = 0. Then for > 3 there exists a strong solution (u;) to the problem (1.1.1)-(1.4.2) such thatjuj 1 2 ru;rjuj +1 2 2L 2 (0;T ;L 2 (R 3 )). Proof. Multiplying (1.1.1) byu, integrating overR 3 and applying Young’s inequality implies 1 2 d dt kruk 2 2 + 2 kuk 2 2 + Z R 3 juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z R 3 rjuj +1 2 2 dx 1 2 kuruk 2 2 : (1.4.12) To boundkuruk 2 2 , we write for M > 1 kuruk 2 2 = Z R 3 juj 2 jruj 2 dx = Z fjuj p Mg juj 2 jruj 2 dx + Z fjuj> p Mg juj 2 jruj 2 dx =:I 1 +I 2 : We estimate I 1 = Z fjuj p Mg juj 2 jruj 2 dxM Z R 3 jruj 2 dx =Mkruk 2 2 ; and, using that > 3, I 2 = Z fjuj> p Mg juj 2 jruj 2 dx = Z fjuj> p Mg juj 1 juj 3 jruj 2 dx< 1 M 3 2 Z R 3 juj 1 jruj 2 dx: Now, we set M := () 2 3 + 1; then > 1 M 3 2 . We get from (1.4.12) that d dt kruk 2 2 M kruk 2 2 : (1.4.13) 36 Grönwall lemma gives for any t 0 kru(t)k 2 kru 0 k 2 e M t ; implying the existence of strong solutions. Theorem 1.4.10 is proven. There have also been several works analyzing the behavior of solutions of the problem (1.1.1)-(1.1.3), (1.4.2). In the latest article [WC], Wen and Chai prove the decay of weak solutions,kuk L 2 (R 3 ) , for any > 1 and > 0. The decay of higher order derivatives,kr m uk 2 , for 7=2 < 5 when u 0 2H m (R 3 ), m 0, is shown in [JZD]. Schonbek [S] and Schonbek and Wiegner [SW] investigated the optimal decay of the higher-order derivatives of the solutions of the Navier-Stokes equations by developing the Fourier splitting method. Using Gevrey estimates, Oliver and Titi [OT] proved the optimal upper and lower bounds of the higher-order derivatives of the solutions. However, these methods are not applicable to the damped Navier-Stokes equations due the lack of uniform estimates of the damping term, especially for large. Up to now there have not been any bounds onku(t)k 2(R 3 ) whenu 0 2H 1 (R 3 ). We have shown that for 3< < 5, the normku(t)k 2 decays on t 1. Theorem 1.4.11. For 3< < 5 the strong solution (u;p) to the problem (1.1.1)-(1.1.3), (1.4.2) satisfies for any t 1 ku(t)k 2 C p t ; where the constant C depends only onku 0 k H 1;;, and . Proof. Now, multiplying (1.1.1) by u and integrating overR 3 we get 1 2 d dt kuk 2 2 +kruk 2 2 +kuk +1 +1 = 0; from where for any t 0 ku(t)k 2 2 ku 0 k 2 2 ; 37 and Z t 0 kru(s)k 2 2 ds 1 2 ku 0 k 2 2 : (1.4.14) Applying standard methods, it is easy to see from (1.4.13) and (1.4.14) that for t 1 kru(t)k 2 2 C 0 t ; (1.4.15) for some constant C 0 =C 0 (ku 0 k 2 ;;;). Applying the Leray projectorP to the equation (1.1.1), we can rewrite it as u t +Au +B(u;u) +P(juj 1 u) = 0; (1.4.16) where B(u;v) =P(urv). We have kAuk 2 ku t k 2 +kB(u;u)k 2 +kPjuj 1 uk 2 : (1.4.17) To prove the required bound ofkAuk 2 , we need to show that each term in (1.4.17) is bounded by C= p t. Multiplying (1.1.1) by u t , integrating overR 3 , and using Hölder and Young inequalities we obtain 2 d dt kruk 2 2 + + 1 d dt kuk +1 +1 + 1 2 ku t k 2 2 1 2 kuruk 2 2 : (1.4.18) Adding up (1.4.18) and (1.4.12) gives + 1 2 d dt kruk 2 2 + + 1 d dt kuk +1 +1 + 2 kAuk 2 2 + 1 2 ku t k 2 2 + Z R 3 juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z R 3 rjuj +1 2 2 dxC 1 kuruk 2 2 ; (1.4.19) 38 where C 1 = 1=(2) + 1=2. Using estimate onkuruk 2 2 as in (1.4.13), we obtain + 1 2 d dt kruk 2 2 + + 1 d dt kuk +1 +1 + 1 2 ku t k 2 2 C 1 Mkruk 2 2 : (1.4.20) The Gagliardo-Nirenberg inequality says that for 3< < 5, kuk +1 Ckruk 3(1) 2(+1) 2 kuk 5 2(+1) 2 (1.4.21) for some absolute constant C. Integrating (1.4.20) on [0;t] and using (1.4.14) and (1.4.21) give Z t 0 ku t (s)k 2 2 ds ( + 1)kru 0 k 2 2 + 2 + 1 ku 0 k +1 +1 + 2C 1 M Z t 0 kru(s)k 2 2 ds ( + 1)kru 0 k 2 2 + 2C + 1 kru 0 k 3(1) 2 2 ku 0 k 5 2 2 + C 1 M ku 0 k 2 2 =C 2 ; (1.4.22) where the constant C 2 =C 2 (ku 0 k H 1;;;). Now, applying @ t to (1.1.1), multiplying the resulting equation by u t , integrating overR 3 and using Hölder and Young inequalities we obtain 1 2 d dt ku t k 2 2 +kru t k 2 2 + Z R 3 juj 1 ju t j 2 dx + ( 1) 4 Z R 3 juj 3 @ @t juj 2 2 dx = Z R 3 (u t r)uu t dx Z R 3 (ur)u t u t dx Z R 3 rp t u t dx = Z R 3 (u t r)uu t dx = Z R 3 (u t r)u t udx 2 kru t k 2 2 + 1 2 kuu t k 2 2 : Using the same idea as above to absorb 1 2 kuu t k 2 2 in kjuj 1 2 ju t jk 2 , we get d dt ku t k 2 2 M ku t k 2 2 : (1.4.23) 39 Therefore, from (1.4.22) and (1.4.23) we obtain for t 1 ku t k 2 C 3 p t (1.4.24) for some constant C 3 =C 3 (ku 0 k H 1;;;). Now, Hölder, Gagliardo-Nirenberg, Young inequalities and estimate (1.4.15) imply for t 1 kB(u;u)k 2 kuruk 2 kruk 3 kuk 6 CkAuk 1=2 2 kruk 3=2 2 4 kAuk 2 +Ckruk 3 2 4 kAuk 2 + C 4 t 3=2 4 kAuk 2 + C 4 p t (1.4.25) for some C 4 = C 4 (ku 0 k H 1;;;). Finally, from Gagliardo-Nirenberg, Young inequalities and (1.4.15) we have for t 1 kPjuj 1 uk 2 kuk 2 CkAuk 3(1) +7 2 kuk (+1)(+3) +7 +1 4 kAuk 2 +Ckuk (+1)(+3) 2(5) +1 4 kAuk 2 +Ckruk 3(1)(+3) 4(5) 2 kuk +3 4 2 4 kAuk 2 + C t 3(1)(+3) 8(5) 4 kAuk 2 + C 5 p t (1.4.26) for some constant C 5 =C 5 (ku 0 k H 1;;;). Plugging in estimates (1.4.24), (1.4.25), and (1.4.26) into (1.4.17) gives 2 kAuk 2 C 3 +C 4 +C 5 p t : Theorem 1.4.11 is proven. 40 Chapter 2 Boussinesq equations with damping 2.1 Introduction In this chapter, we consider the asymptotic behavior of solutions to the damped incompressible 3D Boussinesq equations without diffusivity u t u + (ur)u +juj 1 u +r =e 3 ; t + (ur) = 0; div u = 0: u(x; 0) =u 0 (x); (x; 0) = 0 (x) (2.1.1) (2.1.2) (2.1.3) (2.1.4) in a bounded domain R 3 and inR 3 . Here > 0, > 0, and 1 are real parameters. The unknown functions are velocity u = u(x;t) = (u 1 (x;t);u 2 (x;t);u 3 (x;t)), x = (x 1 ;x 2 ;x 3 ), temperature = (x;t), and pressure = (x;t). Also e 3 = (0; 0; 1) is the unit vector along the vertical coordinate x 3 . When = 0, the system (2.1.1)-(2.1.4) turns into the usual 3D Boussinesq equations. Recently, there has been a lot of progress made on the existence, uniqueness, and persistence of regularity for 2D Boussinesq equations, mostly in the case of positive viscosity and vanishing diffusivity and in the case of 41 vanishing viscosity and positive diffusivity. The first results on the global existence have been obtained in [HL], where the global existence and persistence were proven in the Sobolev spaces H s H s1 for integers 3. In [HKZ1] and [HKZ2], Hu, Kukavica, and Ziane obtained results for s = 2 and 1<s< 3, respectively. In [Ju], Ju addressed the question of long time behavior of solutions; some results in [Ju] have been recently improved in [KW]. Compared to the methods used to study the 2D Boussinesq equations, the three-dimensional case is complicated by difficulties related to the application of the Brezis-Gallouët-Wainger type inequalities. We avoid them by using the ideas of optimal regularity. InR 3 , we need the following condition to ensure the existence of weak solutions: lim jxj!1 ju(x;t)j = 0: (2.1.5) For a smooth, bounded, connected open set R 3 with smooth boundary@ , we study the problem with the standard no-penetration boundary condition un = 0; x2@ (2.1.6) and the Neumann type boundary condition curl un = 0; x2@ ; (2.1.7) where curl u =ru and n denotes the outward unit normal vector with respect to the domain . We recall the classical spaces H =fv2L 2 ( ) 3 j div v = 0 in ; vn = 0 on @ g; V =fv2H 1 0 ( ) 3 j div v = 0 in g: 42 The Stokes operator A: D(A)7!H with the domain D(A) =H 2 \V is defined by A =P whereP is the Leray projector inL 2 ( ) on the spaceH. Defining the standard bilinear operatorB(u;v) =P(urv), u;w2V, B : VV 7!V 0 , where V 0 is dual space of V, we rewrite (2.1.1) as u t +Au +B(u;u) +P(juj 1 u) =P(e 3 ): (2.1.8) Definition 2.1.1. Let R 3 be open bounded set with smooth boundary. The pair (u;) is called a weak solution of the problem (2.1.1)-(2.1.7) if for any T > 0: (1) (u;)2 (L 2 (0;T ;V ( ) 3 )\L 1 (0;T ;H( ) 3 )\L +1 (0;T ;L +1 ( ) 3 ))L 1 (0;T ;L 2 ( ) 3 ); (2) for all 2 (C 1 ([0;T ] )) 3 withr = 0 Z T 0 (u;@ t )(ru;r) (juj 1 uurue 3 ;) dt = (u(T );(T )) (u 0 ;(0)); (3) for all 2C 1 ([0;T ] ) Z T 0 [(;@ t ) (u )] dt = ((T ); (T )) ( 0 ; (0)): The existence of weak solutions to the problems (2.1.1)- (2.1.5) and (2.1.1)-(2.1.4), (2.1.6)-(2.1.7) for any 1 and (u 0 ; 0 )2 L 2 L 2 with div u 0 = 0, readily follows from Theorem 1.4.4. However, the question of uniqueness of weak solutions is still an open problem. This chapter is organized as follows: in Sections from 2.2 to 2.6, we study long-time behavior of weak solutions in L 2 L 2 , H 1 L 2 , H 2 H 1 , H 1 H 1 , H 2 H 2 , respectively. 43 2.2 Global estimates in L 2 L 2 Theorem 2.2.1 (Whole space). Suppose 1, (u 0 ; 0 )2L 2 L 2 (R 3 ) with div u 0 = 0. Then all weak solutions (u;) of the problem (2.1.1)-(2.1.5) satisfy for any t> 0 and r> 0 k(t)k 2 =k 0 k 2 ; (2.2.1) ku(t)k 2 C(t + 1); Z t+r t kru()k 2 2 d + Z t+r t ku()k +1 +1 dC(rt +t +r 2 +r + 1): Proof. Taking inner product of (2.1.1) with , we obtain 1 2 d dt kk 2 2 + Z R 3 (ur)dx = 1 2 d dt kk 2 2 = 0: Integrating from 0 to t, we get (2.2.1). Taking inner product of (2.1.1) with u gives 1 2 d dt kuk 2 2 +kruk 2 2 +kuk +1 +1 = Z R 3 e 3 udxkk 2 kuk 2 =k 0 k 2 kuk 2 ; (2.2.2) or d dt kuk 2 k 0 k 2 ; implying after integration ku(t)k 2 ku 0 k 2 +k 0 k 2 t: (2.2.3) 44 Now, integrating (2.2.2) from t to t +r and using (2.2.3), we obtain Z t+r t kru()k 2 2 d + Z t+r t ku()k +1 +1 d 1 2 ku(t)k 2 +k 0 k 2 Z t+r t ku()k 2 d ku 0 k 2 2 + k 0 k 2 2 t +k 0 k 2 ku 0 k 2 r +k 0 k 2 2 rt + k 0 k 2 2 2 r 2 (2.2.4) completing the proof. Remark 2.2.1. It is easy to see that in generalk(t)k p =k 0 k p for any t> 0 and p> 1 in both open bounded and in the whole spaceR 3 . Theorem 2.2.2 (Bounded domain). Suppose 1, (u 0 ; 0 )2HL 2 ( ). Then all weak solutions of the problem (2.1.1)-(2.1.4), (2.1.6)-(2.1.7) satisfy for any t> 0 and r> 0 k(t)k 2 =k 0 k 2 ; (2.2.5) ku(t)k 2 2 C(e 1t + 1); and for any r> 0 Z t+r t kA 1=2 u()k 2 2 d + 2 Z t+r t kuk +1 +1 dC(e 1t +r + 1); where 1 is the lowest eigenvalue of A. Proof. Taking inner product of (2.1.2) with , we obtain 1 2 d dt kk 2 2 = Z t dx = Z (ur)dx = 1 2 Z ur( 2 )dx = 1 2 Z (ru) 2 dx 1 2 Z @ (un) 2 dS = 0 45 implying (2.2.5). Note that we can apply Poincaré inequality (see [E]) in a bounded domain : kA p uk 2 pq 1 kA q uk 2 for 0q<p 1; where 1 is the lowest eigenvalue of A; in particular,kuk 2 1=2 1 kA 1=2 uk 2 . Taking inner product of (2.1.1) with u, we obtain 1 2 d dt kuk 2 2 +kA 1=2 uk 2 2 +kuk +1 +1 kk 2 kuk 2 1 2 kuk 2 2 + 1 2 1 kk 2 2 2 kA 1=2 uk 2 2 + 1 2 1 kk 2 2 : Thus, d dt kuk 2 2 +kA 1=2 uk 2 2 + 2kuk +1 +1 1 1 k 0 k 2 2 ; (2.2.6) implying d dt kuk 2 2 + 1 kuk 2 2 1 1 k 0 k 2 2 : Multiplying this equation by e 1t , we have d dt [e 1t kuk 2 2 ] k 0 k 2 2 1 e 1t ; i.e. e 1t ku(t)k 2 2 ku 0 k 2 2 k 0 k 2 2 1 Z t 0 e 1 d = k 0 k 2 1 2 (e 1t 1); implying ku(t)k 2 2 ku 0 k 2 2 e 1t + k 0 k 2 1 2 1e 1t : (2.2.7) 46 Integrating (2.2.6) from t to t +r, and using (2.2.7) we obtain Z t+r t kA 1=2 u()k 2 2 d + 2 Z t+r t kuk +1 +1 dku(t)k 2 2 + rk 0 k 2 2 1 ku 0 k 2 2 e 1t + k 0 k 2 1 2 1e 1t + rk 0 k 2 2 1 : (2.2.8) completing the proof. 2.3 Global estimates in H 1 L 2 Theorem 2.3.1 (Whole space). Suppose that > 3, (u 0 ; 0 )2H 1 L 2 (R 3 ) and div u 0 = 0. Then all weak solutions (u;) to the problem (2.1.1)-(2.1.5) satisfy for all t> 0 kru(t)k 2 C(t + 1): (2.3.1) Moreover, for any t;r> 0 Z t+r t ku()k 2 2 dC(rt +t +r 2 +r + 1): (2.3.2) Proof. Taking inner product of (2.1.1) withu, we estimate that 1 2 d dt kruk 2 2 +kuk 2 2 +kjuj 1 2 jrujk 2 2 + 4( 1) ( + 1) 2 Z R 3 rjuj +1 2 2 dx = Z R 3 (uru) udx + Z R 3 e 3 udxkuruk 2 kuk 2 +kk 2 kuk 2 2 kuk 2 2 + 1 kuruk 2 2 + k 0 k 2 2 : (2.3.3) To boundkuruk 2 2 , we write for M > 1 kuruk 2 2 = Z R 3 juj 2 jruj 2 dx = Z fjuj p Mg juj 2 jruj 2 dx + Z fjuj> p Mg juj 2 jruj 2 dx =:I 1 +I 2 : 47 We estimate I 1 = Z fjuj p Mg juj 2 jruj 2 dxM Z R 3 jruj 2 dx =Mkruk 2 2 ; and, using that > 3, I 2 = Z fjuj> p Mg juj 2 jruj 2 dx = Z fjuj> p Mg juj 1 juj 3 jruj 2 dx< 1 M 3 2 Z R 3 juj 1 jruj 2 dx: Now, we set M := () 2 3 + 1; then > 1 M 3 2 . Therefore, from (2.3.3) we get 1 2 d dt kruk 2 2 + 2 kuk 2 2 M kruk 2 2 + k 0 k 2 2 : (2.3.4) Denotex(t) =kru(t)k 2 2 + 1; thenx 0 Kx forK := max 2M ; 2k0k 2 2 , therefore,x(t)x(0)e Kt for any t 0. Therefore, for 0t 1, kru(t)k 2 2 e K (kru 0 k 2 2 + 1) 1: (2.3.5) Using (2.2.4), we have Z t 1 kru()k 2 2 dC(t 2 + 1): Integrating (2.3.4) from 0 to t, we get for any t> 1 kru(t)k 2 2 kru 0 k 2 2 + 2k 0 k 2 2 t + 2M Z t 0 kru()k 2 2 d kru 0 k 2 2 + 2k 0 k 2 2 t + 2M e K (kru 0 k 2 2 + 1) 1 +C(t 2 + 1) C(t + 1) 2 ; (2.3.6) 48 proving (2.3.6). Moreover, for any t;r> 0 from (2.3.4) we obtain using (2.2.4) Z t+r t ku()k 2 2 d kru 0 k 2 2 + 2k 0 k 2 2 2 r + 2M 2 Z t+r t kru()k 2 2 d kru 0 k 2 2 + 2k 0 k 2 2 2 r + 2M 3 ku 0 k 2 2 + k 0 k 2 2 t +k 0 k 2 ku 0 k 2 r +k 0 k 2 2 rt + k 0 k 2 2 2 r 2 which concludes the proof. Unlike in the proof of Theorem 2.3.1, u cannot be used as a test function for the Dirichlet boundary value problem. This was the reason for us to introduce Neumann boundary condition (2.1.7). For the convenience of reading, we recall the Lemma 2.3.2 (Uniform Grönwall Lemma). Let g and y be two non-negative locally integrable functions on (t 0 ;1) such that for all tt 0 dy(t) dt g(t)y(t) +h(t) for Z t+r t g(s)dsa 1 ; Z t+r t h(s)dsa 2 ; Z t+r t y(s)dsa 3 ; where r, a 1 , a 2 , and a 3 are positive constants. Then for all tt 0 y(t +r) a 3 r +a 2 e a1 : The proof of the lemma can be found in [T]. Theorem 2.3.3 (Bounded domain). Suppose > 3, (u 0 ; 0 )2VL 2 ( ). Then all weak solutions (u;) to the problem (2.1.1)-(2.1.4), (2.1.6)-(2.1.7) satisfy for any t> 0 kA 1=2 (t)k 2 C: (2.3.7) 49 Moreover, for any t;r> 0 Z t+r t kAu()k 2 2 dC(e 1t +r + 1): Proof. We will be using the following calculus identity: Z vcurl wdx = Z wcurl vdx Z @ (vw)ndS: (2.3.8) Denote ! = curl u. Note that for divergence-free u u = curl ! = curl curl u; (2.3.9) and that (2.3.8) and the boundary condition (2.1.6) imply for any v Z @ (!v)ndS = Z @ (!n)vdS = 0: (2.3.10) Applying curl operator to (2.1.1) gives ! t ! +curl (ur)u +curljuj 1 u = curl e 3 : (2.3.11) Taking inner product of (2.3.11) with ! implies 1 2 d dt k!k 2 2 (!;!) + (curl (ur)u;!) +(curljuj 1 u;!) = (curl e 3 ;!): (2.3.12) 50 Using (2.3.10) and (2.3.9) gives (!;!) = Z u udx =kuk 2 2 ; (curl (ur)u;!) = Z (ur)u udx; (curl e 3 ;!) = Z e 3 udx: (2.3.13) Finally, (curljuj 1 u;!) = Z curljuj 1 u!dx = Z juj 1 ucurl !dx Z @ (!juj 1 u)ndS = Z juj 1 u udx = Z juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z rjuj +1 2 2 dx Z @ juj 1 (ur)undS Z @ juj 1 (curl un)udS = Z juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z rjuj +1 2 2 dx: (2.3.14) Plugging in (2.3.13) and (2.3.14) into (2.3.12) gives 1 2 d dt k!k 2 2 +kuk 2 2 + Z juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z rjuj +1 2 2 dx = Z (ur)u udx Z e 3 udx 4 kuk 2 2 + 1 kuruk 2 2 + 4 kuk 2 2 + 1 kk 2 2 : (2.3.15) Using damping in the same way as in the proof of Theorem 2.3.1, we get 1 2 d dt k!k 2 2 + 2 kuk 2 2 M kruk 2 2 + k 0 k 2 2 : (2.3.16) 51 Denotex(t) =kru(t)k 2 2 +1; sincek!k 2 =kruk 2 , wehavex 0 Kx forK := max 2M ; 2k0k 2 2 , therefore, x(t)x(0)e Kt , i.e. for any 0t 1 we have x(t)x(0)e K ; so for any 0t 1 kA 1=2 u(t)k 2 2 e K (kA 1=2 u 0 k 2 2 + 1) 1; (2.3.17) Using (2.2.8), we have Z t+1 t kru()k 2 2 d ku 0 k 2 2 e 1t + k 0 k 2 2 2 1 3 1e 1t + k 0 k 2 2 1 2 ku 0 k 2 2 + k 0 k 2 2 2 1 3 + k 0 k 2 2 1 2 ; therefore, the uniform Grönwall’s lemma applied to (2.3.16) implies for any t> 1 kA 1=2 u(t)k 2 2 e 2M " 2k 0 k 2 +ku 0 k 2 2 + k 0 k 2 1 2 + k 0 k 2 2 1 # : (2.3.18) Now, integrating (2.3.16) from t to t +r for any r> 0 and using (2.2.8) we obtain Z t+r t kAu()k 2 2 d kA 1=2 u 0 k 2 2 + 2k 0 k 2 2 2 r + 2M 2 Z t+r t kA 1=2 u()k 2 2 d kA 1=2 u 0 k 2 2 + 2k 0 k 2 2 2 r + 2M 3 " ku 0 k 2 2 e 1t + k 0 k 2 1 2 1e 1t + rk 0 k 2 2 1 # (2.3.19) which completes the proof. Remark 2.3.1. Note that (2.3.15) also implies 1 2 d dt kruk 2 2 + 1 M 3 2 ! Z juj 1 jruj 2 dx + 4( 1) ( + 1) 2 Z rjuj +1 2 2 dx M kruk 2 2 + 1 kk 2 2 ; 52 therefore, both Z t 0 kjuj 1 2 jrujk 2 2 d and Z t 0 rjuj +1 2 2 2 d are bounded from above by C(1 +t) where the exact value of the constant C = C(ku 0 k H 1;k 0 k 2 ; 1 ; ;;; ) can be computed from (2.2.4), (2.3.17), and (2.3.18). 2.4 Global regularity in H 2 H 1 Since the Leray projectorP does not commute with the Stokes operator A, the following lemma is needed to deal with the nonlinear term. Lemma 2.4.1. For u;v2D(A), we have krB(u;v)k 2 Ckuk 1=8 2 kAuk 7=8 2 kvk 1=8 2 kAvk 7=8 2 +Ckuk 1=4 2 kAuk 3=4 2 kAvk 2 : Proof. From Leray decomposition, Pv =vr(p +q); v2H 1 ( ) with p2H 2 ( ) satisfying p =rv in ; p = 0 on @ ; and q2H 2 ( ) solving q = 0 in ; @q @n = (vrp)n on @ : 53 Thus, krPvk 2 krvk 2 +Ckpk H 2 +Ckqk H 2krvk 2 +Ckrvk 2 +kvrpk H 1=2 (@ ) C(krvk 2 +kvk H 1 +krpk H 1)Ckvk H 1: Therefore, for any u;v2D(A), by Poincaré inequality krB(u;v)k 2 =krP(uru)k 2 Ckurvk H 1Ckr(urv)k 2 C 0 @ X i;k k@ i u j @ j v k k 2 + X i;k ku j @ ij v k k 2 1 A : (2.4.1) Now, using Hölder and Gagliardo-Nirenberg inequalities, we estimate X i;k k@ i u j @ j v k k 2 kruk 4 krvk 4 Ckuk 1=8 2 kAuk 7=8 2 kvk 1=8 2 kAvk 7=8 2 and X i;k ku j @ ij v k k 2 kuk 1 kAvk 2 kuk 1=4 2 kAuk 3=4 2 kAvk 2 : The statement of the Lemma follows from (2.4.1). We start by proving local boundedness ofkAu(t)k 2 andkr(t)k 2 . Proposition 2.4.1 (Local regularity in D(A)H 1 ). Suppose > 3, (u 0 ; 0 )2 D(A)H 1 , and let (u;) be any weak solution of the problem (2.1.1)-(2.1.4), (2.1.6)-(2.1.7). Then there exists t 0 > 0 such that for all t2 [0;t 0 ] kAu(t)k 2 2 +kr(t)k 2 2 C(kAu 0 k 2 2 +kr 0 k 2 2 ); (2.4.2) where C =C(ku 0 k 2 ;k 0 k 2 ;;;; ). 54 Proof. Taking inner product of (2.1.8) with A 2 u gives d dt kAuk 2 2 +kA 3=2 uk 2 2 = (P(e 3 );A 2 u) (B(u;u);A 2 u)(P(juj 1 u);A 2 u): (2.4.3) We have (P(e 3 );A 2 u)Ckrk 2 kA 3=2 uk 2 8 kA 3=2 uk 2 2 +Ckrk 2 2 : (2.4.4) To estimatekrk 2 , we applyr to (2.1.2) and take inner product of the resulting equation withr to get 1 2 d dt krk 2 2 kruk 1 krk 2 2 CkA 3=2 uk 3=4 2 kA 1=2 uk 1=4 2 krk 2 2 16 kA 3=2 uk 2 2 +CkA 1=2 uk 2=5 2 krk 16=5 2 ; (2.4.5) where we used Agmon’s inequalitykuk 1 CkAuk 3=4 2 kuk 1=4 2 . After applying Lemma 2.4.1, we see that (B(u;u);A 2 u)krB(u;u)k 2 kuk H 3Ckuk 1=4 2 kAuk 7=4 2 kA 3=2 uk 2 8 kA 3=2 uk 2 2 +Ckuk 1=2 2 kAuk 7=2 2 : (2.4.6) Now, (P[juj 1 u];A 2 u) =(P[juj 1 u];PPu) =(P[juj 1 u]; Pu) = (@ j P[juj 1 u];@ j Pu) kr[juj 1 u]k 2 kA 3=2 uk 2 8 kA 3=2 uk 2 2 +Ckr[juj 1 u]k 2 2 : (2.4.7) Using kuk 1 CkAuk 1=2 2 kuk 1=2 6 CkAuk 1=2 2 kruk 1=2 2 ; 55 we see that kr[juj 1 u]k 2 2 = Z jruj 2 juj 22 dxkruk 2 2 kuk 22 1 Ckruk 2 2 kAuk 1 2 kruk 1 2 =CkAuk 1 2 kruk +1 2 ; therefore, (2.4.7) can be written as (P[juj 1 u];A 2 u) 8 kA 3=2 uk 2 2 +CkAuk 1 2 kruk +1 2 : (2.4.8) Since we proved in Theorems 2.2.2 and 2.3.3 thatku(t)k 2 andkru(t)k 2 are uniformly bounded for t> 0 by the constants depending on the initial data and parameters only, adding up (2.4.3) and (2.4.5), and combining (2.4.4), (2.4.6), and (2.4.8) we get d dt (kAuk 2 2 +krk 2 2 ) + 2 kA 3=2 uk 2 2 Ckrk 2 2 +Ckuk 1=2 2 kAuk 7=2 2 +CkAuk 1 2 kruk +1 2 +CkA 1=2 uk 2=5 2 krk 16=5 2 C +Ckrk 16=5 2 +CkAuk max(1; 7 2 ) 2 C +C(kAuk 2 2 +krk 2 2 ) max( 1 2 ; 7 4 ) : (2.4.9) This finishes the proof of the proposition. Theorem 2.4.2. Suppose (u 0 ; 0 ) 2 D(A)H 1 ( ). Then all weak solutions (u;) to the problem (2.1.1)-(2.1.4), (2.1.6)-(2.1.7) satisfy kAuk 2 C for 3< 5: Proof. Local regularity of u t 2L 1 (0;t 1 ;H). From (2.1.8), ku t k 2 kAuk 2 +kB(u;u)k 2 +kuk 2 +kPe 3 k 2 : (2.4.10) 56 Since by Hölder and Gagliardo-Nirenberg inequalities kB(u;u)k 2 kA 1=2 uk 2 kuk 1 kA 1=2 uk 2 kAuk 3=4 2 kuk 1=4 2 and kuk 2 kAuk 3(1) 4 2 kuk +3 4 2 ; we have ku t k 2 kAuk 2 +kA 1=2 uk 2 kAuk 3=4 2 kuk 1=4 2 +kAuk 3(1) 4 2 kuk +3 4 2 +k 0 k 2 : (2.4.11) The local regularity follows from (2.2.7), (2.3.7), and (2.4.2). Global regularity of u t 2L 1 (0;1;H)\L 2 (0;1;V ). Applying@ t to (2.1.1) and using (2.1.2), we get @ t u t u t + (ur)u t + (u t r)u +(juj 1 u) t +r t =(ur)e 3 : (2.4.12) Taking inner product of (2.4.12) with u t and using Hölder and Young inequalities we obtain 1 2 d dt ku t k 2 2 +kru t k 2 2 + Z juj 1 ju t j 2 dx + ( 1) 4 Z juj 3 @ @t juj 2 2 dx = ((u t r)u;u t ) ((ur)u t ;u t ) (r t ;u t ) ((ur)e 3 ;u t ) = ((u t r)u;u t ) ((ur)e 3 ;u t ) = ((u t r)u t ;u) + (u;ru 3;t ) 4 kru t k 2 2 + 1 kuu t k 2 2 + 4 kru t k 2 2 +kuk 2 2 2 kru t k 2 2 + 1 kuu t k 2 2 +k 0 k 2 1 kuk 2 2 : (2.4.13) or 1 2 d dt ku t k 2 2 + Z juj 1 ju t j 2 dx 1 kuu t k 2 2 +k 0 k 2 1 kuk 2 2 : 57 Using the same idea as in the proof of the Theorem 2.3.1 to absorb 1 kuu t k 2 2 inkjuj 1 2 ju t jk 2 , we obtain 1 2 d dt ku t k 2 2 M ku t k 2 2 +k 0 k 2 1 kuk 2 2 ; implying the global regularity withku t k 2 C. Global regularity of u2L 1 (0;t;D(A)). Clearly, from Gagliardo-Nirenberg and Young inequalities, kB(u;u)k 2 kuk 1 kA 1=2 uk 2 kAuk 3=4 2 kuk 1=4 2 kA 1=2 uk 2 4 kAuk 2 +Ckuk 2 kA 1=2 uk 4 2 ; (2.4.14) and for 3< 5 kuk 2 CkAuk 3(1) +7 2 kuk (+1)(+3) +7 +1 CkAuk 3(1) +7 2 kruk 3(1) 2(+1) 2 kuk 5 2(+1) 2 (+1)(+3) +7 4 kAuk 2 +Ckuk (+1)(+3) 2(5) +1 4 kAuk 2 +Ckruk 3(1)(+3) 4(5) 2 kuk +3 4 2 : (2.4.15) Therefore, for 3< 5, kAuk 2 ku t k 2 +kB(u;u)k 2 +kuk 2 +kk 2 ku t k 2 + 4 kAuk 2 +Ckuk 2 kA 1=2 uk 4 2 + 4 kAuk 2 +Ckruk 3(1)(+3) 4(5) 2 kuk +3 4 2 +k 0 k 2 ; (2.4.16) implying kAuk 2 Cku t k 2 +Ckuk 2 kA 1=2 uk 4 2 +Ckruk 3(1)(+3) 4(5) 2 kuk +3 4 2 +Ck 0 k 2 ; or that kAuk 2 C: 58 We now turn to estimates onkrk 2 . Applyingr to (2.1.2) gives (r) t + (ur)r +rur = 0: Taking inner product of the resulting equation withr we get 1 2 d dt krk 2 2 kruk 1 krk 2 2 ; or kr(t)k 2 kr 0 k 2 e R t 0 kru()k1d : The well-posedness of the 2D Boussinesq equations has been proven by application of the Brezis-Gallouët- Wainger inequality: Theorem 2.4.3 (Brezis-Gallouët-Wainger inequality, [BW]). Letf2W l;q (R n ) withlq>n, 1q1, and let kp =n, 1<p<1. Ifkfk W k;p 1, then kfk 1 C 1 + log 1=p (1 +kfk W l;q)) : In three-dimensional case, to apply the inequality to boundkruk 1 , we need estimates on either kuk W 3;1 or onkuk W 1;3 which are difficult to obtain. Instead, we will be using Theorem 2.4.4 (Optimal parabolic regularity of the Stokes problem, [GS]). Let beR 3 or an open bounded set with smooth boundary, 1<q;s<1, 0<T1. Let u be a weak solution of the u t u +rp =f; div u = 0; u(0) =u 0 ; 59 where f2L s (0;T ;L q ( )). Denote A q =u +rp and for 0<< 1 define D ;s q = ( v2L q ( ):kvk D ;s q =kvk q + Z 1 0 kt 1 A q e tAq vk s q dt t 1=s <1 ) : Let u 0 (x)2D 1 1 s ;s q . Then there exists a constant C(s;q; ) such that ku t k q;s +kuk q;s +krpk q;s C(s;q; ) kfk q;s +ku 0 k D 1 1 s ;s q : Theorem 2.4.5. Suppose (u 0 ; 0 )2 D(A)H 1 ( ) and > 3. Then all weak solutions (u;) to the problem (2.1.1)-(2.1.4), (2.1.6)-(2.1.7) satisfy for t>t 0 > 0 kr(t)k 2 Ce Ct : Proof. Let f =(ur)ujuj 1 u +e 3 and take q = 3( + 1) ; s = + 1 : Then from (2.3.18) and (2.2.8) we obtain k(ur)uk q;s = Z t 0 k(ur)uk +1 3(+1) d +1 Z t 0 kuk +1 6(+1) 1 kruk +1 6 d +1 C Z t 0 kuk 1 2 kuk 6 kuk +1 2 d +1 C Z t 0 kruk 2 kuk +2 2 d +1 C Z t 0 kuk 2 2 d +2 2(+1) Z t 0 kruk 2 2 2 d 2 2(+1) C(t + 1) +1 (2.4.17) 60 for some constant C = C(;;; 1 ;ku 0 k H 1;k 0 k 2 ) which can be computed exactly from (2.2.8) and (2.3.19). For the damping term, kjuj 1 uk s q =kuk +1 3(+1) = Z juj +1 2 6 dx 2(+1) 6(+1) juj +1 2 2 6 rjuj +1 2 2 2 ; so, from Remark 2.3.1 we obtain kjuj 1 uk q;s = Z t 0 kuk +1 3(+1) d +1 Z t 0 rjuj +1 2 2 2 d +1 C(t + 1) +1 : (2.4.18) Clearly, ke 3 k q;s t 1=s k 0 k3(+1) =Ct +1 (2.4.19) for constant C =k 0 k3(+1) . Summing up (2.4.17), (2.4.18), (2.4.19), we have that for any > 3 kfk q;s =k(ur)u +juj 1 u +e 3 k q;s C(t + 1) +1 (2.4.20) for some constantC whose exact value can be computed from (2.4.17), (2.4.18), (2.4.19). It follows from Theorem 2.4.4 that kuk q;s C(t + 1) +1 ; (2.4.21) so u2L 1 0;t;L 3(+1) with Z t 0 ku()k3(+1) dC(t + 1); therefore,ru2L 1 0;t;W 1; 3(+1) withkruk L 1 0;t;W 1; 3(+1) C(t + 1). Now, to boundkruk 1 , we recall: 61 Theorem 2.4.6 (Morrey’s inequality, [E]). Assume 3 < p 1. Then there exists a constant C, depending only on p, such that kvk 1 kvk C 0;1 3 p Ckvk W 1;p (2.4.22) for all v2C 1 \L p (R 3 ). Applying Morrey’s inequality (2.4.22) toru (note that 3( + 1)= > 3), we obtain kruk 1 Ckuk W 2; 3(+1) : Therefore, Z t 0 kru()k 1 dC(t + 1); and thus kr(t)k 2 kr 0 k 2 e R t 0 kru()k1d Ce Ct : Remark 2.4.1. In the recent paper [KW] it is shown that the unique weak solution of the 2D Boussinesq equation with zero diffusivity in H 2 H 1 is bounded bykk H 1 Ce Ct . As Theorem 2.4.5 shows, the same estimate holds for the 3D equation with damping. 2.5 Global regularity in H 1 H 1 The most complicated thing in obtaining the global regularity in D(A)H 1 was the bound on Z t 0 kru()k 1 d: 62 However, the estimates in the previous section containedkA 3=2 uk 2 , and those estimates cannot be used now in H 1 . To study the global regularity in H 1 H 1 , we use the ideas of spectral decomposition. Theorem 2.5.1. Suppose 3 < 5, (u 0 ; 0 )2 VH 1 ( ). Then all weak solutions of the problem (2.1.1)-(2.1.7) satisfy for any t> 0 krk 2 Ce Ct 2 : Proof. Since kr(t)k 2 kr 0 k 2 exp Z t 0 kru()k 1 d ; (2.5.1) we need to bound R t 0 kru()k 1 d. Define the family of spectral projection operatorsfP m g 1 m=1 as P 1 v = X j<2 (u;w j )w j ; P m u = X 2 m1 j<2 m (u;w j )w j ; for m 2; where the familyfw j g 1 j=1 D(A) is orthonormal system in H and Aw j = j w j ; j = 1; 2;:::: Applying P m to (2.1.8), we obtain 1 2 kP m u(t)k 2 2 +(P m Au;P m u) =(P m P(juj 1 u);P m u) + (P m [P(e 3 )B(u;u)];P m u): Note that (P m Au;P m u) = X 2 m1 j<2 m j(u;w j )j 2 2 m1 kP m uk 2 2 ; m 1: 63 We have 1 2 kP m u(t)k 2 +2 m1 kP m uk 2 kP m P(juj 1 u)k 2 +kP m P(e 3 )k 2 +kP m B(u;u)k 2 : Now, 2kP m u(t)k 2 + Z t 0 2 m kP m u()k 2 d 2kP m u 0 k 2 + +2 Z t 0 kP m P(ju()j 1 u())k 2 d + 2 Z t 0 (kP m P(()e 3 )k 2 +kP m B(u();u())k 2 )d; (2.5.2) Multiplying (2.5.2) by 2 m=2 and using kP m A s uk 2 =kA s P m uk 2 . 2 ms kP m uk; m 1; s 0; (2.5.3) we get Z t 0 2 m+ m 2 kP m u()k 2 d .kA 1=2 P m u 0 k 2 + Z t 0 kA 1=2 P m P(ju()j 1 u())k 2 d + Z t 0 (kA 1=2 P m P(()e 3 )k 2 +kA 1=2 P m B(u();u())k 2 )d: (2.5.4) 64 Now, squaring both sides of (2.5.4), we obtain 1 X m=1 Z t 0 2 m+ m 2 kP m u()k 2 d 2 . 1 X m=1 kA 1=2 P m u 0 k 2 2 + 1 X m=1 Z t 0 kA 1=2 P m P(ju()j 1 u())k 2 d 2 + 1 X m=1 Z t 0 kA 1=2 P m P(()e 3 )k 2 d 2 + 1 X m=1 Z t 0 kA 1=2 P m B(u();u())k 2 )d 2 =kA 1=2 u 0 k 2 2 + 1 X m=1 Z t 0 kA 1=2 P m P(ju()j 1 u())k 2 d 2 + 1 X m=1 Z t 0 kA 1=2 P m P(()e 3 )k 2 d 2 + 1 X m=1 Z t 0 kA 1=2 P m B(u();u())k 2 )d 2 : (2.5.5) Using Minkowski inequality, Z t 0 2 m+ m 2 kP m u()k 2 d 1=2 .kA 1=2 u 0 k 2 + 1 X m=1 Z t 0 kA 1=2 P m P(ju()j 1 u())k 2 d 2 ! 1=2 + 1 X m=1 Z t 0 k(A 1=2 P m P(()e 3 )k 2 d 2 ! 1=2 + 1 X m=1 Z t 0 kA 1=2 P m B(u();u())k 2 )d 2 ! 1=2 kA 1=2 u 0 k 2 + Z t 0 1 X m=1 kA 1=2 P m P(ju()j 1 u())k 2 2 ! 1=2 d + Z t 0 1 X m=1 kA 1=2 P m P(()e 3 )k 2 2 ! 1=2 d + Z t 0 1 X m=1 kA 1=2 P m B(u();u())k 2 2 ! 1=2 d =kA 1=2 u 0 k 2 + Z t 0 kA 1=2 P(ju()j 1 u())k 2 d + Z t 0 kA 1=2 P(()e 3 )k 2 d + Z t 0 kA 1=2 B(u();u())k 2 2 d: (2.5.6) 65 The Agmon’s inequality in 3D states that kuk 1 Ckuk 1=4 2 kuk 3=4 H 2 ; therefore, we have, using (2.5.3), krP m uk 1 krP m uk 1=4 2 kArP m uk 3=4 2 .krP m uk 1=4 2 (2 m krP m uk 2 ) 3=4 2 3m 4 krP m uk 2 = 2 3m 4 kA 1=2 P m uk 2 . 2 3m 4 2 m 2 kP m uk 2 = 2 5m 4 kP m uk 2 : (2.5.7) Therefore, from (2.5.6) and (2.5.7), Z t 0 kru()k 1 d Z t 0 1 X m=1 krP m u()k 1 d Z t 0 1 X m=1 2 5m 4 kP m uk 2 d . 1 X m=1 2 m 4 Z t 0 2 m+ m 2 kP m ()k 2 d 1 X m=1 Z t 0 2 m+ m 4 krP m u()k 1 d 2 ! 1=2 .kru 0 k 2 + Z t 0 krP(()e 3 )k 2 d + Z t 0 krB(u();u())k 2 d + Z t 0 krP(u 1 u)k 2 d: (2.5.8) Now, we have Z t 0 krP(()e 3 )k 2 d Z t 0 kr()k 2 d; (2.5.9) 66 and, using Lemma 2.4.1, (2.2.7) and (2.3.19), Z t 0 krB(u();u())k 2 d Z t 0 ku()k 1=4 2 kAu()k 7=4 2 d Z t 0 kAu()k 2 2 d 7=8 Z t 0 ku()k 2 2 d 1=8 C t 7=8 + 1 : (2.5.10) for some constant C. Using Gagliardo-Nirenberg inequality kuk 1 CkAuk 1=2 2 kuk 1=2 6 CkAuk 1=2 2 kruk 1=2 2 ; we see that krP[juj 1 u]k 2 kr[juj 1 u]k 2 = Z jruj 2 juj 1 juj 1 dx 1=2 juj 1 2 jruj 2 kuk 1 2 1 C juj 1 2 jruj 2 kAuk 1 4 2 kruk 1 4 2 ; so, using Remark 2.3.1, we estimate Z t 0 rP[juj 1 u] 2 d Z t 0 kjuj 1 2 jrujk 2 kAuk 1 4 2 kruk 1 4 2 d sup 0t kru()k 1 4 2 Z t 0 kjuj 1 2 jrujk 2 2 d 1=2 Z t 0 kAuk 1 2 2 d 1=2 C(t + 1) (2.5.11) whenever 1 2 2; i.e. when 3< 5: Combining (2.5.9), (2.5.10), and (2.5.11) and plugging in (2.5.8), we get Z t 0 kru()k 1 dC(t + 1) + Z t 0 kr()k 2 d =C + Z t 0 (C +kr()k 2 )d: (2.5.12) 67 , and so k(t)k 2 exp C Z t 0 (1 +krk 2 )d : (2.5.13) Denote x(t) =C + Z t 0 (C +kr()k 2 )d; then x 0 (t)C 1 e x(t) ; therefore, for some t 0 > 0, for t2 [0;t 0 ] C Z t 0 (1 +krk 2 )d =x(t) ln 1 1C 1 t : From (2.5.13) it follows that for t2 [0;t 0 ] k(t)k 2 ln 1 1C 1 t : Since (u 0 ; 0 )2 VH 1 and (u;)2 L 1 (0;t 0 ;VH 1 ), it follows that we can assume without loss of generality that u(t 0 )2D(A): Thus from the theorems from previous section, u2L 1 (t 0 ;1;D(A)) and 2L 1 (t 0 ;1;H 1 ) proving the global regularity. Remark 2.5.1. Theorem 2.5.1 is also valid inR 3 for it’s obviously valid onT 3 , and by Littlewood-Paley frequency decomposition, the proof automatically works onR 3 . 2.6 Global regularity in H 2 H 2 Global regularity in H 2 D(A) can be obtained from the estimate (2.5.12) on R t 0 kru()k 1 d. 68 Theorem 2.6.1. Suppose that > 3, (u 0 ; 0 )2D(A)H 2 ( ) and div u 0 = 0. Then all weak solutions (u;) to the problem (2.1.1)-(2.1.7) satisfy for all tt 1 > 0 kr 2 k 2 Ce Ct : Proof. Applying @ 2 x1 to (2.1.2) and taking inner product of the resulting equation with @ 2 x1 , we obtain d dt k@ 2 x1 k 2 2 =2((@ 2 x1 u)r + 2(@ x1 u)r(@ x1 );@ 2 x1 ) 2krk 6 kAuk 3 kr 2 k 2 + 4kruk 1 kr 2 k 2 2 .kr 2 k 2 2 kAuk 3 +kuk 1 kr 2 k 2 2 (kAuk 3 +kruk 1 )kr 2 k 2 2 : Clearly, similar estimates hold for all other second derivatives, therefore, d dt kr 2 k 2 2 C(kAuk 3 +kruk 1 )kr 2 k 2 2 : (2.6.1) We have shown in (2.5.12) that R t 0 kru()k 1 d C(t + 1) 1=2 is bounded for any > 3. Also, using (2.4.21) we get Z t 0 kAu()k 3 d Z t 0 kAu()k +1 3(+1) + 1 d =kAuk s q;s +t<C(1 +t): Therefore, Grönwall inequality implies kr 2 (t)k 2 2 kr 2 0 k 2 2 e R t 0 (kAu()k3+kru()k1)d Ce Ct : Clearly, this estimate holds for any t>t 1 . 69 Reference List [B] G.Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. math. 104 (1991), 435–446 [BM] A.Bertozzi, A.Majda, Vorticity and Incompressible Flows, Cambridge U. Press, Cambridge (2002) [BW] H.Brezis, S.Wainger, A note on limiting cases of Sobolev embeddings and convolution inequal- ities, Comm. Partial Differential Equations 5 (1980), no. 7, 773–789 [CJ] X.Cai, Q.Jiu, Weak and strong solutions for the incompressible Navier-Stokes equations with damping, J. Math. Anal. Appl., 343 (2008), 799–809 [E] L.Evans, Partial differential equations, Providence, R.I., American Mathematical Society (2010) [F] G.B.Folland, Real Analysis, 2nd Edition, Wiley (1999) [G] E.Gagliardo, Ulteriori proprietà di alcune classi di funzioni in pù variabili, Ricerche Mat. 7 (1959), 24–51 [GS] Y.Giga, H.Sohr, AbstractL p estimates for the Cauchy problem with applications to the Navier- Stokes equations in exterior domains, Journal of Functional Analysis, Volume 102, Issue 1 (1991), 72–94 [H] E.Hopf, "Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nach., 4 (1950/51), 213–231 [HL] T.Hou, C.Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. 12 (2005), 1–12 [HKZ1] W.Hu, I.Kukavica, M.Ziane, On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys. 54 (2013), no. 8, 081507 [HKZ2] W.Hu, I.Kukavica, M.Ziane, Persistence of regularity for the viscous Boussinesq equations with zero diffusivity, Asymptot. Anal. 91 (2015), 111–124 [JZD] Y.Jia, X.Zhang, B.Dong, The asymptotic behavior of solutions to three-dimensional Navier- Stokes equations with nonlinear damping, Nonlinear Anal. RWA, 12 (2011), 1736–1747 [Ju] N.Ju, Global regularity and long-time behavior of the solutions to the 2D Boussinesq equations without diffusivity in a bounded domain, J. Math. Fluid Mech. 19 (2017), 105–121 [K] T.Kato, On classical solutions of two dimensional nonstationary Euler equations, Arch. Ra- tional Mech. Anal. 25 (1967), 188–200 70 [KW] I.Kukavica, W.Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity, J Dyn Diff Equat (2019), 1–17 [L] J.Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), 193–248 [N] L.Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115–162 [OT] M.Oliver, E.S.Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations, J. Funct. Anal., 172 (2000), 1–18 [Si] W.Sickel, Superposition of functions in Sobolev spaces of fractional order. A survey, Partial differential equations, Banach Center Publ., 27, Part 1, 2, Polish Acad. Sci., Warsaw, (1992), 481–497 [S] M.Schonbek, Large time behaviour of solutions to the Navier-Stokes equations in spaces, Comm. Partial Differential Equations, 20 (1995), 103–117 [SW] M.Schonbek, M.Wiegner, On the decay of higher order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinburgh, 126 (1996), 677–685 [T] R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edi- tion, Appl. Math. Sci. Ser. Vol 68, Springer-Verlag, Berlin, New York (1997) [Te] R.Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal. 20 (1975), 32–43 [WC] R.Wen, S.Chai, Decay and the asymptotic behavior of solutions to the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 101 (2020) [WXY] J.Wu, X. Xu, Z. Ye, Global smooth solutions to the n-dimensional damped models of incom- pressible fluid mechanics with small initial datum, J. Nonlinear Science, 25(1) (2015), 157–192 [Zhang] N.Zhang, A new global well-posedness for the damped Boussinesq equations, Journal of Math- ematical Physics 60, 031507 (2019) [ZWL] Z.Zhang, X.Wu, M.Lu, On the uniqueness of strong solution to the incompressible Navier- Stokes equations with damping, J. Math. Anal. Appl., 377 (2011), 414–419 [Z] X.Zhong, A note on the uniqueness of strong solution to the incompressible Navier-Stokes equations with damping, Electronic Journal of Qualitative Theory of Differential Equations, 15 (2019), 1–4 [Zh] Y.Zhou, Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping, Appl. Math. Lett., 25 (11) (2012), 1822-1825 [ZZ] Y.Zhou, Y. Zhu, A class of large solutions to the 3D incompressible MHD and Euler equations with damping, Acta. Math. Sin.-English Ser., 34(1) (2018), 63–78 71

Abstract (if available)

Abstract In Chapter 1 of the dissertation, we present a simple proof of the existence of a global strong solution to the Cauchy problem of the 3D incompressible Navier-Stokes equations with damping α|u|ᵝ⁻¹u for any α > 0, β > 3 with $u_0 \in H^1(R^3)$. We also obtain new results on the decay of the solution. In Chapter 2, we study the global regularity and long-time behavior of the solutions to the 3D Boussinesq equations with nonlinear damping α|u|ᵝ⁻¹u (α > 0, β ≥ 1) for the flow of an incompressible fluid with positive viscosity and zero diffusivity.

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Creator Temirgaliyeva, Zhanerke (author)

Core Title On some nonlinearly damped Navier-Stokes and Boussinesq equations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 08/11/2020

Defense Date 08/11/2020

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

Tag Boussinesq equations,global solutions,local solutions,long-time behavior of solutions,Navier-Stokes equations,nonlinear damping,OAI-PMH Harvest,regularity

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ziane, Nabil (committee chair), Jang, Juhi (committee member), Kukavica, Igor (committee member), Newton, Paul (committee member)

Creator Email temirgal@usc.edu,zhanerke@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-364604

Unique identifier UC11666657

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Rights Temirgaliyeva, Zhanerke

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