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Regularity problems for the Boussinesq equations

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Regularity problems for the Boussinesq equations

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Content Regularity problems for the Boussinesq equations by Weinan Wang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Applied Mathematics) August 2020 Copyright 2020 Weinan Wang Dedication Dedicated to my grandfather ii Acknowledgements First and foremost, I am greatly indebted to my advisor Professor Igor Kukavica for his constant encouragement and support and for giving me the opportunity to freely explore new ideas and elds. This dissertation will never be completed without his guidance and thoughtful advice. I also want to thank Professors Roger Ghanem, Juhi Jang, Sergey Lototsky, and Mohammed Ziane for their comments and discussions. I am especially grateful to Professors Juhi Jang and Mohammed Ziane for their support and encouragement during my past several years. I would also like to thank Professor Guillermo Reyes Souto for his kind help. Further thanks go to my collaborators, including Zongyuan Li, Qingtang Su, and Haitian Yue for numerous useful and inspiring discussions and conversations. I am thankful to my friends in graduate school at USC, including Jiajun Luo and Man Luo for their help and many fun memories in the gym and in Chinatown. I owe a great deal to my family who have constantly supported me ever since. They have always been my side with unconditional support during my ups and downs, regardless of the distance. Lastly, I am very thankful to my lovely anc ee Wen Feng for her love, inspiration, and pa- tience. iii Table of Contents Dedication ii Acknowledgements iii Abstract v Chapter 1: Global persistence of Sobolev regularity for the 2D fractional Boussi- nessq equations 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Notation and the main result on global persistence . . . . . . . . . . . . . . . . . . 3 1.3 An L q inequality for G and a Kato-Ponce type commutator estimate . . . . . . . . 7 1.4 The Sobolev persistence for 1<s . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 The Sobolev persistence for s> . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 2: Long time behavior of solutions to the 2D Boussinesq equations with zero diusivity 30 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Long time behavior for periodic boundary conditions . . . . . . . . . . . . . . . . . 32 2.3 The caseR 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.4 Bounds with the Lions boundary condition . . . . . . . . . . . . . . . . . . . . . . 50 2.5 Bounds with the Dirichlet boundary condition . . . . . . . . . . . . . . . . . . . . . 54 Chapter 3: Global regularity for the 3D modied Boussinesq equations 56 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Notation and the main result on global persistence . . . . . . . . . . . . . . . . . . 57 3.3 Proof of Theorem 3:2:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4 Uniquenesss in L 2 (R 3 )L 2 (R 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Bibliography 65 iv Abstract This thesis consists of three main parts. The rst part is concerned with the global persistence of Sobolev regularity for the 2D fractional Boussinesq equations. In this chapter, we address the persistence of regularity for the 2D -fractional Boussinesq equations with positive viscosity and zero diusivity in general Sobolev spaces, i.e., for (u 0 ; 0 )2 W s;q (R 2 )W s;q (R 2 ), where s > 1 andq2 (2;1). We prove that the solution (u(t);(t)) exists and belongs toW s;q (R 2 )W s;q (R 2 ) for all positive timet for a range of exponentsq depending on, where2 (1; 2) is arbitrary. The second part addresses the long time behavior of solutions to the 2D Boussinesq equations, where we consider long time behavior of solutions to the 2D Boussinesq equations with zero diusivity in the cases of the torus,R 2 , and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the normk(u;)k H 2 H 1 is bounded by a single exponential, improving earlier bounds. In the last chapter, we address the global regularity problem for the 3D modied Boussinesq equations. In a recent paper [Y], Ye proved the global persistence of regularity for a 3D Boussinesq model inH s (R 3 )H s (R 3 ) withs> 5=2. We show that the global persistence and uniqueness still hold when s> 3=2. v Chapter 1 Global persistence of Sobolev regularity for the 2D fractional Boussinessq equations 1.1 Introduction The Boussinesq equations are given by a coupled system of the Navier{Stokes equations and a diusion equation for the temperature or the density of uid and they are a common model in the study of heat conduction. In this chapter, we address the persistence of regularity for the 2D fractional Boussinesq equations u t + u +uru +r =e 2 t +ur = 0 ru = 0 in Sobolev spaces. Here, u is the velocity satisfying the 2D Navier-Stokes equations [CF, DG, FMT, R, T2, T3] driven by, which represents the density or temperature of the uid, depending on the physical context. Also,e 2 = (0; 1) is the unit vector in the vertical direction and 1<< 2. The global existence and persistence of regularity has been a topic of high interest since the seminal work of Chae [C] and of Hou and Li [HL], who proved the global existence of a unique 1 solution in the case of Laplacian, = 2. Namely, the global persistence holds for (u 0 ; 0 ) in H s H s1 for integers s 3 [HL], while we have the global persistence in H s H s for integers s 3 by [C]. The global existence and uniqueness in the low regularity space H 1 L 2 was established by Lunasin et al in [LLT]. The persistence in H s H s1 for the intermediate values 1 < s < 3 was then settled in [HKZ1, HKZ2]. For other results on the global existence and persistence of solutions, see [ACW, BS, BrS, CD, CG, CLR, CN, CW, DP1, ES, HK1, HK2, HS, KTW, KWZ, LPZ, SW, T1]. The main diculty when studying the persistence of regularity in the Sobolev spaces W s;q W s1;q when q6= 2 is the lack of availability of the energy equation, which is one of the essential features of the Boussinesq system. This problem was studied in [KWZ], where it was proven that the persistence holds if (s 1)q> 2. In the present chapter, we consider the fractional dissipation in the range < 2, addressing the persistence inW s;q (R 2 )W s;q (R 2 ), i.e., we prove that if (u 0 ; 0 )2W s;q (R 2 )W s;q (R 2 ), then (u(;t);(;t))2W s;q (R 2 )W s;q (R 2 ) for allt 0. The rst result is contained in Theorem 1.2.1 and asserts the global persistence fors andq> (2)=(s1). The more dicult cases> is addressed in Theorem 1.5.1, where we obtain the result for q 2=s. The main device in the proofs of both theorems is the generalized vorticity G =!@ 1 (I ) =2 : (1.1.1) This change of variable is inspired by the one introduced by Hmidi, Keraani, and Rousset [HKR]. Here we need to modify it to avoid problems with low frequencies (our data are not square integrable). We show in (1.2.6) that the modied vorticity G dened in (1.1.1) satises the equation G t +urG + G = [ ~ R ;ur] ( ~ I)@ 1 (1.1.2) 2 where ~ R = @ 1 (I ) =2 with = () 1=2 and ~ = (I ) 1=2 . Compared to the original change of variable in [HKR], we obtain a new term N = ( ~ I)@ 1 , for which however we show in Lemma 1.2.2 below that it is smoothing of degree 1. The reason why this change of variable is suitable for low frequencies is due to the inhomogeneity in the second term of (1.1.2). Also, an important part of the proof of Sobolev persistence is based on the observation that a fractional derivative of the commutator term in (1.1.2) is a sum of two terms, which are also of commutator type and are thus suitable for the use of a Kato-Ponce type inequality; cf. (1.4.5) below. The chapter is organized as follows. In Section 1.2, we state the rst main theorem on the persistence for s and introduce the change of the vorticity variable. We also prove the smoothing property of the operatorN. The next section contains a variant of a Kato-Ponce lemma suitable for the operator ~ R arising in (1.1.2). Lemma 1.3.3 contains the bound for the vorticity and its modied version. The proof of the rst main theorem is then provided in Section 1.4. Finally, the last section contains the second main result on the persistence for s > and its proof. The rst main theorem on the persistence is needed in the proof of the second one when we establish a bound onk() 1=2 uk L 1 in (1.5.6) below. 1.2 Notation and the main result on global persistence We consider solutions of the Boussinesq system u t + u +uru +r =e 2 (1.2.1) t +ur = 0 (1.2.2) ru = 0; (1.2.3) 3 where the operator is dened by = () =2 ; 1<< 2; or, using the Fourier transform, ( f)^() =jj ^ f(); 2R 2 : (1.2.4) The following is the rst main result of the chapter. Theorem 1.2.1. Let 1<s< 2 and q2 (2;1) be such that q> 2 s 1 : (1.2.5) Assume thatku 0 k W s;q <1 withru 0 = 0 andk 0 k W s;q <1. Then there exists a unique solution (u;) to the equations (1.2.1){(1.2.3) such thatu2C [0;T ];W s;q (R 2 ) and2C [0;T ];W s;q (R 2 ) for all positive T . Applying the curl operator to (1.2.1), we obtain the vorticity equation ! t + ! +ur! =@ 1 : Dene ~ = (I ) 1=2 and set G =! ~ R ; where ~ R =@ 1 ~ =@ 1 (I ) =2 : 4 The equation satised byG is obtained by replacing! withG + ~ R and combining the resulting equation with (1.2.2). We get G t +urG + G = ~ R t ur ~ R ~ R +@ 1 = [ ~ R ;ur] ( ~ I)@ 1 : (1.2.6) Therefore, the equation for the generalized vorticity G reads G t +urG + G = [ ~ R ;ur]N; (1.2.7) where we set N = ( ~ I)@ 1 : (1.2.8) The operator N is a Fourier multiplier with the symbol m() = jj 1 (1 +jj 2 ) =2 1 : It is possible to check that the symbol satises the assumptions of the H ormander-Mikhlin theorem and thuskNk L qCkk L q for 1< q <1. However, as asserted in the next lemma, a stronger statement holds. Namely, the operator N dened in (1.2.8) is of the derivative order1, i.e., it is smoothing of order 1. Lemma 1.2.2. Consider the Fourier multiplier T ~ m with the symbol ~ m() = (jj 2 + 1) 1=2 m(): Then T ~ m is a H ormander-Mikhlin operator satisfying kT ~ m fk L q.kfk L q; f2L q ; (1.2.9) 5 for 1< q<1. An equivalent way of stating (1.2.9) is kNfk L q +krNfk L q.kfk L q; f2L q ; q2 (1;1): Proof of Lemma 1.2.2. It suces to prove that the symbol ~ m() = 1 (1 +jj 2 ) =2 jj (1 +jj 2 ) 1=2 satises the H ormander-Mikhlin condition j@ ~ m()j C(jj) jj ; 2N 2 0 ; 2R 2 nf0g: Since the symbol 1 =(1 +jj 2 ) 1=2 is a H ormander-Mikhlin symbol, it is sucient to prove that m() = (1 +jj 2 ) 1=2 (jj (1 +jj 2 ) =2 ) satises the H ormander-Mikhlin condition. In order to check this, we write m() = 2 Z 1 0 (1 +jj 2 ) 1=2 (t +jj 2 ) 1=2 dt and then verify that the condition holds for the low and high frequencies, i.e., whenjj. 1 and jj& 1 respectively. Next, we recall a version of the Kato-Ponce inequality from [KWZ]. 6 Lemma 1.2.3 ([KWZ]). Let s2 (0; 1) and f;g2S(R 2 ). For 1 < q <1 and j2f1; 2g, the inequality k[ s @ j ;g]fk L qCkfk L q 1k 1+s gk L ~ q 1 +Ck s fk L q 2kgk L ~ q 2 holds, where q 1 ; ~ q 1 ; ~ q 2 2 [q;1] and q 2 2 [q;1) satisfy 1=q = 1=q 1 + 1=~ q 1 = 1=q 2 + 1=~ q 2 and C =C(q 1 ; ~ q 1 ; ~ q 2 ;q 2 ;s). Finally, we recall from [CC, J] an inequality suitable for treating the fractional coercive term. Lemma 1.2.4 ([CC, J]). Consider the operator dened in (1.2.4) onR 2 . If; s 2L p , where p 2, then Z R 2 jj p2 s dx 2 p Z R 2 ( s=2 (jj p=2 )) 2 dx; (1.2.10) for all s2 (0; 2). 1.3 AnL q inequality forG and a Kato-Ponce type commutator estimate The following lemma provides an L q bound for the modied vorticity G. Lemma 1.3.1. Assume that u 0 ; 0 2W s;q (R 2 ), where s> 1 and q> 2. Then we have kGk L qCe Ct ; t 0; (1.3.1) and k!k L qCe Ct ; t 0; (1.3.2) where C =C(k! 0 k L q;k 0 k L q). Moreover, we have Z t 0 k =2 (jGj q=2 )k 2 L 2Ce Ct ; (1.3.3) 7 for all t 0. The main step in the proof of Lemma 1.3.1 and Theorem 1.2.1 is an inhomogeneous Kato-Ponce type commutator estimate, which is stated next. Lemma 1.3.2. Denote R :=jrj(I ) =2 : Then, for j2f1; 2g and 0, we have k[ ~ R @ j ;g]fk L qCkrgk L r 1k R fk L ~ r 1 +Ck +1 R gk L r 2kfk L ~ r 2 ; where r 1 ; ~ r 1 ; ~ r 2 2 [q;1] and r 2 2 [q;1) satisfy 1=q = 1=r 1 + 1=~ r 1 = 1=r 2 + 1=~ r 2 and C = C(r 1 ; ~ r 1 ; ~ r 2 ;r 2 ;q). Proof of Lemma 1.3.2 (sketch). We follow the strategy from [KP] (cf. also [KWZ]) and consider the commutator in three regions dened by the supports of k below. Namely, we write [ ~ R @ j ;g]f =c 0 3 X k=1 Z Z e ix(+) j +j ( 1 + 1 )( j + j ) (1 +j +j 2 ) =2 jj 1 j (1 +jj 2 ) =2 ^ f()^ g() k jj jj dd =c 0 3 X k=1 Z Z e ix(+) A k (;)dd; (1.3.4) where k :R! [0; 1] are C 1 cut-o functions such that 3 X k=1 k = 1 on [0;1) with supp 1 [1=2; 1=2]; supp 2 [1=4; 3]; supp 3 [2;1] 8 and A k (;) = j +j ( 1 + 1 )( j + j ) (1 +j +j 2 ) =2 jj 1 j (1 +jj 2 ) =2 ^ f()^ g() k jj jj : Thus, the commutator (1.3.4) may be rewritten as [ s1 ~ R @ j ;u j ] = 3 X k=1 Z Z e ix(+) A k (;)dd: We write A 1 as A 1 (;) = j +j ( 1 + 1 )( j + j )(1 +jj 2 ) =2 (1 +j +j 2 ) =2 jj +2 jj 1 j (1 +jj 2 ) =2 (1 +jj 2 ) =2 jj +2 ^ f()( +1 Rg)^() 1 jj jj = 1 (;) ^ f()( +1 Rg)^(): It is elementary to show that j 1 jC; as well as more generally j@ @ 1 j C(jj;jj) (jj +jj) jj+jj ; ;2N 2 0 : By the Coifman-Meyer theorem, we get Z Z e ix(+) A 1 (;)dd L q .kfk L r 1k +1 Rgk L ~ r 1 ; 9 where 1=q = 1=r 1 + 1=~ r 1 . For A 3 , we write A 3 (;) = j +j ( 1 + 1 )( j + j ) (1 +j +j 2 ) =2 jj 1 j (1 +jj 2 ) =2 ^ f()^ g() 3 jj jj = (1 +jj 2 ) =2 jjjj +1 j +j ( 1 + 1 )( j + j ) (1 +j +j 2 ) =2 jj 1 j (1 +jj 2 ) =2 ( Rf)^()(rg)^() 3 jj jj = 3 (;)( Rf)^()(rg)^() 3 jj jj : Setting (t) = j +tj ( 1 +t 1 )( j +t j ) (1 +j +tj 2 ) =2 ; t2 [0; 1] we have 0 (t) = j +tj 2 ( +t)( 1 +t 1 )( j +t j ) (1 +j +tj 2 ) =2 + j +tj 1 ( j +t j ) (1 +j +tj 2 ) =2 + j +tj j ( 1 +t 1 ) (1 +j +tj 2 ) =2 + j +tj ( 1 +t 1 )( j +t j )( +t) (1 +j +tj 2 ) =2+1 : Note that in the region 3 > 0, we havejj 2jj. Therefore, j 3 jC; as well as more generally j@ @ 3 j C(jj;jj) (jj +jj) jj+jj ; ;2N 2 0 : By the Coifman-Meyer theorem, we get Z Z e ix(+) A 3 (;)dd L q .krgk L q 1k Rfk L q 2; 10 where 1=q = 1=q 1 +1=q 2 . ForA 2 , we use the complex interpolation inequality. Since the argument is the same as in [KP], we omit the proof. By combining the estimates forA 1 ,A 2 , andA 3 , we get k[ ~ R @ j ;g]fk L q.krgk L q 1k R fk L q 2 +k R rgk L q 3kfk L q 4; where the parametersq 1 ;q 2 ;q 3 ;q 4 2 [q;1] satisfy 1=q = 1=q 1 +1=q 2 = 1=q 3 +1=q 4 and the implicit constant depends on q 1 , q 2 , q 3 , q 4 , and . Next, we establish the L q bound for the modied vorticity G. Proof of Lemma 1.3.1. Since s> 1, we have W s;q (R 2 )L 1 (R 2 ), and thus 0 2L q ; q2 [q;1]: Using the L q conservation property for the density equation (1.2.2), we get k(t)k L qk 0 k L q; q2 [q;1]: (1.3.5) In order to estimatekGk L q, we multiply the equation (1.2.7) withjGj q2 G and integrate obtaining 1 q d dt kGk q L q + Z ( G)jGj q2 Gdx = Z NjGj q2 Gdx + Z [ ~ R ;ur]jGj q2 Gdx =I 1 +I 2 : (1.3.6) For I 1 , we have I 1 kNk L qkjGj q2 Gk L q=(q1).kk L qkGk q1 L q .k 0 k L qkGk q1 L q .kGk q1 L q ; (1.3.7) 11 where we used H older's inequality. Since u is divergence-free, we may rewrite the commutator as [ ~ R ;ur] = ~ R u j @ j u j @ j ~ R = (@ j ~ R )(u j )u j (@ j ~ R ) = [@ j ~ R ;u j ]: Observe that @ j ~ R is an operator of order 2. Thus, by Lemma 1.3.2 with = 0, we have I 2 k[ ~ R ;ur]k L qkGk q1 L q =k[@ j ~ R ;u]k L qkGk q1 L q . (k R k L a 1kruk L b 1 +kk L a 2k R ruk L b 2 )kGk q1 L q . (k R k L a 1k!k L b 1 +kk L a 2k R !k L b 2 )kGk q1 L q ; with the Lebesgue exponents above satisfying 1=q = 1=a 1 + 1=b 1 = 1=a 2 + 1=b 2 anda 1 ;a 2 ;b 1 ;b 2 2 (q;1). Therefore, choosing a 1 =a 2 =q=( 1) and b 1 =b 2 =q=(2), I 2 . (k ~ R k L q=(1)k!k L q=(2) +kk L q=(1)k ~ R !k L q=(2))kGk q1 L q : Now, by the fractional Gagliardo-Nirenberg inequality applied tojGj q=2 , we have kGk L r.kGk (2r4r+4q)=2r L q k =2 (jGj q=2 )k 4(rq)=rq L 2 ; qr 2q=(2); (1.3.8) from where kGk L q=(2).kGk (2)= L q k =2 (jGj q=2 )k 4(1)=q L 2 : (1.3.9) Also, using the triangle inequality k!k L q=(2)kGk L q=(2) +k ~ R k L q=(2).kGk L q=(2) +kk L q=(2).kGk L q=(2) + 1 .kGk (2)= L q k =2 (jGj q=2 )k 4(1)=q L 2 + 1 12 we get I 2 . (k!k L q=(2) +k ~ R !k L q=(2))kGk q1 L q .kGk (2)=+q1 L q k =2 (jGj q=2 )k 4(1)=q L 2 +kGk q1 L q : (1.3.10) Replacing (1.3.7) and (1.3.10) in (1.3.6) and using (1.2.10) on the coercive term, we obtain 1 q d dt kGk q L q + 2 q Z ( =2 (jGj q=2 )) 2 dx.kGk q1 L q +kGk (2)=+q1 L q k =2 (jGj q=2 )k 4(1)=q L 2 : Since 4( 1)=q < 2, we may use Young's inequality with exponents q=(q 2 + 2) and q=2( 1) to get d dt kGk q L q + Z ( =2 (jGj q=2 )) 2 dx.kGk q1 L q +kGk q L q; (1.3.11) where the implicit constant depends on the initial data. The inequality (1.3.1) then follows by applying the Gronwall inequality, while (1.3.2) is a consequence of (1.3.1) and the triangle inequality. Finally, (1.3.3) holds by using (1.3.1) in (1.3.11) and integrating. It is important that we may bootstrap the above statement and obtain the conclusion on the behavior of L q norm of G, and thus of !, for all q>q. Lemma 1.3.3. Assume that u 0 ; 0 2 W s;q (R 2 ), where s 1 and q2 (2;1). Then for every q2 (q;1) and t 0 > 0 we have kGk L qCe Ct ; tt 0 and k!k L qCe Ct ; tt 0 ; 13 where C =C(k! 0 k L q;k 0 k L q; q;t 0 ). Moreover, we have Z t 0 k =2 (jG q=2 j)k 2 L 2Ce Ct ; for all t 0 where C =C(k! 0 k L q;k 0 k L q; q;t 0 ). Proof of Lemma 1.3.2. We rst prove that the statement holds for all q2 [q; 2q=(2)], and the rest follows by an iteration argument. Using (1.3.3) witht =t 0 , for which we may assumet 0 1, we obtain n t2 (0;t 0 ] :k =2 (jG q=2 (t)j)k 2 L 2C o 1 C for any C > 0 suciently large. It is easy to deduce then that there exists t2 (0;t 0 ) such that k =2 (jG q=2 j)( t)k L 2C: Since also kG( t)k L qC; we get by (1.3.8) kG( t)k L qC sinceq q 2q=(2). Applying Lemma 1.3.1 but withq replaced with q, we obtain the state- ment for q in this range. Continuing by induction, we get then the conclusion for all q2 [q;1), and the lemma is established. 1.4 The Sobolev persistence for 1<s In this section, we prove our rst main result, Theorem 1.2.1. 14 Proof of Theorem 1.2.1. We start by multiplying (1.2.1) withjuj q2 u and integrating the resulting equation with respect to x obtaining 1 q d dt kuk q L q + Z ( u)juj q2 udx = Z rjuj q2 udx + Z e 2 juj q2 udx since due to the divergence-free condition foru we have R (uru)juj q2 udx = 0. By Lemma 1.2.4 and H older's inequality, we get 1 q d dt kuk q L q + 2 q 2 X j=1 k =2 (ju j j q=2 )k 2 L 2krk L qkuk q1 L q +kk L qkuk q1 L q : (1.4.1) Using the Calder on-Zygmund and Sobolev embedding theorems, we obtain krk L q.k!k L q(kuk L q +k!k L q) +kk L q: Combining the L q conservation property for the density equation with (1.4.1) gives 1 q d dt kuk q L q + 2 q 2 X j=1 k =2 (ju j j q=2 )k 2 L 2.k!k L qkuk q L q +k!k 2 L qkuk q1 L q +kuk q1 L q .e Ct kuk q L q +e Ct kuk q1 L q ; where we used (1.3.2). This leads to kuk L q.(t); where we denote (t) =C exp (C exp(Ct)): (1.4.2) 15 Next, we consider theL q norm of higher order derivatives. Applying s1 to (1.2.7), multiplying the resulting equation byj s1 Gj q2 s1 G and integrating, we get 1 q d dt k s1 Gk q L q + Z ( s1 G)j s1 Gj q2 s1 Gdx = Z s1 (urG)j s1 Gj q2 s1 Gdx + Z s1 ([ ~ R ;ur])j s1 Gj q2 s1 Gdx Z s1 Nj s1 Gj q2 s1 Gdx =J 1 +J 2 +J 3 : (1.4.3) For J 1 , we use Lemma 1.2.3 to estimate J 1 = Z s1 (urG)u s1 rG j s1 Gj q2 s1 Gdx k s1 (urG)u s1 rGk L qk s1 Gk q1 L q . (k s1 Gk L r 4kuk L r 3 +kGk L r 1k s1 !k L r 2 )k s1 Gk q1 L q . k s1 Gk L r 4k!k L r 3 +kGk L r 1 (k s1 Gk L r 2 +k s1 ~ R k L r 2 ) k s1 Gk q1 L q ; for any r 1 ;r 2 ;r 3 ;r 4 2 (q;1) such that 1=q = 1=r 1 + 1=r 2 = 1=r 3 + 1=r 4 . One may choose r 1 =r 2 =r 3 =r 4 = 2q. Note that k s1 ~ R k L r 2.kk L r 2. 1 and thus, using Lemma 1.3.3, we get J 1 .e Ct (k s1 Gk L 2q + 1)k s1 Gk q1 L q : 16 By (1.3.8), we obtain J 1 .e Ct k s1 Gk (1)= L q k =2 (j s1 Gj) q=2 k 2=q L 2 +e Ct k s1 Gk q1 L q : For J 2 , we write s1 [ ~ R ;ur] = s1 ~ R (ur) s1 (ur) ~ R = s1 ~ R (ur) ur s1 ( ~ R ) +ur s1 ( ~ R ) s1 (ur) ~ R = s1 ~ R @ j u j u j @ j s1 ( ~ R ) +u j @ j s1 ( ~ R ) s1 @ j u j ~ R ; (1.4.4) where we used the divergence-free condition (1.2.3) in the last step. The rst two and the last two terms on the far right side of (1.4.4) form commutators, and we may write s1 [ ~ R ;ur] = [ s1 ~ R @ j ;u j ] [ s1 @ j ;u j ] ~ R : (1.4.5) For the second commutator in (1.4.5), we apply Lemma 1.2.3 and obtain k[ s1 @ j ;u j ] ~ R k L q.k ~ R k L p 1k s uk L p 2 +k s1 ~ R k L p 3kruk L p 4; 17 where 1=q = 1=p 1 + 1=p 2 = 1=p 3 + 1=p 4 and p i 2 (q;1) for i = 1; 2; 3; 4. Thus, by Lemma 1.3.2, we have J 2 k s1 [ ~ R ;ur]k L qk s1 Gk q1 L q . (kruk L q 1k s1 R k L q 2 +k s1 R ruk L q 3kk L q 4 )k s1 Gk q1 L q + (k R k L p 1k s uk L p 2 +kk L p 3k R s uk L p 4 )k s1 Gk q1 L q =J 21 +J 22 : (1.4.6) Now, we use the conservation property (1.3.5) for the density and the fact that the operator s1 ~ R is of H ormander-Mikhlin type, and we get J 21 . (kruk L q 1 +kruk L q 3 )k s1 Gk q1 L q . (k!k L q 1 +k!k L q 3 )k s1 Gk q1 L q .e Ct k s1 Gk q1 L q ; where we choseq 1 =q 3 = 2q and used Lemma 1.3.1 in the last step. ForJ 22 , we choosep 2 = 2q= and p 4 = 2q=(2), then by the conservation of density we have J 22 . (k s1 !k L 2q= +k!k L 2q=(2))k s1 Gk q1 L q . k s1 Gk L 2q= +k s1 ~ R k L 2q= +kGk L 2q=(2) +k s1 ~ R k L 2q=(2) k s1 Gk q1 L q . k s1 Gk L 2q= +e Ct + 1 k s1 Gk q1 L q .k s1 Gk 2(1)= L q k =2 (j s1 Gj q=2 )k 2(2)=q L 2 + (e Ct + 1)k s1 Gk q1 L q : For J 3 , we use Lemma 1.2.2 and obtain J 3 k s1 Nk L qk s1 Gk q1 L q .kk L qk s1 Gk q1 L q .k s1 Gk q1 L q : 18 Combining the estimates of J 1 , J 2 , and J 3 and using Young's inequality, we get d dt k s1 Gk q L q + 1 C k =2 (j s1 Gj q=2 )k 2 L 2.e Ct k s1 Gk q L q +e Ct k s1 Gk q1 L q : (1.4.7) Setting X =k s1 Gk q L q; X =k =2 (j s1 Gj q=2 )k 2 L 2; we may rewrite (1.4.7) as d dt X + 1 C X.e Ct (X +X (q1)=q ): Therefore, by the Gronwall lemma we get k s1 Gk L q(t); t 0; where is dened in (1.4.2). Similarly to Lemma 1.3.3, we have k s1 Gk L q(t); t 0; for all q2 [q;1). Thus we get k s1 !k L qk s1 Gk L q +k s1 ~ R k L q(t): 19 Next, we consider the evolution ofk s k L q. We apply s to the equation (1.2.2), multiply it by j s j q2 s , and integrate obtaining 1 q d dt k s k q L q + Z s (ur)j s j q2 s dx = 0: Therefore, using Lemma 1.2.3, 1 q d dt k s k q L q = Z s (ur)j s j q2 s dx = Z ( s (ur)u s r)j s j q2 s dx .k s (ur)u s rk L qk s k q1 L q . (k s uk L s 1krk L s 2 +kuk L 1k s k L q)k s k q1 L q ; under the conditions 1=q = 1=s 1 + 1=s 2 and s 1 ;s 2 2 (q;1). Choosing s 1 = 2q=(2) and s 2 = 2q=, since s, we get 1 q d dt k s k q L q. k s1 !k L 2q=(2)kk 1 L q k s k L q +kuk L 1k s k L q k s k q1 L q . (k s1 Gk L 2q=(2) +k s1 ~ R k L 2q)k s k q+1 L q +kuk L 1k s k q L q . (k s1 Gk L 2q=(2) +kk L 2q)k s k q+1 L q +kuk L 1k s k q L q . (k s1 Gk L 2q=(2) + 1)k s k q+1 L q +kuk L 1k s k q L q; (1.4.8) where = (2 +q)=sq2 (0; 1) by the condition (1.2.5). Let q2 [q;1), then by the Gagliardo- Nirenberg inequality kuk L 1.kuk 13 L q k s1 (u)k 3 L q .k!k 13 L q k s1 !k 3 L q .e Ct(13) k s1 !k 3 L q .e Ct ; 20 where 3 = 1= q(s 1). Also, by the Gagliardo-Nirenberg inequality, we have k s1 Gk L 2q=(2).k s1 Gk 12 L q k =2 (j s1 Gj q=2 )k 22=q L 2 ; where 0< 2 = 1 1=< 1. Hence, continuing from (1.4.8), we get 1 q d dt k s k q L q.Ck s1 Gk 12 L q k =2 (j s1 Gj q=2 )k 22=q L 2 k s k q+1 L q +k s k q+1 L q +Ckuk L 1k s k q L q: Now, in order to conclude the proof, let Y =k s k q= L q W =kuk q L q: Then, for any > 1=(1), by Young's inequality and Lemma 1.3.1, we have d dt (X +Y +W ) + 1 C X. (W + 1)k!k L q +X (12)=q Y (q+1)=q( 1) X 2=q ; whence 1 q d dt (X +Y +W ) + 1 C X.e Ct W + (X +Y + 1): The proof is then concluded by a simple application of the Gronwall inequality. 21 Remark 1.4.1. Note that the identity (1.4.5) only uses the additivity of s1 and the fact that it commutes with the dierential operators. Thus, for any multiplier operator T , we have T [R;ur] = [TR@ j ;u j ] [T@ j ;u j ]R: The proof uses the fact that u is divergence-free. 1.5 The Sobolev persistence for s> We now consider the persistence of regularity when s > , where we always assume 1 < < 2. The following is our second main result. Theorem 1.5.1. Suppose that s> and q 2=s. Assume thatku 0 k W s;q <1 withru 0 = 0 andk 0 k W s;q <1. Then there exists a unique solution (u;) to the equations (1.2.1){(1.2.3) such that u2C [0;T ];W s;q (R 2 ) and 2C [0;T ];W s;q (R 2 ) for all positive T . Denote =s> 0. Proof of Theorem 1.5.1. As in (1.4.3), we have 1 q d dt k s1 Gk q L q + Z ( s1 G)j s1 Gj q2 s1 Gdx = Z s1 (urG)j s1 Gj q2 s1 Gdx + Z s1 ([ ~ R ;ur])j s1 Gj q2 s1 Gdx Z s1 Nj s1 Gj q2 s1 Gdx =J 1 +J 2 +J 3 : 22 For J 1 , we use Lemma 1.2.3 to estimate J 1 = Z s1 (urG)u s1 rG j s1 Gj q2 s1 Gdx k s1 (urG)u s1 rGk L qk s1 Gk q1 L q . (k s1 Gk L r 2kuk L r 1 +kGk L r 1k s1 !k L r 2 )k s1 Gk q1 L q . k s1 Gk L r 2k!k L r 1 +kGk L r 1k s1 Gk L r 2 +kGk L r 1k s1 ~ R k L r 2 k s1 Gk q1 L q .e Ct (k s1 Gk L r 2 +k k L r 2 )k s1 Gk q1 L q ; for any r 1 ;r 2 2 (q;1) such that 1=q = 1=r 1 + 1=r 2 . We restrict r 2 2 (q; 2q=(2)) (1.5.1) so that we may use the inequality (1.3.8) obtaining kGk L r 2.kGk 11 L q k =2 (jGj q=2 )k 21=q L 2 where 1 = 2(r 2 q)=r 2 . Note that (1.5.1) implies r 2 < 2q=(2q) if q< 2, and we have k k L r 2.kk 12 L q k s k 2 L q.k s k 2 L q; (1.5.2) with 2 = (2=q 2=r 2 +)=s. Thus, by (1.3.9) and Lemma 1.3.1, we obtain J 1 .e Ct k s1 Gk 11 L q k =2 (j s1 Gj q=2 )k 21=q L 2 +e Ct k s k 2 L qk s1 Gk q1 L q : 23 The term J 2 is rewritten using (1.4.5) as J 2 = Z [ s1 ~ R @ j ;u j ]j s1 Gj q2 s1 Gdx Z [ s1 @ j ;u j ] ~ R j s1 Gj q2 s1 Gdx =J 21 +J 22 : For the rst term, we have J 21 . (kruk L r 1k s1 R k L r 2 +k s1 R ruk L r 3kk L r 4 )k s1 Gk q1 L q .e Ct (k k L r 2 +k !k L r 3 )k s1 Gk q1 L q ; (1.5.3) where r 3 ;r 4 2 (q;1) are such that 1=r 3 + 1=r 4 = 1=q. Fork k L r 2 , we use (1.5.2), while for k !k L r 3 , we have by the triangle inequality k !k L r 3.k Gk L r 3 +k ~ R k L r 3.kGk 13 L q k s1 Gk 3 L q +k (s2+1)+ k L r 3 .e Ct k s1 Gk 3 L q +k (s2+1)+ k L r 3; where 3 = (2=q 2=r 3 +)=(s 1), as long as r 3 is suciently close to q. From (1.5.3) we thus obtain J 21 e Ct k s k 2 L q +k s1 Gk 3 L q + 1 k s1 Gk q1 L q (1.5.4) if s 2 1, and J 21 e Ct k s k 2 L q +k s1 Gk 3 L q +k s k (s2+1)=s L q k s1 Gk q1 L q ; s> 2 1; 24 where we used [BM] to estimatek s2+1 k L r 3 forr 3 suciently close toq. Since 2 (s 2 + 1)=s, we obtain that (1.5.4) holds even if s > 2 1 as long as r 3 > q is suciently close to q. For J 22 , we recall (1.4.6), by which J 22 . k R k L r 1k s uk L r 2 +kk L 1k R s uk L q k s1 Gk q1 L q .e Ct (k s uk L r 2 +k R s uk L q) .e Ct (k s1 Gk L r 2 +k s1 ~ R k L r 2 +k R s uk L q) .e Ct (k s1 Gk L r 2 +k k L r 2 +k R s uk L q) .e Ct (k s1 Gk L r 2 +k k L r 2 +k R s1 !k L q): The right hand side does not lead to any new terms compared to the estimate for J 1 . Therefore, J 2 .e Ct k s1 Gk 11 L q k =2 (j s1 Gj q=2 )k 21=q L 2 +e Ct k s k 2 L qk s1 Gk q1 L q +e Ct k s1 Gk 3+q1 L q : Next, we treat the term J 3 . When s 2, we have J 3 .k s1 Nk L qk s1 Gk q1 L q .kk L qk s1 Gk q1 L q .k s1 Gk q1 L q ; while if s 2, J 3 .k s1 Nk L qk s1 Gk q1 L q .k s2 k L qk s1 Gk q1 L q .kk 2=s L qk s k (s2)=s L q +k s1 Gk q1 L q .k s k (s2)=s L q k s1 Gk q1 L q : We thus conclude d dt k s1 Gk q L q + 1 C k =2 (j s1 Gj q=2 )k 2 L 2 .e Ct k s1 Gk q L q +e Ct k s1 Gk q1 L q +e Ct k s k 2 L qk s1 Gk q1 L q +k s k ((s2)=s)+ L q k s1 Gk q1 L q : (1.5.5) 25 Next, we considerk s k L q. First, we have by Sobolev embedding, with q = 2=( 1), kuk L 1.k 1 uk L q .k Gk L q +k 1 ~ R k L q .e Ct ; (1.5.6) where we used Theorem 1.2.1 in the third inequality. Thus, by Lemma 1.2.3, 1 q d dt k s k q L q = Z s (ur)j s j q2 s dx = Z ( s (ur)u s r)j s j q2 s dx .k s (ur)u s rk L qk s k q1 L q . (k s uk L s 1krk L s 2 +kuk L 1k s k L q)k s k q1 L q . k s1 !k L s 1krk L s 2 +e Ct k s k L q k s k q1 L q (1.5.7) where s 1 ;s 2 2 (q;1) are such that 1=s 1 + 1=s 2 = 1=q. At this point, we employ an inequality from [BM], which gives krk L s 2.kk 11=s L s 2 k s k 1=s L q .k s k 1=s L q (1.5.8) where 1=s 2 = 1=sq = (1= s 2 )(1 1=s), assuming that s 2 sq which is equivalent to s 1 qs s 1 : From (1.5.7) and (1.5.8) we then obtain 1 q d dt k s k q L q. k s1 !k L s 1k s k 1=s L q +e Ct k s k L q k s k q1 L q 26 whence d dt k s k L q.k s1 !k L s 1k s k 1=s L q +e Ct k s k L q: (1.5.9) If s 1 2q 2 we may apply (1.3.8) and obtain k s1 !k L s 1k s1 Gk L s 1 +k s1 ~ R k L s 1 .k s1 Gk 13 L q k =2 (j s1 Gj q=2 )k 23=q L 2 +k k L s 1 .k s1 Gk 13 L q k =2 (j s1 Gj q=2 )k 23=q L 2 +kk 14 L q k s k 4 L q .k s1 Gk 13 L q k =2 (j s1 Gj q=2 )k 23=q L 2 +k s k 4 L q; where 3 = 2(s 1 q)=s 1 and 4 = (s 2=s 1 + 2=q)=2. Therefore, by (1.5.9), d dt k s k L q.k s1 Gk 13 L q k =2 (j s1 Gj q=2 )k 23=q L 2 k s k 1=s L q +k s k 4+1=s L q +e Ct k s k L q: (1.5.10) Now, in order to conclude the proof, let > 0, and denote X =k s1 Gk q L q; Y =k s k q= L q ; W =kuk q L q; Z =k =2 (j s1 Gj q=2 )k 2 L 2: 27 Then (1.5.5) and (1.5.10) may be rewritten as d dt X + 1 C Z.e Ct X +e Ct X (q1)=q +e Ct Y 2 =q X (q1)=q +Y ((s2)=s)+ =q X (q1)=q (1.5.11) and d dt Y .X (13)=q Z 3=q Y 1+ =sq =q +Y 4=q+ =sq+1 =q +e Ct Y; (1.5.12) respectively. (We use here that if (d=dt)k s k L qf, then _ Y fY 1 =q .) Adding (1.5.11) and (1.5.12), we obtain d dt (X +Y ) + 1 C Z.e Ct X +e Ct Y 2 =q X (q1)=q +Y ((s2)=s)+ =q X (q1)=q +X (13)=q Z 3=q Y 1+ =sq =q +Y 4=q+ =sq+1 =q +e Ct Y: In order to apply the Gronwall lemma, it is sucient that the conditions 3 q < 1 2 q + q 1 q 1 s 2 s + q + q 1 q 1 1 q + sq q 0 4 q + sq q 0 hold. While the rst condition is automatic, the next three may be summarized as s s 1 min ( 1 2 ; s s 1 + ) 28 while the last one reads s 1 1 4 : Setting s 1 =qs=(s 1), it is easy to check that a sucient condition for the existence of proper is q 2=s, as claimed. 29 Chapter 2 Long time behavior of solutions to the 2D Boussinesq equations with zero diusivity 2.1 Introduction We consider the asymptotic behavior of solutions to the Boussinesq equations without diusivity u t u +uru +r =e 2 (2.1.1) t +ur = 0 (2.1.2) ru = 0 (2.1.3) in a bounded domain R 2 ,T 2 , andR 2 . Here,u is the velocity satisfying the 2D Navier-Stokes equations [CF, DG, FMT, R, T1, T2, T3] driven by, which represents the density or temperature of the uid, depending on the physical context. Also, e 2 = (0; 1) is the unit vector in the vertical direction. Recently, there has been a lot of progress made on the existence, uniqueness, and persistence of regularity, mostly in the case of positive viscosity and vanishing diusivity, considered here, while the same question with both vanishing viscosity and diusivity is an important open problem. The initial results on the global existence in the regularity class have been obtained by Hou 30 and Li [HL], who proved the global existence and persistence in the class H s H s1 for integer s 3. Independently, Chae [C] considered the classH s H s and proved the global persistence in H 3 H 3 . The class H s H s1 has subsequently been studied in the case of a bounded domain, where Larios et al proved in [LLT] the global existence and uniqueness for s = 1 and then by Hu et al, who proved in [HKZ1] the persistence for s = 2. The remaining range 1 < s < 3 was then resolved in [HKZ2] in the case of periodic boundary conditions. For other works on the global existence and persistence in Sobolev and Besov classes, see [ACW, BS, BrS, CD, CG, CN, CW, DP1, HK1, HK2, HS, KTW, KWZ, LPZ]. In a recent paper [J], Ju addressed the important question of long time behavior of solutions. He proved that in the case of Dirichlet boundary conditions on a bounded domain , theH 2 ( ) H 1 ( ) norm grows at most asCe Ct 2 , whereC > 0 is constant. In the present chapter, we consider this question for this and other boundary conditions. When the domain is nite, we prove that actually the H 2 H 1 norm is increasing as a single exponential. We conjecture that this bound is sharp. This is because it is not expected that the solutions of the Boussinesq equation decay. However, note that the rate of increase of the gradient of the density is bounded by the exponential integral of the L 1 norm of the gradient, i.e., kr(t)k L 2. exp Z t 0 kru(s)k L 1ds kr 0 k L 2; cf. (2.2.37) below, and if u is not decaying, we should expect the integral to be bounded from below by a constant multiple of t. In addition to the behavior ofk(u;)k H 2 H 1, we also address the long time behavior of the vorticity. In the case of the torus, we nd constant upper bounds for the vorticity and the gradient of the vorticity for all L p norms. This result relies on the uniform upper bound forkuk H 2 established in [J] as well as on a Nash-Moser type result on the growth of the vorticity, stated as Lemma 2.2.2 below and which we believe is of independent interest. 31 The chapter is structured as follows. In Section 2.2, we rst address the case of periodic boundary conditions. In this case, the exponential bound for the gradient of the density is obtained by establishing a constant upper bound forkruk L p. For this purpose, we rst obtain a uniform upper bound for all the L p norms of the vorticity, a result based on a Nash-Moser type iteration. To do the same for the gradient of the vorticity, it is not suitable to proceed with direct estimates. Instead, we recall the concept of the generalized vorticity (cf. (2.2.20) below), which reduces the number of the derivatives in the density by one. In Section 2.3, we consider the case of the unbounded domain R 2 . Here, the energy does not decay and in fact, the quantityku(;t)k L 2 grows linearly in time. Applying a similar procedure as in Section 2.2, we obtainku(;t)k H 2 =O(t 1=2 ) as well as an information on the growth ofkk H 1. In addition, we obtain upper bounds fork!k L p and p 3=2 kr!k L p, which are uniform in p. In the nal two sections, we address the case of a smooth bounded domain with either Lions or Dirichlet boundary conditions. For the Lions boundary conditions, we obtainkrk L 2Ce Ct , using a dierent technique than the one for periodic boundary conditions. In addition, we obtain a uniform constant upper bound fork!k L p. Similarly, the last section contains the results in the case of Dirichlet boundary conditions, where we obtain an exponential upper bound forkrk L 2.e Ct , improving the main result in [J]. 2.2 Long time behavior for periodic boundary conditions In this section, we consider the Boussinesq system (2.1.1){(2.1.3) in the case of the torus T 2 , i.e., assuming that u and are 1-periodic. We assume for simplicity that R T 2 u(;t) = 0 for all t 0; the general case can be addressed with the same methods; cf. Remark 2.2.3 below. The system is supplemented with the initial condition (u(; 0);(; 0)) = (u 0 ; 0 )2H 2 (T 2 )H 1 (T 2 ) 32 with u 0 divergence-free. By [HKZ1], there exists a global solution (u(t);(t)) which belongs to H 2 H 1 . Also, by [J], we have ku(t)k H 2C; t 0: (2.2.1) In the following statement, we provide an upper bound for the growth of the component of the normk(u;)k H 2 H 1. Also, we establish a uniform upper bound on the quantitiesk!(;t)k L p and p 3=2 kr!(;t)k L p for all p 2. Theorem 2.2.1. Assume that (u 0 ; 0 )2 H 2 (T 2 )H 1 (T 2 ) satisesru 0 = 0 and R T 2 u 0 = 0. Then we have k(t)k H 1Ce Ct ; t 0 for a constant C =C(ku 0 k H 2;k 0 k H 1). Moreover, k!(t)k L pC; tt 0 ; p2 [2;1] and kr!(t)k L pCp 3=2 ; tt 0 ; p2 [2;1); (2.2.2) where t 0 0 depends onku 0 k L 2. Note that (2.2.1) and (2.2.2) imply kuk W 2;pCp 5=2 ; tt 0 ; p2 [2;1): In the proof, we need the following statement on the long time behavior of solutions to the Navier-Stokes equations, which is of independent interest. 33 Lemma 2.2.2. Consider the Navier-Stokes system u t u +uru +r =f ru = 0; supplemented with a divergence-free initial condition u(; 0) = u 0 2 L 2 (T 2 ) such that R T 2 u 0 = 0 and R T 2 f(;t) = 0 for t 0. If, for some 0, we have kfk L 1 ([0;1);L p (T 2 )) p M; 2p<1; (2.2.3) where M 1, then there exists t 0 > 0 depending only onku 0 k L 2 such that k!(;t)k L pCM; tt 0 ; 2p1; (2.2.4) where C is a universal constant. Moreover, for every t 0 > 0, there exists a constant C depending only onku 0 k L 2 and t 0 such that (2.2.4) holds. The proof uses ideas from [K, Lemma 3.1], where = 0 was considered. Lemma 2.2.2 is needed below with = 1=2. Proof of Lemma 2.2.2. First, we prove (2.2.4) for some t 0 > 0, leaving the last assertion to the end of the proof. Without loss of generality, M 2. The energy inequality reads 1 2 d dt kuk 2 L 2 +kruk 2 L 2kfk L 2kuk L 2; (2.2.5) from where, using the Poincar e inequality, d dt kuk L 2 + 1 C kuk L 2kfk L 2: 34 Applying the Gronwall inequality and shifting time, we may assume, without loss of generality, that ku(t)k L 2CM; t 0: (2.2.6) Note that the size of the time shift depends only onku 0 k L 2 andM. Next, the vorticity! =ru satises ! t ! +ur! =rF ; (2.2.7) where F = (F 1 ;F 2 ) = (f 2 ;f 1 ). For p = 2; 4; 8;:::, dene p = Z ! p ; where all the integrals in this section are assumed to be over T 2 . First, the enstrophy inequality reads 1 2 0 2 +kr!k 2 L 2kFk L 2kr!k L 2 =kfk L 2kr!k L 2 1 2 kfk 2 L 2 + 1 2 kr!k 2 L 2; from where, using kr!k 2 L 2 k!k 4 L 2 kuk 2 L 2 = 2 2 kuk 2 L 2 ; which follows fromk!k L 2 =kruk L 2kuk 1=2 L 2 kuk 1=2 L 2 =kuk 1=2 L 2 kr!k 1=2 L 2 , we obtain 0 2 + 2 2 Ckuk 2 L 2 kfk 2 L 2: Therefore, by (2.2.6) andkfk L 2.M, 0 2 + 2 2 CM 2 CM 2 ; 35 and thus there exists a universal constant t 1 0 such that 2 (t)CM 2 ; tt 1 : Now, let p2f2; 4; 8;:::g. Testing the vorticity equation (2.2.7) with ! 2p1 , we get 1 2p 0 2p + (2p 1) Z ! 2p2 jr!j 2 = Z @ j F j ! 2p1 =(2p 1) Z F j ! 2p2 @ j ! (2p 1)kFk L 2pk! p1 k L 2p=(p1)k! p1 r!k 2 2p 1 2 Z ! 2p2 jr!j 2 +Cpkfk 2 L 2pk! p1 k 2 L 2p=(p1) ; from where 1 2p 0 2p + 2p 1 2 Z ! 2p2 jr!j 2 Cpkfk 2 L 2pk!k 2p2 L 2p : (2.2.8) Using Nash's inequality, kvk L 2.kvk 1=2 L 1 krvk 1=2 L 2 +kvk L 1 (2.2.9) withv =! p , we getk! p k L 2.k! p k 1=2 L 1 kr(! p )k 1=2 L 2 +k! p k L 1 whencek!k 4p L 2p .p 2 k!k 2p L pk! p1 r!k 2 L 2 + k!k 4p L p. Therefore, k! p1 r!k 2 L 2 k!k 4p L 2p Ck!k 4p L p Cp 2 k!k 2p L p : Applying this inequality on the second term in (2.2.8), we get 1 2p 0 2p + 2 2p C 4 p Cp 2 p Cpkfk 2 L 2p (p1)=p 2p ; (2.2.10) whence, by (2.2.3), 0 2p + 2 2p C 4 p C 2 p Cp 2+2 M 2 (p1)=p 2p : Note that if 2p C 0 max n 2 p ;p 2(1+)p=(p+1) 2p=(p+1) p M 2p=(p+1) o ; (2.2.11) 36 then 0 2p + 2 2p C 2 p 0; which means that once p is bounded, 2p is rapidly decreasing as long as it is suciently large. By increasing the constants, we may assume that 2 (t)C 0 M 2 ; tt 1 and C 0 1. Denote p k = 2 k , for k2N. Now, dene recursively a sequence M 1 ;M 2 ;M 3 ;::: such that M k+1 =C 0 max n p k M 2 k ;p 2(1+)p k =(p k +1) k M 2p k =(p k +1) k M 2p k =(p k +1) o ; k = 1; 2;::: (2.2.12) (the reason forp k in front ofM 2 k , comparing (2.2.12) with (2.2.11), is that it appears on the right side of (2.2.13) below). Also, let M 1 =C 0 M 2 : We shall dene a sequence 0t 1 t 2 such that 2 k(t)M k ; tt k withft k g 1 k=1 uniformly bounded. To construct this sequence, we proceed inductively, and assume that t k has been set. As long as 2 k+1M k+1 , we have 0 2 k+1 + 2 2 k+1 CM 2 k 0: 37 Solving this inequality, we obtain the existence of t k+1 t k such that 2 k+1(t) 2 k M 2 k ; tt k+1 (2.2.13) with t k+1 t k C 2 k : (2.2.14) Note that (2.2.12) and (2.2.13) imply 2 k+1(t)M k+1 ; tt k+1 : By the summability of the right side of (2.2.14) ink, the sequencet k with the indicated properties has been constructed. In particular, 2 kM k ; tT 0 ; where T 0 = lim k t k <1. It remains to obtain a suitable upper bound for M k . For this purpose, we construct a domi- nating sequence R 1 ;R 2 ;R 3 ;:::. Let R k+1 =C 1 p k R 2 k ; k = 1; 2;::: (2.2.15) with a constant C 1 C 0 to be determined and with = 2 + 2. Also, set R 1 =C 1 2 M 2 : (2.2.16) 38 First, using induction, it is easy to check that (2.2.15) and (2.2.16) imply R k = (2 C 1 ) 2 k 1 M 2 k ; k = 1; 2; 3;:::: (2.2.17) Next, we claim that M k R k ; k = 1; 2;:::: (2.2.18) It is clear that (2.2.18) holds for k = 1. Assuming that (2.2.18) holds for k2N, we get M k+1 =C 0 max n p k M 2 k ;p 2(1+)p k =(p k +1) k M 2p k =(p k +1) k M 2p k =(p k +1) o C 0 max n p k R 2 k ;p 2(1+)p k =(p k +1) k R 2p k =(p k +1) k M 2p k =(p k +1) o C 1 p k R 2 k =R k+1 : (2.2.19) The second inequality in (2.2.19) is obtained by a direct verication. Since we have now established M k (2 C 1 ) 2 k 1 M 2 k ; k = 1; 2; 3;:::; by (2.2.17) and (2.2.18), we get M 1=2 k k 2 C 1 M; k = 1; 2; 3;:::; and the rst part of the lemma is established. As for the last assertion, let t 0 > 0 be arbitrary. Applying the Gronwall lemma on (2.2.5), we get (2.2.6) for t t 0 =2, where C depends onku 0 k L 2 and t 0 . By shifting time by t 0 =2, we have (2.2.6) for t 0. Similarly, we can choose t k = t 0 =2 k+1 for k = 1; 2;::: and the constants then depend onku 0 k L 2 and t 0 . 39 An important device in the proof of Theorem 2.2.1 is the modied vorticity =!R; (2.2.20) introduced in [KW1] where R =@ 1 ~ 2 =@ 1 (I ) 1 with ~ = (I ) 1=2 . This, in turn, is a modication of the change of variable introduced in [SW, HKR]. The quantity satises t +ur = [R;ur]N; (2.2.21) where N = ( ~ 2 I)@ 1 (2.2.22) is a smoothing operator of order1 (cf. [KW1]), i.e., the operatorrN in the Calder on-Zygmund class. Using that u is divergence-free, the rst term on the right hand side of (2.2.21) may be rewritten as [R;ur] =Ru j @ j u j @ j R =@ j R(u j )u j @ j R = [@ j R;u j ]: (2.2.23) Also, for any multiplier operator T , we have T [R;ur] = [TR@ j ;u j ] [T@ j ;u j ]R (2.2.24) (cf. [KW1]). In both identities (2.2.23) and (2.2.24), which may be veried by a direct calculation, it is essential that u is divergence-free. 40 Proof of Theorem 2.2.1. We assume ku 0 k H 2;k 0 k H 1C: (2.2.25) By the Gagliardo-Nirenberg inequality kvk L p.p 1=2 kvk 2=p L 2 krvk 12=p L 2 +kvk L 2 with v = 0 and by (2.2.25), we get k 0 k L p.p 1=2 ; p2 [2;1) and thus k(t)k L p.p 1=2 ; t 0; p2 [2;1): (2.2.26) Using (2.2.26) and applying Lemma 2.2.2 with = 1=2, there exists t 1 0 such that k!k L pC; tt 1 ; p2 [2;1]; (2.2.27) which by the triangle inequality implies kk L p. 1; tt 1 ; p2 [2;1]: (2.2.28) Since C is allowed to depend onku 0 k L 2, we may assume that t 1 > 0 is arbitrarily small. 41 In order to boundr!, we consider evolution of the modied vorticity (2.2.20). Applying @ k to (2.2.21), multiplying the resulting equation byj@ k j 2p2 @ k , integrating and summing in k leads to 1 2p d dt X k k@ k k 2p L 2p X k Z (@ k )j@ k j 2p2 @ k dx = X k Z @ k (u j @ j )j@ k j 2p2 @ k dx + X k Z @ k ([R;ur])j@ k j 2p2 @ k dx X k Z @ k Nj@ k j 2p2 @ k dx =J 1 +J 2 +J 3 (2.2.29) with no summation convention applied to the index k in this proof. For p2f2; 4; 8;:::g, denote p = X k Z j@ k j p : (2.2.30) Note that the second term on the left hand side of (2.2.29) equals 2p 1 p 2 X k Z @ j (j@ k j p )@ j (j@ k j p ) D p ; where D = X k Z @ j (j@ k j p )@ j (j@ k j p ) = X k kr(j@ k j p )k 2 L 2: Regarding J 1 , we use the divergence-free condition on u to write J 1 = 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 2 rk L 4p X k j@ k j 2p1 k L 4p=(2p1).kruk L 2 X k krk 2p L 4p : 42 Therefore, J 1 .k!k L 2krk 2p L 4p . X k k@ k k 2p L 4p : Using the Gagliardo-Nirenberg inequality, we have 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 =k@ k k p L 2p kr(j@ k j p )k L 2 for k = 1; 2, and thus J 1 D 4p +Cp X k k@ k k 2p L 2p D 4p +Cp 2p : (2.2.31) Next, for the second term J 2 , we have J 2 =(2p 1) X k Z [R;ur]j@ k j 2p2 @ kk dx = 2p 1 p X k Z [R;ur]j@ k j p2 @ k @ k (j@ k j p )dx .k[R;ur]k L 2p X k kr(j@ k j p )k L 2kj@ k j p1 k L 2p=(p1): The rst factor is estimated as k[R;ur]k L 2pkR(u j @ j )k L 2p +ku j @ j Rk L 2p =kR@ j (u j )k L 2p +ku j @ j Rk L 2p .pkuk L 2p +kuk L 1k(rR)k L 2p.pkuk L 1kk L 2p.p 3=2 ; where we used (2.2.1) and (2.2.26) in the last inequality. Therefore, we obtain J 2 .p 3=2 X k kr(j@ k j p )k L 2kj@ k j p1 k L 2p=(p1).p 3=2 X k kr(j@ k j p )k L 2k@ k k p1 L 2p p 3=2 D 1=2 X k k@ k k p1 L 2p D 4p +Cp 4 X k k@ k k 2p2 L 2p D 4p +Cp 4 (p1)=p 2p : (2.2.32) 43 For J 3 , we use that the operator N, dened in (2.2.22), is a smoothing operator or order1 (cf. [KW1]). Thus J 3 . X k k@ k Nk L 2pkj@ k j 2p1 k L 2p=(2p1).p X k kk L 2pkj@ k j 2p1 k L 2p=(2p1) .p 3=2 X k k@ k k 2p1 L 2p .p 3=2 (2p1)=2p 2p : (2.2.33) By replacing the estimates (2.2.31), (2.2.32), and (2.2.33) in (2.2.29), we get 1 p 0 2p + 1 p DCp 2p +Cp 4 (p1)=p 2p +Cp 3=2 (2p1)=2p 2p ; p 2: Using (2.2.9) with v =j@ k j p , we obtain 1 p 0 2p + 2 2p C 4 p Cp 2 p Cp 2p +Cp 4 (p1)=p 2p +Cp 3=2 (2p1)=2p 2p ; p 2; and thus, absorbing the last term on the right side and multiplying the resulting inequality by p, 0 2p + 2 2p C 2 p C 2 p +Cp 2 2p +Cp 5 (p1)=p 2p ; p 2: (2.2.34) In order to start the induction, we also need an estimate for 2 . In this case, we have D = X k Z @ jk @ jk = X k kr(@ k )k 2 L 2 krk 4 L 2 kk 2 L 2 : Then the same derivation as above shows that 0 2 + krk 4 L 2 kk 2 L 2 C 2 +C; (2.2.35) 44 from where, using (2.2.28) with p = 2, 0 2 + 2 2 C 2 +C: Applying the Gronwall inequality, this implies that there exists t 2 t 1 such that krk L 2C; tt 2 : Going back to the inequality (2.2.34), x p 2, and note that if for any t 0 we have 2p C maxfp 2 2 p ;p 5 2p=(p+1) p g; for a suciently large constant C, half of the second term on the left hand side dominates the terms on the right hand side and thus 0 2p + 2 2p 2 2 p 0: As in the proof of Lemma 2.2.2, this implies the existence of t 3 t 2 such that krk L pC; tt 3 ; p2 [2;1]: In particular, we get kr!k L p.p 3=2 ; tt 3 ; p2 [2;1); (2.2.36) sincekrRk L p.pkk L p.p 3=2 . The inequalities (2.2.27) and (2.2.36) then imply kruk L 1C; tt 3 : 45 Since d dt krk 2 L 2.kruk L 1krk 2 L 2; (2.2.37) we get krk L 2Ce Ct ; t 0; and the assertion is proven. Remark 2.2.3. It is not dicult to extend Theorem 2.2.1 to the case when we do not assume R T 2 u 0 = 0. In this case, we get R T 2 u .t + 1. Based on the energy inequality 1 2 d dt kuk 2 L 2 +kruk 2 L 2Ckuk L 2 we getku(t)k L 2.t + 1 for t 0. Also, as in the proof above, we getk!k L p. (t + 1) 1=2 for all p2 [2;1] and thus alsok!k L p. (t + 1) 1=2 for all tt 1 for some t 1 0. Again proceeding as above, we getkrk L p. (t + 1) 1=2 rst forp = 2 and then for allp2 [2;1] fort suciently large. 2.3 The case R 2 In this section, we consider the case of the whole space R 2 . Theorem 2.3.1. Assume that (u 0 ; 0 )2H 2 (R 2 )H 1 (R 2 ), whereru 0 = 0. Then we have kuk H 2C(t + 1) 1=2 ; t 0 46 and krk L 2Ce C(t+1) +1 log(t+2) ; t 0 for a constant C =C(ku 0 k H 2;k 0 k H 1), where = 1 Y j=1 1 1 2 j = 0:28878 : Moreover, k!(t)k L p. (t + 1) 1=p+(12=p) ; tt 0 ; p2 [2;1] and kr!(t)k L p.p 3=2 + (t + 1) 1=2 ; tt 0 ; p2 [2;1) for some t 0 0. Remark 2.3.2. The reason for a dierent bound than in Theorem 2.2.1 is a lack of the Poincar e inequality, which is available in other settings in this chapter. If an additional damping term u, where > 0, is added to the left side of the equation (2.1.1), then the bounds are identical to those in Theorem 2.2.1, with constants depending on . Proof of Theorem 2.3.1. The energy inequality 1 2 d dt kuk 2 L 2 +kruk 2 L 2Ckuk L 2 implies ku(t)k L 2.t + 1; t 0: 47 Similarly, the L 2 inequality for the vorticity reads 1 2 d dt k!k 2 L 2 +kr!k 2 L 2C; which implies kru(t)k L 2 =k!k L 2. (t + 1) 1=2 ; t 0: (2.3.1) Next, we consider the upper bounds fork!k L p andkr!k L p for p 2. Denote p =k!k p L p and x p 2. From the vorticity equation ! t ! +ur! =@ 1 we obtain, as in (2.2.10), the inequality 0 2p + 2 2p C 2 p Cp 3 (p1)=p 2p : As in the proof of Lemma 2.2.2, we conclude by induction that k!k L p. (t + 1) p ; tt 0 ; for p = 2; 4;:::, where 2 k = k Y j=1 1 1 2 j : Therefore, k!k L 1. (t + 1) ; tt 0 : 48 Combined with (2.3.1), we get k!k L p. (t + 1) 1=p+(12=p) ; tt 0 ; p2 [2;1]; from where also kruk L p.p(t + 1) 1=p+(12=p) ; tt 0 ; p2 [2;1): In order to obtain an estimate on the growth ofr!, we consider the generalized vorticity (2.2.20), which satises (2.2.21). As in the periodic case, we set (2.2.30), i.e., for p 2 p = X k Z j@ k j p and obtain 0 2 + 2 2 kk 2 L 2 C 2 +C and 0 2p + 2 2p C 2 p Cp 2 2p +Cp 5 (p1)=p 2p ; p 2 (2.3.2) (cf. (2.2.34) above). kk L 2. (t + 1) 1=2 ; t 0 imply 2 (t).t + 1; t 0: Continuing by induction, we obtain from (2.3.2) p (t).p (t + 1) p=2 ; t 0; p = 2; 2 2 ; 2 3 ;::: 49 with a certain > 0. These inequalities then lead to krk L p. (t + 1) 1=2 ; t 0; p2 [2;1]: From here, we obtainkr!k L pkrk L p +krRk L p. (t + 1) 1=2 +p 3=2 , and thuskD 2 uk L p. p 5=2 (t + 1) 1=2 . Therefore, kruk L 1Ckruk 12=p L p kD 2 uk 2=p L p .Cp(t + 1) 3=p2=p 2 : Choosing a proper value for p, we get kruk L 1. (t + 1) 1 log(t + 1); tt 0 which then implies krk L 2. exp (t + 1) 1+1 log(t + 1) ; t 0; and the theorem is proven. 2.4 Bounds with the Lions boundary condition In this section, we consider the Boussinesq system on a bounded smooth domain R 2 , with the Lions boundary conditions un =! = 0 on @ ; 50 where n denotes the outward unit normal. We use the standard notation corresponding to the Navier-Stokes system [CF, T1, R, HKZ1]. In particular, denote H =fu2L 2 ( ) :ru = 0;un = 0 on @ g; wheren stands for the outward unit normal vector with respect to the domain , which is assumed to be smooth and bounded. Let also V =fu2H 1 ( ) :ru = 0;un = 0 on @ g: The Stokes operatorA: D(A)!H, with the domainD(A) =H 2 ( )\V , is dened byA =P, whereP is the Leray projector in L 2 ( ) on the space H. Theorem 2.4.1. Assume that (u 0 ; 0 )2D(A)H 1 ( ). Then we have kuk H 2C; t 0 (2.4.1) and krk L 2Ce Ct ; t 0 (2.4.2) for a constant C =C(ku 0 k D(A) ;k 0 k H 1). In addition, we have k!(t)k L pC; tt 0 ; p2 [2;1]; where t 0 0 depends onku 0 k L 2 andk 0 k L 2. The global persistence for the Boussinesq system with the Lions boundary conditions was recently addressed by Doering et al. The authors moreover proved thatkuk H 1! 0 as t!1. 51 It is not clear whether the same holds for other boundary conditions considered in the present section. Namely, the important ingredients in [DWZZ] are that = ay +b belongs to the state space and that the vorticity! vanishes on the boundary. From here on, the constantC is allowed to depend onku 0 k D(A) andk 0 k H 1. From [J], we also recall the inequality Z t2 t1 kA 3=2 u(s)k 2 L 2dsC(t 2 t 1 + 1); 0t 1 t 2 : The proof of the assertion (2.4.1) is the same as in [J], which considered the Dirichlet boundary condition. Proof of Theorem 2.4.1. Note that the proof of Lemma 2.2.2 applies here verbatim, and thus we obtain k!(;t)k L pC; tt 0 ; 2p1: (2.4.3) Since t 0 > 0 may be chosen arbitrarily small (cf. Lemma 2.2.2) and by the local existence, we may simply assume that (2.4.3) holds for all t 0. Now, note that the argument starting in (2.2.29) does not apply in this setting due to arising boundary terms. Thus we use an alternative argument, described next. Fix t 0 > 0. Let : R! [0;1) be a smooth non-decreasing function such that 0 on [0;t 0 =2] and 1 on [t 0 ;1]. Then we have @ t ((t)!) ((t)!) = 0 (t)!@ j ((t)u j !) +@ 1 ((t)) =@ 1 ( 0 (t)u 2 )@ 2 ( 0 (t)u 1 )@ j ((t)u j !) +@ 1 ((t)): 52 Using the parabolic regularity with the right side in divergence form we get, for all t 0, Z t 0 k(s)r!(s)k p L pds 1=p Cp Z t 0 k 0 (s)uk p L pds 1=p +Cp Z t 0 k(s)!(s)u(s)k p L pds 1=p +Cp Z t 0 k(s)(s)k p L pds 1=p Cp Z t 0 k 0 (s)uk p L pds 1=p +Cp Z t 0 k(s)!(s)k p L pds 1=p +Cp Z t 0 k(s)(s)k p L pds 1=p Cp Z t 0 kuk p L pds 1=p +Cp 3=2 t 1=p ; (2.4.4) where 2p<1 bykuk L 1.kuk H 2. 1. Therefore, usingkuk L p. 1, Z t 0 k(s)r!(s)k p L pds 1=p Cp 3=2 t 1=p : Now, for every p2 [2;1), we have kruk L 1Ckruk 12=p L p kD 2 uk 2=p L p +Ckruk L pCpk!k 12=p L p kr!k 2=p L p +Cpk!k L p: In particular, Z t t0 kruk L 1ds. Z t t0 k!k 1=2 L 4 kr!k 1=2 L 4 ds + Z t t0 k!k L 4ds . Z t t0 k!k 4=7 L 4 ds 7=8 Z t t0 kr!k 4 L 4ds 1=8 + Z t t0 k!k L 4ds .t 7=8 t 1=8 +t.t; (2.4.5) where we used (2.4.4) with p = 4 in the last inequality. Integrating (2.2.37), which also holds in this setting, and applying (2.4.5) then gives the inequality (2.4.2). 53 2.5 Bounds with the Dirichlet boundary condition Finally, we address the long time behavior of the Boussinesq system with the classical Dirichlet (non-slip) boundary condition u = 0 on @ ; where is a bounded smooth domain. Recall the standard notation H =fu2 L 2 ( ) :ru = 0;un = 0 on @ g, where n denotes the outward unit normal vector with respect to the domain , and V =H 1 0 ( )\H. The Stokes operator is then dened as in the previous section, i.e., A =P; with the domain D(A) =H 2 ( )\V , whereP is the Leray projector in L 2 ( ) on the space H. Theorem 2.5.1. Assume that (u 0 ; 0 )2D(A)H 1 ( ). Then we have kvk H 2C; t 0 (2.5.1) and krk L 2Ce Ct ; t 0 (2.5.2) for a constant C =C(ku 0 k D(A) ;k 0 k H 1). Proof of Theorem 2.5.1. With = (t) a smooth cut-o function as in the previous section, we have @ t (u) (u) +ur(u) +r(p) = 0 u +e 2 : 54 Using the W 2;4 regularity estimate due to Sohr and Von Wahl [SvW], we get by R t t0 kD 2 uk 4 L 4 .t Z t 0 kD 2 uk 4 L 4 1=4 . Z t 0 kur(u)k 4 L 4ds 1=4 + Z t 0 k 0 uk 4 L 4ds 1=4 + Z t 0 kk 4 L 4ds 1=4 . Z t 0 kuk 4 L 8kruk 4 L 8ds 1=4 + Z t 0 kuk 4 L 4ds 1=2 +t 1=4 . Z t 0 kuk L 2kruk 4 L 2kD 2 uk 3 L 2ds 1=4 + Z t 0 kuk 2 L 2kruk 2 L 2ds 1=4 +t 1=4 .t 1=4 Also, by (2.5.1), we obtain k!(t)k L pC(p); tt 0 ; p2 [2;1): (2.5.3) As in the previous section, the inequalities R t t0 kD 2 uk 4 L 4 .t and (2.5.3) with p = 4 imply Z t t0 kruk L 1ds.t; tt 0 ; and (2.5.2) follows from (2.2.37). 55 Chapter 3 Global regularity for the 3D modied Boussinesq equations 3.1 Introduction In [C1], Chae proposed a modied Navier-Stokes model and addressed the global regularity per- sistence with initial data u 0 2H s (R 3 ) and s> 5=2. The modied Navier-Stokes model reads u t u +R (u!) = 0 ru = 0; where the operatorR = (R 1 ;R 2 ;R 3 ) is the vector of Riesz transforms dened by using the Fourier transform as (R j f)^() = j ijj ^ f(); j = 1; 2; 3: Subsequently, Ye in [Y] proved the global regularity and persistence for the 3D Boussinesq model u t u +R (u!) +RR (e 3 ) = 0 (3.1.1) t +ur = 0 (3.1.2) ru = 0 (3.1.3) 56 in H s (R 3 )H s (R 3 ) for s > 5=2. For the 2D Boussinesq equations, the global existence and persistence of regularity have been topics of high interest since the seminal work of Chae [C] and of Hou and Li [HL], who proved the global existence of a unique solution. Namely, the global persistence holds for (u 0 ; 0 ) in H s H s1 for integers s 3 [HL], while we have the global persistence in H s H s for integers s 3 by [C1]. The persistence in H s H s1 for the intermediate values 1 < s < 3 was then settled. For other results on the global existence and persistence of solutions, see [ACW, CW, KWZ, LPZ, W, WY]. The persistence of regularity in the Sobolev spaces W s;q W s1;q when q6= 2 was studied in [KWZ], where it was proven that the persistence holds if (s 1)q > 2. Later, Kukavica and the author of this thesis addressed the global regularity persistence for the fractional Boussinesq equations in [KW1] and long time behavior of the solutions in [KW2]. In this chapter, we prove that the global Sobolev persistence and the uniqueness still hold with initial data (u 0 ; 0 )2H s H s and s> 3=2. The chapter is organized as follows. In Section 3.2, we introduce basic notations and state the main theorem on the persistence. In Section 3.3, we prove Theorem 3:2:1, while the uniqueness of solutions is obtained in Section 3.4. 3.2 Notation and the main result on global persistence We consider the 3D Boussinesq model (3.1.1){(3.1.3) with the initial condition u(x; 0) =u 0 . The following is the main result of this chapter on the global Sobolev persistence. Theorem 3.2.1. Let s > 3=2, and assume thatku 0 k H s <1 withru 0 = 0 andk 0 k H s < 1. Then there exists a unique solution (u;) to the equations (3.1.1){(3.1.3) such that u 2 C [0;1);H s (R 3 ) \L 2 loc [0;1);H s+1 (R 3 ) and 2C [0;1);H s (R 3 ) . The operator is dened by = () =2 ; 1<< 2; 57 or, using the Fourier transform ( f)^() =jj ^ f() for2R 3 . In the next lemma, we recall the product rule for fractional derivatives. Lemma 3.2.2 (Product estimate). Let s> 0. For all f;g2H s (R 3 )\L 1 (R 3 ), the inequality k s (fg)k L q (R 3 ) Ckfk L q 1 (R 3 ) k s gk L ~ q 1 (R 3 ) +Ck s fk L q 2 (R 3 ) kgk L ~ q 2 (R 3 ) holds, where q 1 ; ~ q 1 ; ~ q 2 2 [q;1] and q 2 2 [q;1) satisfy 1=q = 1=q 1 + 1=~ q 1 = 1=q 2 + 1=~ q 2 and C =C(q 1 ; ~ q 1 ; ~ q 2 ;q 2 ;s). In particular, k s (fg)k L 2 (R 3 ) Ckfk L 1 (R 3 ) k s gk L 2 (R 3 ) +Ck s fk L 2 (R 3 ) kgk L 1 (R 3 ) : For the proof, cf. [KP]. In the following lemma, we recall a version of the Kato-Ponce inequality from [KWZ]. Lemma 3.2.3 ([KWZ]). Let s2 (0; 1). For 1 < q <1 and j2f1; 2; 3g and for f;g2S(R 3 ), the inequality k[ s @ j ;g]fk L q (R 3 ) Ckfk L q 1 (R 3 ) k 1+s gk L ~ q 1 (R 3 ) +Ck s fk L q 2 (R 3 ) kgk L ~ q 2 (R 3 ) holds, where q 1 ; ~ q 1 ; ~ q 2 2 [q;1] and q 2 2 [q;1) satisfy 1=q = 1=q 1 + 1=~ q 1 = 1=q 2 + 1=~ q 2 and C =C(q 1 ; ~ q 1 ; ~ q 2 ;q 2 ;s). In particular, k[ s @ j ;g]fk L 2 (R 3 ) Ckfk L 1 (R 3 ) k 1+s gk L 2 (R 3 ) +Ck s fk L 2 (R 3 ) kgk L 1 (R 3 ) for f;g2S(R 3 ). 58 3.3 Proof of Theorem 3:2:1 In this section, we prove Theorem 3.2.1. Next we establish the global existence and the persistence of regularity, while the uniqueness is shown in the next section. Proof of Theorem 3.2.1(existence). Assume thatku 0 k H s;k 0 k H s . 1, where s > 3=2 is xed. Since s > 3=2, we have H s (R 3 ) L 1 (R 3 ). Next, to get the bound for , we multiply the equation (3.1.2) byjj q2 and integrate the resulting equation obtaining 1 q d dt kk q L q = 0; (3.3.1) where we used Z urjj q2 dx = 0: (3.3.2) Thus, we have the L q conservation property for the density equation (3.1.2), i.e., k(t)k L qk 0 k L q. 1; q2 [2;1]: (3.3.3) By the L 2 a-priori estimates on the velocity equation, the H 1 energy inequality kuk 2 H 1 + Z t 0 kruk 2 H 1dC(t); t 0 (3.3.4) was obtained in [Y], where one can nd more details. Next, we use the W 2;4 regularity estimate due to Sohr and Von Wahl [SvW] to estimate R t 0 kruk L 1d. Fix t 0 > 0 and let : R! [0;1) be a smooth non-decreasing function such that 0 on [0;t 0 =2] and 1 on [t 0 ;1]. Then we have @ t ((t)u) ((t)u) =R ((t)u!)RR ((t)e 3 ) + 0 (t)u: 59 For any t> 0, we get Z t 0 k(s)u(s)k 2 L 4ds 1=2 Z t 0 kR (u!)k 2 L 4ds 1=2 + Z t 0 kRR (e 3 )k 2 L 4ds 1=2 + Z t 0 k 0 (s)uk 2 L 2ds 1=4 . Z t 0 ku!k 2 L 4ds 1=2 + Z t 0 kk 2 L 4ds 1=2 + Z t 0 kuk 2 L 2ds 1=4 . Z t 0 kuk 2 L 8kruk 2 L 8ds 1=2 +t 1=2 +C(t): By the Gagliardo-Nirenberg inequality, we get Z t 0 k(s)u(s)k 2 L 4ds 1=2 . t 3=8 Z t 0 kD 2 uk 5=8 L 2 ds 1=2 +t 1=2 +C(t) .t 7=8 Z t 0 kD 2 uk 2 L 2ds 5=16 +t 1=2 +C(t).C(t); from where Z t t0 kuk 2 L 4dsC(t): Thus, by interpolation, we have kuk L 1.kuk 1=4 L 4 kruk 3=4 L 4 .kruk 1=4 L 4 kuk 3=4 L 4 . kruk 1=4 L 2 kuk 3=4 L 2 1=4 kuk 3=4 L 4 .C(t)kuk 3=16 L 2 kuk 3=4 L 4 : Therefore, we get Z t 0 kuk L 1d.C(t): (3.3.5) 60 Next, for the evolution ofk s uk L 2, we apply the operator s to the equation (3.1.1), multiply by s u, and integrate the resulting equation obtaining 1 2 d dt k s uk 2 L 2+kr( s u)k 2 L 2 = Z s (R (u!)) s udx Z s (RR (e 3 )) s udx =J 1 +J 2 : We apply the Cauchy-Schwarz inequality and the fractional product rule to estimate J 1 .k s (u!)k L 2k s uk L 2. (k s uk L 2k!k L 1 +k s !k L 2kuk L 1)k s uk L 2 Ck!k L 1k s uk 2 L 2 + 1 2 kr( s u)k 2 L 2 +Ckuk 2 L 1k s uk 2 L 2: For J 2 , we apply the Cauchy-Schwarz inequality J 2 k s k L 2k s uk L 2: Thus, using the estimates for J 1 and J 2 above yields d dt k s uk 2 L 2 + 1 C kr( s u)k 2 L 2. (k!k L 1 +kuk 2 L 1)k s uk 2 L 2 +k s k L 2k s uk L 2: (3.3.6) Next, we consider higher order derivatives of the density . We apply the operator s to the equation (3.1.2), multiply by s , and integrate the resulting equation obtaining 1 2 d dt k s k 2 L 2 = Z [ s ;ur] s dxk[ s ;ur]k L 2k s k L 2: (3.3.7) By Lemma 3.2.3, we have k[ s ;ur]k L 2.kk L 1k s+1 uk L 2 +kuk L 1k s k L 2.kr( s u)k L 2 +kuk L 1k s k L 2: 61 Therefore, 1 2 d dt k s k 2 L 2 1 2 kr( s u)k 2 L 2 + (1 +kuk L 1)k s k 2 L 2: Finally, adding (3.3.6) and (3.3.7) yields d dt (k s uk 2 L 2 +k s k 2 L 2) +k s+1 uk 2 L 2. (k!k L 1 +kuk 2 L 1)k s uk 2 L 2 + (1 +kuk L 1)k s k 2 L 2 +k s uk 2 L 2 +k s k 2 L 2: By the Sobolev embedding, (3.3.4), and (3.3.5), we get Z t 0 kuk 2 L 1d. Z t 0 kuk 2 H 2dC(t) and Z t 0 kuk L 1dC(t): We thus conclude the proof by applying the Gronwall inequality. 3.4 Uniquenesss in L 2 (R 3 )L 2 (R 3 ) In this section, we address the uniqueness. Proof of Theorem 3.2.1(uniqueness). Consider two solutions (u (1) ; (1) ) and (u (2) ; (2 ) of the sys- tem (3.1.1){(3.1.3) and set U =u (1) u (2) = (1) (2) =! (1) ! (2) =ru (1) ru (2) : 62 Subtracting the equations for (u (1) ; (1) ) and (u (2) ; (2 ), we get U t U +R (U! (1) ) +R (u (2) ) +RR (e 3 ) = 0 (3.4.1) t +u (2) r +Ur (1) = 0: (3.4.2) We multiply (3.4.1) with U and integrate the resulting equation obtaining 1 2 d dt kUk 2 L 2 +krUk 2 L 2 = Z R (U! (1) ) Udx Z R (u (2) ) Udx Z (RR (e 3 ))Udx =I 1 +I 2 +I 3 : For I 1 , we apply H older's inequality I 1 CkU! (1) k L 2kUk L 2k! (1) k L 1kUk 2 L 2: Similarly, for I 2 we have I 2 ku (2) k L 2kUk L 2ku (2) k L 1k k L 2kUk L 2Cku (2) k L 1krUk L 2kUk L 2 1 2 krUk 2 L 2 +Cku (2) k 2 L 1kUk 2 L 2: For I 3 , we apply the Cauchy{Schwarz inequality I 3 kk L 2kUk L 2: Combining the estimates for I 1 , I 2 , and I 3 gives 1 2 d dt kUk 2 L 2 +krUk 2 L 2 1 2 krUk 2 L 2 + (ku (2) k 2 L 1 +k! (1) k L 1)kUk 2 L 2 +kk L 2kUk L 2: (3.4.3) 63 Next, for the evolution of kk L 2 (R 3 ) , we multiply (3.4.2) with and integrate the resulting equation yielding 1 2 d dt kk 2 L 2 = Z R 3 Ur (1) dxkUk L 6kr (1) k L 3kk L 2 krUk L 2kr (1) k L 3kk L 2 1 2 krUk 2 L 2 +Ckr (1) k 2 L 3kk 2 L 2: (3.4.4) Adding (3.4.3) and (3.4.4), we obtain d dt (kUk 2 L 2 +kk 2 L 2) (ku (2) k 2 L 1 +k! (1) k L 1)kUk 2 L 2 +Ckk L 2kUk L 2 +Ckr (1) k 2 L 3kk 2 L 2: Next, k! (1) k L 1 is integrable in time. Indeed, for any T > 0 and s > 3=2, by the Sobolev embedding we get Z T 0 k! 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Abstract (if available)

Abstract This thesis consists of three main parts. The first part is concerned with the global persistence of Sobolev regularity for the 2D fractional Boussinesq equations. In this chapter, we address the persistence of regularity for the 2D fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces. We prove that the solution exists for all positive time. The second part addresses the long time behavior of solutions to the 2D Boussinesq equations, where we consider long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, the whole space, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity, improving earlier bounds. In the last chapter, we address the global regularity problem for the 3D modified Boussinesq equations. In a recent paper, Ye proved the global persistence of regularity for a 3D Boussinesq model with s > 5/2. We show that the global persistence and uniqueness still hold when s > 3/2.

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Creator Wang, Weinan (author)

Core Title Regularity problems for the Boussinesq equations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 07/06/2020

Defense Date 05/15/2020

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Tag Boussinesq equations,fluid dynamics,global regularity,Navier-Stokes equations,OAI-PMH Harvest,Sobolev spaces

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

Creator Email wangwein@usc.edu,weinanwang2013@gmail.com

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