An Inviscid Regularization for the Surface Quasi-Geostrophic Equation

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1 An Inviscid Regularization for the Surface Quasi-Geostrophic Equation BOUALEM KHOUIDER University of Victoria AND EDRISS S. TITI University of California, Irvine Weizmann Institute of Science Abstract Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite-time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. c 2007 Wiley Periodicals, Inc. 1 Introduction We consider the surface quasi-geostrophic (SQG) equations with periodic boundary conditions on a basic periodic square D Œ0; 1 2 R C div.v/ D 0;. /1=2 D ; r? D v;.x;0/d 0.x/ in ; dx D 0; dx D 0; v dx D 0: Here is the horizontal Laplacian operator and. / 1=2 is 2 pseudodifferential operator defined in the Fourier space by. / 4 1=2 u.k/ D Communications on Pure and Applied Mathematics, Vol. LI, (2008) c 2007 Wiley Periodicals, Inc.

2 1332 B. KHOUIDER AND E. S. TITI jkjou.k/. The first equation in (1.1) describes the evolution of the potential temperature at the surface of the ocean for a quasi-geostrophic flow, i.e., a first-order perturbation of a geostrophically balanced mean state, a state where the horizontal pressure gradient is balanced by the vertical component of the Coriolis force. Details on the derivation of, and more discussion on, the system in (1.1) can be found in [5, 14] and references therein. In (1.1), v represents the incompressible horizontal velocity at the surface and is the stream function. The system of equations in (1.1) is interesting in itself since it models an important geophysical problem. However, it has also been the focus of interesting mathematical work [3, 4, 5, 12], since the evolution of r resembles the evolution of the vorticity in the three-dimensional Euler equations. This is despite the two-dimensional nature of the equation, and the misleading impression that evolves like the vorticity in the two-dimensional Euler equations. Preliminary numerical simulations conducted in [12] revealed that the SQG equations (1.1) with smooth initial data develop sharp fronts in the level contours of and suggested the possibility of formation of a finite-time singularity in r. A more careful simulation conducted in [6, 7] revealed the absence of such singularity and attributed the observation in the simulation of [12] to a growth of the type e et. Indeed, it was rigorously proven in [8] that the scenario of blowup suggested in [12] is not possible. We present here a new inviscid regularization (2.1), inspired by the inviscid simplified Bardina model of turbulence [2] (see also the inviscid version of the Navier-Stokes-Voight model [13]), for the SQG equations (1.1). This new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, for r to blow up as the regularization parameter tends to 0. As opposed to viscous regularizations, used extensively in analytical studies (see, e.g., [3]), the inviscid regularization employed here conserves a modified energy (2.2). In fact, instead of smoothing the solution by dissipating energy at small scales, here the small scales are simply prevented from getting too much energy via a penalty method due to the modified energy (see (2.2) below). Therefore, the new regularization provides a systematic practical procedure for finite-time blowup testing for (1.1). The rest of the paper is organized as follows: The regularized problem is introduced in Section 2 where minimal regularity requirements guarantee the conservation of the modified energy (2.2). In Section 3 we prove long-time existence and uniqueness for the regularized problem (2.1). In Section 4 we prove that if the initial data is smooth (in H m ), then the solution for the regularized problem (2.1) remains smooth (in H m ) for all time. In Section 5, we prove that the regularized solution of (2.1) converges to a weak solution of the SQG equations (1.1) when the regularizing parameter goes to 0. We also prove that if the original SQG equations have a regular (smooth) solution, then the regularized solution necessarily converges strongly to this regular solution. We finally prove in Section 5 that a

3 INVISCID REGULARIATION FOR SQG EQUATION 1333 necessary and sufficient condition, satisfied by the regularized solution of (2.1), for the solution of the SQG equations to blow up in finite time T is sup lim inf kr.t/k L 2 D >0; t2œ0;t /!0 where Œ0; T / is the maximal interval of existence of solutions of (1.1). Numerical tests of this approach will be reported in a forthcoming work. 2 The Regularized Problem Let be a small positive parameter (length scale). Consider the following inviscid regularization of (1.1), subject to periodic boundary conditions: Q C div.v / D 0 in Œ0; T / 1=2 D ;.1 2 / D Q ; r? D v ;.x; 0/ D 0.x/ in ; dx D 0; dx D 0: Unless otherwise stated, the superscript is dropped in Sections 2 through 4 to simplify our presentation. 2.1 Energy Conservation and Minimal Regularity We define the modified energy for the solution of the regularized problem in (2.1) as (2.2) E.t/ D. 2.x;t/C 2jr.x;t/j 2 /dx: It is easy to show that if the solution ;v; for the regularized problem (2.1) is smooth, then the energy E.t/ of the system is conserved. This statement will be made rigorous below in the proof of Theorem 3.1. Moreover, we notice that if is in the Sobolev space H 1./ D fu 2 L 2./ W R.u2 Cjruj 2 /dx < C1g, R then D. / 1=2 belongs to the Sobolev space H 2 Dfu 2 H 1./ W j@ x i ;x j uj 2 dx < C1g, xi ;x j is any derivative of second order. This in turn implies that v Dr? is in H 1./. Therefore, under the periodic boundary conditions, the integral (2.3) div.v/ dx D 1 div.v 2 /dx D 0 2

4 1334 B. KHOUIDER AND E. S. TITI which implies that 0 D d dt D d dt Q.x; t/.x;t/dx. 2.x;t/C 2jr.x;t/j 2 /dx; where Q D 2. This is made rigorous in the following lemma: LEMMA 2.1 Let 2 HP 1./ Dfu 2 H 1./; R udx D 0g and v 2 HP 1./ HP 1./I then div.v/ 2 HP 1./, where HP 1./ is the dual space of HP 1./. Moreover, for v 2 HP 1./ HP 1./ fixed,! div.v/ is a linear continuous operator from HP 1./ to HP 1./. PROOF: Let 2 HP 1./; then jhdiv.v/;ij D ˇ.x/v.x/ r.x/dx ˇ C kvk L 4./ kk L 4./ krk L 2./ (by Hölder s inequality) C kvk 1=2 L 2./ krvk1=2 L 2./ kk1=2 L 2./ krk1=2 L 2./ krk L 2./; from the two-dimensional Gagliardo-Nirenberg-Ladyzhenskaya interpolation inequality (see, e.g., [1, 4, 9, 10]). Therefore kdiv.v/k P H 1 C kvk 1=2 L 2./ krvk1=2 L 2./ kk1=2 L 2./ krk1=2 L 2./ : As a consequence we have the following corollary: COROLLARY 2.2 Let v 2 HP 1./ HP 1./ such that div v D 0 and 2 HP 1./. Then (2.4) hdiv.v/;id0: 3 Global Existence for the Regularized Problem Here we prove that the regularized problem in (2.1) admits a global smooth solution for all time if the initial condition 0 is smooth. More precisely, we have the following theorem:

5 THEOREM 3.1 Let 0 2 initial value problem INVISCID REGULARIATION FOR SQG EQUATION 1335 P H 1./ (i.e., Q 0 D.1 2 / 0 t Q D div.v/; v Dr? ;. / 1=2 D ;.0/ D 0 ;.1 2 / D ; Q dx D 0; has a unique global solution 2 C 1.. 1; C1/; HP 1.// (or Q 2 C 1.. 1; C1/; HP 1.//). P H 1.//I then the PROOF: Given the relations Q D.1 2 /, v Dr?, and. / 1=2 D, we can write div.v/ D F. / Q WD div..r?. / 1=2.1 2 / 1 / Q.1 2 / 1 /: Q Thus, by Lemma 2.1, we have a functional differential equation of the form d (3.1) dt Q D F. /; Q.0/ Q D 0 Q ; in the space HP 1./. We first show short-time existence and uniqueness. For this, it is enough to establish that the functional F. / Q is locally Lipschitz as a map from HP 1./ into HP 1./. Wehave kf. Q 1 / F. Q 2 /k P H 1 Dkdiv. 1 v 1 / div. 2 v 2 /k P H 1 kdiv. 1.v 1 v 2 //k P H 1 Ckdiv /v 2 /k P H 1 C kv 1 v 2 k 1=2 kr.v L 2 1 v 2 /k 1=2 k L 2 1 k 1=2 kr L 2 1 k 1=2 L 2 Ckv 2 k 1=2 krv L 2 2 k 1=2 k L k 1=2 kr. L /k 1=2 L 2 Now we invoke the Poincaré inequality [1, 4, 10, 11], (3.2) kk L 2 1=2 1 krk L HP 1./: Here 1 is the first eigenvalue of with domain D. / D H 2./ \ H P. /, which leads to kf. Q 1 / F. Q 2 /k P H 1 C.kr.v 1 v 2 /k L 2kr 1 k L 2 Ckrv 2 k L 2kr. 1 2 /k L 2/: But, given that the functional operator! v Dr? Œ. / 1=2 is an isomorphism from HP 1 into HP 1 HP 1 and that! Q D.1 2 / is a bounded

6 1336 B. KHOUIDER AND E. S. TITI operator from HP 1 into HP 1, and given the Poincaré inequality (3.2), we have the following norm equivalences: krvk L 2 krk L 2 kkh P kk Q 1 H P : 1 Therefore, kf. Q 1 / F. Q 2 /kh P C.k Q 1 1 Q 2 kh P /.k Q 1 1 kh P Ck Q 1 2 kh P /: 1 Consequently, the functional differential equation (3.1) has short-time existence and uniqueness about t D 0. Suppose Œ0; T / is the maximal positive interval of existence such that Q 2 C 1.Œ0; T /; HP 1.//. To show global existence for (3.1), it is enough to show that the norm kk Q H P 1 stays bounded on the maximal interval of existence. Indeed, we have on Œ0; T / d ; dt Q D 1 d.jj 2 C 2jrj 2 /dx; 2 dt and by virtue of Corollary 2.2, equation (2.4) implies that d. 2.x;t/C 2jr.x;t/j 2 /dx D 0: dt That is, the energy E.t/ defined in (2.2) is indeed conserved. Therefore. 2 C 2jrj 2 /dx D.0 2 C 2jr 0 j 2 /dx (3.3) k 0 k 2 for all 2.0; 1 : H 1 This implies that the L 2 norms of both and its gradient remain bounded. This means that the HP 1 norm of is bounded or, equivalently, the norm of Q in HP 1 is bounded. A similar argument holds for the negative time interval. This concludes the proof of Theorem 3.1. Remark 3.2. It follows from the energy estimate in (3.3) above that we have the following bounds: (3.4) krk L 2 1 k 0k P H 1 ; (3.5) kk L 2 k 0 kh P : 1 Therefore in case the solution for the original problem in (1.1) develops a singularity in finite time, and if this singular weak solution is the limit of the regularized solution when! 0, then we at most expect kr k 1 L 2 D O :

7 INVISCID REGULARIATION FOR SQG EQUATION Higher Regularity In this section we discuss the higher regularity and prove a maximum principle for the regularized problem in (2.1). We start by proving the following regularity result. The idea of the proof is similar to the presentation in [11] for the Euler equations. THEOREM 4.1 (Regularity) Let 0 2 H m./, m 1I then the solution for the regularized problem (2.1) is.t/ 2 C 1 Œ. 1; C1/; HP m (or.t/ Q 2 C 1 Œ. 1; C1/; HP ). PROOF: The case m D 1 follows from Theorem 3.1. For m>1, we proceed as in the proof of Theorem 3.1. We first show local existence and uniqueness in H m ; then we prove that the kk H m norm remains finite for any finite interval of time. It is easy to see that if m 2 then 2 HP m which implies that v 2 HP m HP m div.v/ 2 P H : Q 2 P H./ Indeed, by applying the Gagliardo-Nirenberg and Ladyzhenskaya interpolation inequality, as in Lemma 2.1, we have jd div.v/j 2 dx D jd.v r/j 2 dx C C C kd0 kd0 kd0 jd k v rd k j 2 dx kd k vk 2 L 4./ krdm k 2 k 2 L 4./ kd k vk L 2./krD k vk L 2./ krd m k 2 k L 2./ krrd m k 2 k L 2./ kvk 2 H m./ kk2 H m./ : Moreover, a similar procedure as in the proof of Theorem 3.1 applied to the functional differential equation d Q D F. / dt Q div.v/

8 1338 B. KHOUIDER AND E. S. TITI leads to kf. Q 1 / F. Q 2 /k P H kdiv../v 1 v 2 / 1 k P H Ckdiv.v/ /k P H C k 1 k P H m kv 1 v 2 k P H m Ck 1 2 k P H m kv 2 k P H m : This completes the proof of short-time existence and uniqueness for equation (3.1) in HP. To show global existence in H m, it suffices to prove that ˆ.t/ DkD m 1.t/k 2 L 2./ C 2krD m 1.t/k 2 L 2./ remains bounded in any finite interval of time. We proceed without proof using mathematical induction. The case m D 1 is provided by Theorem 3.1. Assume by induction that 2 C 1 Œ. 1; C1/; H m 1 \ HP 1. If 2 HP m \ HP 1, then D m 1 2 HP 1. Thus D m 1 2 HP 1 and we 1 2 D m 1 /;D m 1 D hd m 1.v r/;d m 1 i; d ˆ.t/ D dt D D 1 2 D m 1 kd0 kd1 C m 1 k D k v r.d m k 1 /D m 1 dx C m 1 k D k v r.d m k 1 /D m 1 dx v r.d m 1 /D m 1 dx kd1 kd1 C m 1 k D k v r.d m k 1 /D m 1 dx div.v.d m 1 / 2 /dx C m 1 k D k v r.d m k 1 /D m 1 dx:

9 INVISCID REGULARIATION FOR SQG EQUATION 1339 Here we used the fact that hdiv.vd m 1 /;D m 1 id0by Corollary 2.2. Therefore, d dt ˆ.t/ C kd k vk L 2 kd m 1 k rk L 4 kd m 1 k L 4 C C kd1 kd1 C. / kvk H k kd m 1 k rk 1=2 krd m 1 k rk 1=2 L 2 L 2 kd m 1 k 1=2 krd m 1 k 1=2 L 2 L 2 kd1 kvk H k/ kk H m 1krk H m 1 kvk H k kd1 ˆ.t/ WD.t/ˆ.t/: Here.t/ D C. /. P kd1 kvk H k/. Therefore, by using Gronwall s lemma we obtain t ˆ.t/ ˆ.0/ exp.s/ds ; which remains bounded on any finite interval of time, since.t/ is bounded by the induction assumption. This completes the proof of the regularity theorem. 5 Weak Convergence and Conditions for Blowup for the SQG In this section we prove the convergence of the solution of the regularized problem (2.1) to a weak solution of the original SQG equation (1.1) as tends to 0. We also show that if the original problem has a regular (smooth) solution, then the solution for the regularized problem converges strongly to this regular solution. Moreover, we show that when these two results are combined with the energy estimate obtained in Section 2.1, a necessary and sufficient condition for the solution of the original SQG equation to blow up on a finite time interval Œ0; T / is that the gradient of the regularized solution satisfies (5.1) sup lim inf 2kr k 2 D >0: Œ0;T /!0 C L 2 Next, the weak convergence of the solution of the regularized problem to a weak solution of the original problem (1.1) is discussed. THEOREM 5.1 Let T > 0 fixed. Then the set of solutions, 0< 1, for the regularized problem (2.1) with initial condition 0 2 HP 1 is weakly compact in L 2..;T //. Moreover, if a subsequence of, >0, converges weakly in L 2..;T // to N 2 L 2..;T // when! 0, then N is a weak solution for the SQG equations (1.1). 0

10 1340 B. KHOUIDER AND E. S. TITI The weak compactness follows directly from the energy estimate in (3.3). It remains to prove that if! N weakly in L 2./, then N is a weak solution for the SQG equation in (1.1). For this purpose we use the following lemma due to Constantin, Cordoba, and Wu [3]. LEMMA 5.2 Let T>0be fixed. The nonlinear map! B.; / D.r?. / 1=2 /r is weakly continuous on L 2./. PROOF: See appendix B of Constantin, Cordoba, and Wu [3] and references therein. PROOF OF THEOREM 5.1: Let.x;t/ be a smooth test function. Let be a sequence of solutions for the regularized SQG equations (2.1) weakly convergent in L 2..;T // to some limit.wehave N T T t dx dt 2 r r T t dx dt D B. ; / dx dt: Observe that ˇ ˇ 2 T rr t dx dt ˇ T 2kr.t/k L 2./ kr t.t/k L 2./dt C k 0 kh P! 0 when! 0: 1 Here we used the energy conservation property derived in the proof of Theorem 3.1 and the first upper bound estimate (3.4) in Remark 3.2. It remains to show that T T (5.2) B. ; / dx dt! B. ; N / N dx dt: Without loss of generality we can assume that the test function is of the form D.t/e ikx : It is shown in appendix B of [3] that the nonlinearity B satisfies (5.3) k. / 1.B. 1 ; 2 / B. 2 ; 2 //k w C k 1 2 k w.1 C log.1 Ck 1 2 k 1 w //.k 1k L 2./ Ck 2 k L 2.//; where kk w is the weak norm given by kk w D sup j2 2 nf0g j.j/j: O Let. / 1 B..t/;.t// D Ob ;k.t/e ikx k2 2 nf0g

11 and We have (5.4) T INVISCID REGULARIATION FOR SQG EQUATION 1341 T D. / 1 B. N.t/; N.t// D k2 2 nf0g.b. ; / B. N ; N // dx dt Ob k.t/e ikx :. / 1.B. ; / B. N ; N // dx dt T Djkj 2. b ;k O b O k /.t/dt: Combining (5.3), (3.5), and the fact that k k w k k L 2,wehave jb ;k O.t/ b O k.t/j C.k 0 kh P ; k.t/k N 1 L 2/: Therefore, (5.2) follows from Lemma 5.2 and the dominated convergence theorem of Lebesgue. Thus T T N t dx dt D B. ; N / N dx dt; i.e., the weak limit N is a weak solution for the original SQG equation in (1.1) on Œ;T. Remark 5.3. We make the following important observation, which provides a sufficient condition for the limiting weak solution for the original SQG equation to blow up in finite time. Recall the energy conservation property, Œ..t// 2 C 2jr.t/ j 2 dx D Œ0 2 C 2jr 0 j 2 dx: If then either sup Œ0;T / lim inf!0 2jr.t/j 2 dx D >0; (i) does not converge in the norm (i.e., does not converge strongly) to N in L 2./, or (ii) N does not conserve energy, i.e., N 2 dx 0 2 dx:

12 1342 B. KHOUIDER AND E. S. TITI Notice, however, that the weak limit N obeys the stability condition kk N L 2 k 0 k L 2: Indeed, 0 k k N 2 Dk k 2 2h ; ick N k N 2 L 2 L 2 L 2 k 0 k 2 C 2kr L 2 0 k 2 2h ; ick N k N 2 : L 2 L 2 When! 0 this yields (because of the weak convergence) 0 k 0 k 2 kk N 2 : L 2 L 2 Now we prove the following strong convergence theorem for the regularized problem to the strong solution of the original SQG equation when this latter exists and is regular enough. This in turn guarantees that we have blowup of the SQG solution if and only if the weak limit in Theorem 5.1 blows up in finite time. THEOREM 5.4 Let the initial condition 0 be in H 2./ \ HP 1./. Let N 2 H 2 \ HP 1 be a regular solution for the original SQG equations in (1.1) on a finite time interval Œ;T ;T > 0. Then the solution of the regularized problem (2.1) converges strongly to N when! 0. More precisely, we have lim k..t/ N.t//k 2 C 2kr..t/ N.t//k 2 D 0 uniformly in Œ;T :!0 L 2 L 2 PROOF: For simplicity in exposition, we restrict the discussion to Œ0; T. Let N.x;t/ 2 C 1 ŒŒ0; T ; H 2 \ HP 1 be a strong solution of the original SQG equation in (1.1) with the given initial data, and let 2 H 2 \ HP 1 be the corresponding solution for the regularized problem (2.1). We N / C div..nv v / / N div..nv v /. N // C div.nv. N // D 0; at least in L 2, according to the proof of Theorem 3.1 and the regularity Theorem 4.1. First note that div.nv v. N //. N /dx D div.nv. N //. N /dx D 0: Therefore, the action of the expression in (5.5) on. N / yields 1 d. 2 dt N / 2 dx 2. N t t /. N /dx C 2 N t. N /dx C Œ.Nv v / r. N N /dx D 0

13 INVISCID REGULARIATION FOR SQG EQUATION 1343 or 1 d 2 dt. N / 2 dx C 2 1 d jr. 2 dt N /j 2 dx 2 r N t r. N /dx C Œ.Nv v / r. N N /dx D 0; i.e., (5.6) 1 d. 2 dt N / 2 dx C 2 1 d jr. 2 dt N /j 2 dx C 2 r div.nv /r. N N /dx C Œ.Nv v / r. N N /dx D 0; which implies (5.7) 1 d. 2 dt N / 2 C 2jr. N /j 2 dx 2kr div.nv N /k L 2 kr. N /k L 2 Ckr N k L 1 k.nv v /k L 2 k. N /k L 2: But 2kr div.nv /k N L 2 kr. N /k L 2 2kr div.nv /k N L 2.krk N L 2 Ckr /k L 2/ 2C N 1 C 1. /; where the upper bound estimate kr k L 2 1 k 0kH P in Remark 3.2 is used. 1 Therefore, 1 d..x;t/ N.x;t// 2 C 2jr..x;t/ 2 dt N.x;t//j 2 dx. / C C kr N.x;t/k L 1k. N.x;t/.x;t//k 2 L 2. / C C kr N.t/k L 1 k. N.t/.t//k 2 L 2 C 2kr. N.t/.t//k 2 L 2 ;

14 1344 B. KHOUIDER AND E. S. TITI which yields..x;t/ N.x;t// 2 C 2ˇˇr..x;t/ N.x;t//ˇˇ2 dx (5.8) t. /T C C kr.s/k N L 1 k..s/ N.s//k 2 L 0 2 C 2kr..s/ N.s//k 2 L ds: 2 Therefore, by using Grönwall s lemma, we have k..t/ N.t//k 2 C 2kr..t/ N.t//k 2 L 2 L 2 T T. / exp C kr.s/k N L 1 ds! 0 when! 0: 0 Finally, we show that the condition anticipated in (5.1) is indeed necessary and sufficient for the original problem to have a singular solution. Therefore, we provide here a systematic and practical procedure relying only on the behavior of the regularized problem (2.1) for detecting the eventual blowup in finite time of the smooth solutions for the SQG equations in (1.1). More precisely, we have the following result: THEOREM 5.5 The solution N of the original SQG problem develops a singularity in its gradient at time t D T if and only if (5.9) sup lim inf 2kr k 2 D >0: Œ0;T /!0 C L 2 PROOF: First we show that the condition is sufficient. Assume that a sequence of solutions for the regularized problem (2.1) converges weakly to a weak solution N 2 L 2./ for the original problem (1.1). As stated in Remark 5.3, if sup lim inf 2kr k 2 D >0; Œ0;T /!0 C L 2 then either does not converge strongly to N or N is not a regular solution. Theorem 5.4, however, guarantees that a regular solution is necessarily a strong limit of. Therefore (5.9) is a sufficient condition for blowup in finite time. Now assume that N 2 HP 1./ is a regular solution for the SQG equations on a maximal interval of existence Œ0; T / such that lim sup krk N L 2 DC1: t!t According to Theorem 5.4, for 0 t<t fixed, we have lim!0 kr.t/ r N.t/k L 2 D 0:

15 INVISCID REGULARIATION FOR SQG EQUATION 1345 Let t be sufficiently close to T so that kr.t/k N L 2 >ı= where ı>0is fixed and >0sufficiently small so that kr.t/ r.t/k N L 2 <ı=2;wehave kr.t/k L 2 kr.t/k N L 2 kr.t/ r.t/k N L 2 > ı 2 : Therefore lim sup lim inf kr k L 2 D >0: t!t!0 Acknowledgments. The work of B.K. was partly supported by a grant from the National Sciences and Engineering Research Council of Canada. The work of E.S.T. was supported in part by National Science Foundation Grant No. DMS , Binational Science Foundation Grant No , and Israel Science Foundation Grant No. 120/06. Bibliography [1] Adams, R. A. Sobolev spaces. Pure and Applied Mathematics, 65. Academic [Harcourt Brace Jovanovich], New York London, [2] Cao, Y.; Lunasin, E. M.; Titi, E. S. Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4 (2006), no. 4, [3] Constantin, P.; Cordoba, D.; Wu, J. On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50 (2001), special issue, no. 1, [4] Constantin, P.; Foias, C. Navier-Stokes equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, [5] Constantin, P.; Majda, A. J; Tabak, E. Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7 (1994), no. 6, [6] Constantin, P.; Nie, Q.; Schörghofer, N. Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241 (1998), no. 3, [7] Constantin, P.; Nie, Q.; Schörghofer, N. Front formation in an active scalar equation. Phys. Rev. E (3) 60 (1999), no. 3, [8] Cordoba, D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. (2) 148 (1998), no. 3, [9] Ladyzhenskaya, O. A. The mathematical theory of viscous incompressible flow. 2nd English ed., revised and enlarged. Mathematics and Its Applications, 2. Gordon and Breach, New York London Paris, [10] Ladyzhenskaya, O. A. The boundary value problems of mathematical physics. Applied Mathematical Sciences, 49. Springer, New York, [11] Majda, A. J.; Bertozzi, A. L. Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, [12] Majda, A. J.; Tabak, E. G. A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). Phys. D 98 (1996), no. 2-4,

16 1346 B. KHOUIDER AND E. S. TITI [13] Oskolkov, P. A. The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Boundary value problems of mathematical physics and related questions in the theory of functions, 7. ap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), [14] Pedlosky, J. Geophysical fluid dynamics. Springer, New York, BOUALEM KHOUIDER EDRISS S. TITI University of Victoria University of California, Irvine Department of Mathematics Department of Mathematics and Statistics Multipurpose Science PO BO 3045 STN CSC & Technology Building Victoria, B.C. V8W 3P4 Irvine, CA CANADA and math.uvic.ca Weizmann Institute of Science Department of Computer Science and Applied Mathematics P.O. Box 26 Rehovot ISRAEL weizmann.ac.il Received January 2007.