Critical SQG in bounded domains

Transcription

1 Criical SQG in bounded domains Peer Consanin and Mihaela Ignaova ABSTRACT. We consider he criical dissipaive SQG equaion in bounded domains, wih he square roo of he Dirichle Laplacian dissipaion. We prove global a priori inerior C α and Lipschiz bounds for large daa.. Inroducion The Surface Quasigeosrophic equaion SQG of geophysical origin [8] was proposed as a wo dimensional model for he sudy of inviscid incompressible formaion of singulariies [5], [9]. While he global regulariy of all soluions of SQG whose iniial daa are smooh is sill unknown, he original blowup scenario of [9] has been ruled ou analyically [3] and numerically [8], and nonrivial examples of global smooh soluions have been consruced [4]. Soluions of SQG and relaed equaions wihou dissipaion and wih non-smooh piece-wise consan iniial daa give rise o inerface dynamics [7], [3] wih poenial finie ime blow up [5]. The addiion of fracional Laplacian dissipaion produces globally regular soluions if he power of he Laplacian is larger or equal han one half. When he linear dissipaive operaor is precisely he square roo of he Laplacian, he equaion is commonly referred o as he criical dissipaive SQG, or criical SQG. This acive scalar equaion [5] has been he objec of inensive sudy in he pas decade. The soluions are ranspored by divergence-free velociies hey creae, and are smoohed ou and decay due o nonlocal diffusion. Transpor and diffusion do no add size o a soluion: he soluion remains bounded, if i sars so []. The space L R is no a naural phase space for he nonlinear evoluion: he nonlineariy involves Riesz ransforms and hese are no well behaved in L. Unforunaely, for he purposes of sudies of global in ime behavior of soluions, L is unavoidable: i quanifies he mos imporan informaion freely available. The equaion is quasilinear and L criical, and here is no wiggle room, nor a known beer smaller space which is invarian for he evoluion. One mus work in order o obain beer informaion. A pleasan aspec of criicaliy is ha soluions wih small iniial L norm are smooh [6]. The global regulariy of large soluions was obained independenly in [] and [] by very differen mehods: using harmonic exension and he De Georgi mehodology of zooming in, and passing from L o L and from L o C α in [], and consrucing a family of ime-invarian moduli of coninuiy in []. Several subsequen proofs were obained please see [] and references herein. All he proofs are dimension-independen, bu are in eiher R d or on he orus T d. The proofs of [] and [] were based on an exension of he Córdoba-Córdoba inequaliy [4]. This inequaliy saes ha Φ fλf ΛΦf poinwise. Here Λ = is he square roo of he Laplacian in he whole space R d, Φ is a real valued convex funcion of one variable, normalized so ha Φ = and f is a smooh funcion. The fracional Laplacian in he whole space has a very singular inegral represenaion, and his can be used o obain. In [] specific nonlinear maximum principle lower bounds were obained and used o prove he global regulariy. A ypical example is Df = fλf Λ f c θ L f 3 poinwise, for f = i θ a componen of he gradien of a bounded funcion θ. This is a useful cubic lower bound for a quadraic expression, when θ L θ L is known o be bounded above. The criical SQG Key words and phrases. SQG, global regulariy, nonlinear maximum principle, bounded domains. MSC Classificaion: 35Q35, 35Q86.

2 PETER CONSTANTIN AND MIHAELA IGNATOVA equaion in R is θ + u θ + Λθ = 3 where u = Λ θ = R θ 4 and =, is he gradien roaed by π. Because of he conservaive naure of ranspor and he good dissipaive properies of Λ following from, all L p norms of θ are nonincreasing in ime. Moreover, because of properies of Riesz ransforms, u is essenially of he same order of magniude as θ. Differeniaing he equaion we obain he sreching equaion + u + Λ θ = u θ. 5 In he absence of Λ his is he same as he sreching equaion for hree dimensional voriciy in incompressible Euler equaions, one of he main reasons SQG was considered in [5], [9] in he firs place. Taking he scalar produc wih θ we obain + u + Λq + Dq = Q 6 for q = θ, wih Q = u θ θ u q. The operaor + u + Λ is an operaor of advecion and fracional diffusion: i does no add size. Using he poinwise bound we already see ha he dissipaive lower bound is poenially capable of dominaing he cubic erm Q, bu here are wo obsacles. The firs obsacle is ha consans maer: he wo expressions are cubic, bu he useful dissipaive cubic lower bound Dq K q 3 has perhaps oo small a prefacor K if he L norm of θ is oo large. The second obsacle is ha alhough u = R θ has he same size as θ modulo consans in all L p spaces < p <, i fails o be bounded in L by he L norm of θ. In order o overcome hese obsacles, in [] and [], insead of esimaing direcly gradiens, he proof proceeds by esimaing finie differences, wih he aim of obaining bounds for C α norms firs. In fac, in criical SQG, once he soluion is bounded in any C α wih α >, i follows ha i is C. More generally, if he equaion has a dissipaion of order s, i.e., Λ is replaced by Λ s wih < s hen if θ is bounded in C α wih α > s, hen he soluion is smooh []. This condiion is sharp, if one considers general linear advecion diffusion equaions, [3]. In [] he smallness of α is used o show ha he erm corresponding o Q in he finie difference version of he argumen is dominaed by he erm corresponding o Dq. In his paper we consider he criical SQG equaion in bounded domains. We ake a bounded open domain R d wih smooh a leas C,α boundary and denoe by he Laplacian operaor wih homogeneous Dirichle boundary condiions and by Λ D is square roo defined in erms of eigenfuncion expansions. Because no explici kernel for he fracional Laplacian is available in general, our approach, iniiaed in [7] is based on bounds on he hea kernel. The criical SQG equaion is θ + u θ + Λ D θ = 7 wih u = Λ D θ = R Dθ 8 and smooh iniial daa. We obain global regulariy resuls, in he spiri of he ones in he whole space. There are quie significan differences beween he wo cases. Firs of all, he fac ha no explici formulas are available for kernels requires a new approach; his yields as a byproduc new proofs even in he whole space. The main difference and addiional difficuly in he bounded domain case is due o he lack of ranslaion invariance. The fracional Laplacian is no ranslaion invarian, and from he very sar, differeniaing he equaion or aking finie differences requires undersanding he respecive commuaors. For he same reason, he Riesz ransforms R D are no specral operaors, i.e., hey do no commue wih funcions of

3 he Laplacian, and so velociy bounds need a differen reamen. In [7] we proved using he hea kernel approach he exisence of global weak soluions of 7 in L. A proof of local exisence of smooh soluions is provided in he presen paper in d =. The local exisence is obained in Sobolev spaces based on L and uses Sobolev embeddings. Because of his, he proof is dimension dependen. A proof in higher dimensions is also possible bu we do no pursue his here. We noe ha for regular enough soluions e.g. θ H he normal componen of he velociy vanishes a he boundary R D θ N = because he sream funcion ψ = Λ D θ vanishes a he boundary and is gradien is normal o he boundary. Le us remark here ha even in he case of a half-space and θ C, he angenial componen of he velociy need no vanish: here is angenial slip. In order o sae our main resuls, le dx = disx, 9 denoe he disance from x o he boundary of. We inroduce he C α space for inerior esimaes: DEFINITION. Le be a bounded domain and le < α < be fixed. We say ha θ C α if θ L and [f] α = supdx α fx + h fx x h α <. The norm in C α is Our main resuls are he following: sup h, h <dx f C α = f L + [f] α. THEOREM. Le θx, be a smooh soluion of 7 on a ime inerval [, T, wih T, wih iniial daa θx, = θ x. Then he soluion is uniformly bounded, sup θ L θ L. <T There exiss α depending only on θ L and, and a consan Γ depending only on he domain and in paricular, independen of T such ha holds. The second heorem is abou global inerior gradien bounds: sup θ C α Γ θ C α 3 <T THEOREM. Le θx, be a smooh soluion of 7 on a ime inerval [, T, wih T, wih iniial daa θx, = θ x. There exiss a consan Γ depending only on such ha sup dx x θx, Γ [sup dx x θ x + ] 4 + θ L 4 x, <T x holds. REMARK. Higher inerior regulariy can be proved also. In fac, once global inerior C α bounds are obained for any α >, he inerior regulariy problem becomes subcriical, meaning ha here is room o spare. This is already he case for Theorem and jusifies hinking ha he equaion is L ineriorcriical. However, we were no able o obain global uniform C α bounds. Moreover, we do no know he implicaion C α C uniformly, and hus he equaion is no L criical up o he boundary. This is due o he fac ha he commuaor beween normal derivaives and he fracional Dirichle Laplacian is no conrolled uniformly up o he boundary. The example of half-space is insrucive because explici kernels and calculaions are available. In his example odd reflecion across he boundary permis he consrucion of global smooh soluions, if he iniial daa are smooh and compacly suppored away from he boundary. The suppor of he soluion remains compac and canno reach he boundary in finie ime, bu he gradien of he soluion migh grow in ime a an exponenial rae. 3

4 4 PETER CONSTANTIN AND MIHAELA IGNATOVA The proofs of our main resuls use he following elemens. Firs, he inequaliy which has been proved in [7] for he Dirichle Λ D is shown o have a lower bound Dfx = fλ D f Λ D f x c f x 5 dx wih c > depending only on. Noe ha in R d, dx =, which is consisen wih. This lower bound valid for general Φ convex, wih c independen of Φ, see 46 provides a srong damping boundary repulsive erm, which is essenial o overcome boundary effecs coming from he lack of ranslaion invariance. The second elemen of proofs consiss of nonlinear lower bounds in he spiri of []. A version for derivaives in bounded domains, proved in [7] is modified for finie differences. In order o make sense of finie differences near he boundary in a manner suiable for ranspor, we inroduce a family of good cuoff funcions depending on a scale l in Lemma 3. The finie difference nonlinear lower bound is Dfx c h θ L fx 3 + c fx 6 dx when f = χδ h θ is large see 48, where χ belongs o he family of good cuoff funcions. Once global inerior C α bounds are obained, in order o obain global inerior bounds for he gradien, we use a differen nonlinear lower bound, Df = fλ D f Λ Df x c fx 3+ α α θ α C α dx α α + c f x dx for large f = χ θ see 6. This is a super-cubic bound, and makes he gradien equaion look subcriical. Similar bounds were obained in he whole space in []. Proving he bounds 6 and 7 requires a differen approach and new ideas because of he absence of explici formulas and lack of ranslaion invariance. The hird elemen of proofs are bounds for RD θ based only on global apriori informaion on θ L and he nonlinear lower bounds on Df for appropriae f. Such an approach was iniiaed in [] and []. In he bounded domain case, again, he mehod of proof is differen because he kernels are no explici, and reference is made o he hea kernels. The boundaries inroduce addiional error erms. The bound for finie differences is ρdfx h δ h RDθx C + θ L dx + h + δ h θx 8 ρ for ρ cdx, wih f = χδ h θ and wih C a consan depending on see 9. The bound for gradien is ρdfx RDθx C + θ L dx + + θx 9 ρ for ρ cdx wih f = χ θ wih a consan C depending on see 7. These are remarkable poinwise bounds clearly no valid for he case of he Laplacian even in he whole space, where Dfx = fx. The fourh elemen of he proof are bounds for commuaors. These bounds 7 [χδ h, Λ D ] θx C h dx θ L, for l dx, see, and [χ, Λ D ] θx C dx θ L, for l dx, see 5, reflec he difficulies due o he boundaries. They are remarkable hough in ha he only price o pay for a second order commuaor in L is dx. Noe ha in he whole space his commuaor vanishes χ =. This nonrivial siuaion in bounded domains is due o cancellaions and bounds on he hea kernel represening ranslaion invariance effecs away from boundaries see 37, 38.

5 Alhough he hea kernel in bounded domains has been exensively sudied, and he proofs of 37 and 38 are elemenary, we have included hem in he paper because we have no found hem readily available in he lieraure and for he sake of compleeness. The paper is organized as follows: afer preliminary background, we prove he nonlinear lower bounds. We have separae secions for bounds for he Riesz ransforms and he commuaors. The proof of he main resuls are hen provided, using nonlinear maximum principles. We give some of he explici calculaions in he example of a half-space and conclude he paper by proving he ranslaion invariance bounds for he hea kernel 37, 38, and a local well-posedness resul in wo appendices.. Preliminaries The L - normalized eigenfuncions of are denoed w j, and is eigenvalues couned wih heir mulipliciies are denoed λ j : w j = λ j w j. I is well known ha < λ... λ j and ha is a posiive selfadjoin operaor in L wih domain D = H H. The ground sae w is posiive and c dx w x C dx 3 holds for all x, where c, C are posiive consans depending on. Funcional calculus can be defined using he eigenfuncion expansion. In paricular β f = λ β j f jw j 4 wih f j = fyw j ydy for f D β = {f λ β j f j l N}. We will denoe by j= Λ s D = s, 5 he fracional powers of he Dirichle Laplacian, wih s and wih f s,d he norm in D Λ s D : f s,d = λ s jfj. 6 I is well-known and easy o show ha Indeed, for f D we have f L = j= D Λ D = H. f fdx = Λ D f L = f,d. 7 We recall ha he Poincaré inequaliy implies ha he Dirichle inegral on he lef-hand side above is equivalen o he norm in H and herefore he ideniy map from he dense subse D of H o D Λ D is an isomery, and hus H D Λ D. Bu D is dense in D Λ D as well, because finie linear combinaions of eigenfuncions are dense in D Λ D. Thus he opposie inclusion is also rue, by he same isomery argumen. Noe ha in view of he ideniy λ s = cs e λ s d, 8 wih = c s e τ τ s dτ, 5

6 6 PETER CONSTANTIN AND MIHAELA IGNATOVA valid for s <, we have he represenaion Λ D s [ f x = c s fx e fx ] s d 9 for f D Λ D s. We use precise upper and lower bounds for he kernel H D, x, y of he hea operaor, e fx = H D, x, yfydy. 3 These are as follows [6],[4],[5]. There exiss a ime T > depending on he domain and consans c, C, k, K, depending on T and such ha c min w x x y, min w y x y, d e x y k 3 H D, x, y C min w x x y, min w y x y, d e x y K holds for all T. Moreover x H D, x, y H D, x, y C holds for all T. Noe ha, in view of H D, x, y = { dx, + x y if dx,, if dx 3 e λ j w j xw j y, 33 j= ellipic regulariy esimaes and Sobolev embedding which imply uniform absolue convergence of he series if is smooh enough, we have ha β H D, y, x = β H D, x, y = e λ j y β w j yw j x 34 for posiive, where we denoed by β and β derivaives wih respec o he firs spaial variables and he second spaial variables, respecively. Therefore, he gradien bounds 3 resul in { y H D, x, y dy C, if dy, H D, x, y + x y, if 35 dy. We also use a bound j= x x H D x, y, C d e x y K 36 valid for cdx and < T, which follows from he upper bounds 3, 3. Imporan addiional bounds we need are x + y H D x, y, dy C e dx K 37 and x x + y H D x, y, dy C e dx K 38 valid for cdx and < T. These bounds reflec he fac ha ranslaion invariance is remembered in he soluion of he hea equaion wih Dirichle boundary daa for shor ime, away from he boundary. We skech he proofs of 36, 37 and 38 in he Appendix.

7 3. Nonlinear Lower Bounds We prove bounds in he spiri of []. The proofs below are based on he mehod of [7], bu hey concern differen objecs finie differences, properly localized or differen assumpions C α. Nonlinear lower bounds are an essenial ingredien in proofs of global regulariy for drif-diffusion equaions wih nonlocal dissipaion. We sar wih a couple lemmas. In wha follows we denoe by c and C generic posiive consans ha depend on. When he logic demands i, we emporarily manipulae hem and number hem o show ha he argumens are no circular. There is no aemp o opimize consans, and heir numbering is local in he proof, meaning ha, if for insance C appears in wo proofs, i need no be he same consan. However, when emphasis is necessary we single ou consans, bu hen we avoid he leers c, C wih or wihou subscrips. LEMMA. The soluion of he hea equaion wih iniial daum equal o and zero boundary condiions, Θx, = H D x, y, dy 39 obeys Θx,, because of he maximum principle. There exis consans T, c, C depending only on such ha he following inequaliies hold: { } dx d Θx, c min, 4 7 for all T, and Θx, C dx 4 for all T. Le < s <. There exiss a consan c depending on and s such ha holds. s Θx, d cdx s REMARK. Λ s D is defined by dualiy by he lef hand side of 4 and belongs o H. Proof. Indeed, Θx, = H D, x, ydy x y dx H D, x, ydy because H D is posiive. Using he lower bound in 3 we have ha x y dx implies w x x y c, and hen, using he lower bound in 3 we obain Inegraing i follows ha w y x y c, H D, x, y cc d e x y k. Θx, cc ω d k d dx k ρ d e ρ dρ. If dx hen he inegral is bounded below by k ρd e ρ dρ. If dx hen ρ implies ha he k exponenial is bounded below by e and so 4 holds. Now 4 holds immediaely from 3 and he upper bound in 3 because he inegral ξ e ξ K dξ < R d 4

8 8 PETER CONSTANTIN AND MIHAELA IGNATOVA if d. Regarding 4 we use and choose appropriaely τ. In view of 4, if hen, when τ T we have and herefore holds. The choice T τ T s Θx, d s Θx, d τ dx τ C Θx,, s Θx, d s τ s dx τ = C τ s T implies 4 provided τ T which is he same as dx C. On he oher hand, Θ is exponenially small if is large enough, so he conribuion o he inegral in 4 is bounded below by a nonzero consan. This ends he proof of he lemma. LEMMA. Le α <. There exiss consan C depending on and α such ha y H D, x, y x y α dy C α 43 holds for T. Indeed, he upper bounds 3 and 35 yield C T dy yh D, x, y x y α dy R + x y d = C 3 α d e x y K x y α dy and, in view of he upper bound in 3, dy w y C and he upper bound in 3, we have dy yh D, x, y x y α dy C 4 R d x y d e x y K x y α dy = C 5 α. This proves 43. We inroduce now a good family of cuoff funcions χ depending on a lengh scale l. LEMMA 3. Le be a bounded domain wih C boundary. For l > small enough depending on here exis cuoff funcions χ wih he properies: χ, χy = if dy l 4, χy = for dy l, k χ Cl k wih C independen of l and χy dy C x y d+j dx j 44 and χy x y d α Cdx α 45 hold for j > d, α < d and dx l. We will refer o such χ as a good cuoff.

9 Proof. There exiss a lengh l such ha if P is a poin of he boundary, and if P y l, hen y if and only if afer a roaion and a ranslaion y d > F y, where y = y,..., y d and F is a C funcion wih F =, F =, F. We ook hus wihou loss of generaliy coordinaes such ha P =, and he normal o a P is,...,,. Now if l < l and dx l and y P l saisfies dy l, hen here exiss a poin Q BP, l such ha x y 6 y Q + dx 6 y Q + dx Indeed, if x P l we ake Q = P because hen x y = x P +P y l l, so x y y Q. Bu also x y dx because here exiss a poin P = p, F p such ha y P = dy l while obviously x P dx l. If, on he oher hand x P < l, hen x is in he neighborhood of P and we ake Q = x. Because y P = y p, y d F p we have dy y d F y dy for y BP, l. We ake a pariion of uniy of he form = ψ + N j= ψ j wih ψ k C Rd, subordinaed o he cover of he boundary wih neighborhoods as above, and wih ψ suppored in dx l 4, idenically for dx l, ψ j suppored near he boundary in balls of size l and idenically on balls of radius l. The cuoff will be aken of he form χ = α + N j= χ j y d F y l α j y, where of course he meaning of y changes in each neighborhood. The smooh funcions χ j z, are idenically zero for z 4 and idenically for z. The inegrals in 44 and 45 reduce o inegrals of he ype y d >F y, y l χ ««y d F y l x y d+j Cldx j Cdx j dy C u χ l du dy R d y Q +dx d+j 9 and y d >F y, y l Cdx α. yχ y d F y l «dy C x y d α χ z dz dy R d y Q +dx d α This complees he proof. We recall from [7] ha he Córdoba-Córdoba inequaliy [4] holds in bounded domains. In fac, more is rue: here is a lower bound ha provides a srong boundary repulsive erm: PROPOSITION. Le be a bounded domain wih smooh boundary. Le s <. There exiss a consan c > depending only on he domain and on s, such ha, for any Φ, a C convex funcion saisfying Φ =, and any f C, he inequaliy Φ fλ s Df Λ s DΦf c fφ dx s f Φf 46 holds poinwise in. The proof follows in a sraighforward manner from he proof of [7] using convexiy, approximaion, and he lower bound 4. We prove below wo nonlinear lower bounds for he case Φf = f, one when f is a localized finie difference, and one when f is a localized firs derivaive. The proof of Proposiion can be lef as an exercise, following he same paern as below.

10 PETER CONSTANTIN AND MIHAELA IGNATOVA THEOREM 3. Le f L be smooh enough C, e.g. and vanish a he boundary, f DΛ s D wih s <. Then Df = fλ s D f Λs D f = γ s d H Dx, y, fx fy dy + γ f x [ s e ] xd = γ s d 47 H Dx, y, fx fy dy + f x Λs D. holds for all x. Here γ = cs wih c s of 9. Le l > be a small number and le χ C, χ be a good cuoff funcion, wih χy = for dy l, χy = for dy l 4 and wih χy C l. There exis consans γ > and M > depending on such ha, if qx is a smooh funcion in L hen if hen fx = χxδ h qx = χxqx + h qx Df = fλ s Dfx Λs Df x γ h s f dx +s q s L + γ f x dx s 48 holds poinwise in when h l 6, and dx l wih { fx, if fx M q L f d x = h dx,, if fx M q L h dx. 49 Proof. We sar by proving 47: fxλ s D fx Λs D f x = c s {[ ] } s fx fxh D, x, yfy f x + H D, x, yf y dy = c s s d { [ HD, x, yfx fy ] [ ]} + f x H D, x, y dy = c s s d { [ HD, x, yfx fy ] dy + f x [ e ] x } = c s s d [ HD, x, yfx fy ] dy + f xλ s D. I follows ha fλ s D f Λs D f x c s ψ τ s d H D, x, yfx fy dy + f xλ s D 5 where τ > is arbirary and ψs is a smooh funcion, vanishing idenically for s and equal idenically o for s. We resric o T, fλ s D f Λs D f x c T s ψ τ s d H D, x, y fx fy dy + f xλ s D 5 and open brackes in 5: fλ s D f Λs D f x f T xc s ψ τ s d H D, x, ydy T fxc s ψ τ s d H D, x, yfydy + f xλ s D fx [ 5 fx Ix Jx] + f xλ s D wih and T Ix = c s ψ s d H D, x, ydy, 53 τ T Jx = c s ψ τ s d H D, x, yfydy T = c s ψ τ s d H D, x, yχyδ h qydy. 54

11 We proceed wih a lower bound on I and an upper bound on J. For he lower bound on I we noe ha, in view of 4 and he fac ha T Ix = c s ψ s Θx, d τ we have mint,d Ix c x ψ τ s d = c τ s τ mint,d x ψuu s du. Therefore we have ha Ix c τ s 55 wih c = c ψuu s du, a posiive consan depending only on and s, provided τ is small enough, τ mint, d x. 56 In order o bound J from above we use 43 wih α =. Now T J c s ψ τ s d δ hh D, x, yχyqydy + T c s ψ τ s d H D, x, y hδ h χyqydy We have ha Indeed, T J = c s ψ s d τ H D, x, y hδ h χyqydy C h 6 dx q L τ s. d e x y K C K x y d so he bound follows from 3 and 45. On he oher hand, T J = c s ψ τ s d δ hh D, x, yχyqydy q L h T ψ τ s d yh D, x, y dy and herefore, in view of 43 T J C h q L ψ 3 s d τ and herefore wih a consan depending only on and s. In conclusion J C 7 h q L τ s 57 C 7 = C ψuu 3 s du Now, because of he lower bound 5, if we can choose τ so ha J C 8 τ s h τ + dx q L. 58 Jx 4 fx Ix hen i follows ha [fλ sdf ΛsDf ] x 4 f xix + f xλ s D. 59

12 PETER CONSTANTIN AND MIHAELA IGNATOVA Because of he bounds 55, 58, if fx 8C 8 c h dx q L, hen a choice τx = C9 q L fx h 6 wih C 9 = c 8C 8 achieves he desired bound. This concludes he proof. We are providing now a lower bound for Df for a differen siuaion. THEOREM 4. Le l > be a small number and le χ C, χ be a good cuoff funcion, wih χy = for dy l, χy = for dy l 4 and wih χy C l. There exis consans γ > and M > depending on such ha, if qx is a smooh funcion in C α wih < α < and hen fx = χx qx, Df = fλ s Dfx Λs Df x γ f d x + q s α s α C α holds poinwise in when dx l, wih { fx, if fx M q L f d x = dx,, if fx M q L dx. dx sα α + γ f x dx s 6 Proof. We follow exacly he proof of Theorem 3 up o, and including he definiion of Ix given in 53. In paricular, he lower bound 55 is sill valid, provided τ is small enough 56. The erm J sars ou he same, bu is reaed slighly differenly, T Jx = c s ψ τ s d H D, x, yfydy T = c s ψ τ s d H 63 D, x, yχy y qy qxdy. In order o bound J we use 45 and 43. T Jx c s ψ τ s d yh D, x, yχyqy qxdy + T c s ψ τ s d H D, x, y χyqy qxdy = J x + J x We have from 3 and 45, as before, T J x = c s ψ s d H D, x, y χyqy qxdy τ Cdx q L τ s. On he oher hand, In view of 43 and so T J x = c s ψ c s dx α q C α +c s q L = J x + J x. τ s d yh D, x, yχyqy qxdy T ψ τ s d x y dx yh D, x, y x y α dy T ψ τ s d x y dx yh D, x, y dy J x C dx α q C α T ψ 3 α s d τ 6 J x C dx α q C α τ α s 64

13 wih C = C ψzz 3 α s dz a consan depending only on and s. Regarding J x we have in view of 35 T J x C q L ψ s τ + e dx K d C3 τ s dx q dx L because, in view of 3 w y x y C dy x y C dy dx on he domain of inegraion. In conclusion Jx C 3 τ s ] [τ α dx α q C α + dx q L. 65 The res is he same as in he proof of Theorem 3: If fx Mdx q L for suiable M, M = 8C 3 c hen we choose τ such ha fx = Mτ α dx α, q C α and his yields fx I 4 Jx, and consequenly, in view of 59 which is hen valid, he resul 6 is proved. We specialize from now on o s =. 4. Bounds for Riesz ransforms We consider u given in 8, u = Λ D θ. We are ineresed in esimaes of u in erms of θ, and in paricular esimaes of finie differences and he gradien. We fix a lengh scale l and ake a good cuoff funcion χ C which saisfies χx = if dx l, χx = if dx l 4, χx Cl, 44 and 45. We ake h l 4. In view of he represenaion we have on he suppor of χ δ h ux = c Λ D = c e d 66 d δ x h x H D x, y, θydy. 67 We spli δ h u = δ h u in + δ h u ou 68 wih ρ δ h ux in = c d δh x x H D x, y, θydy 69 and ρ = ρx, h > a lengh scale o be chosen laer bu i will be smaller han he disance from x o he boundary of : ρ cdx. 7 We represen δ h u in x = u h x + v h x 7 where ρ u h x = c d x Hx, y, χyδ h θy χxδ h θxdy 7 3

14 4 PETER CONSTANTIN AND MIHAELA IGNATOVA and where wih ρ e x = c e x = c ρ e 3 x = c v h x = e x + e x + e 3 x + χxδ h θxe 4 x 73 d d ρ d x H D x + h, y, H D x, y, χyθydy, 74 x H D x + h, y, H D x, y h, χyθydy, 75 x H D x, y, χy + h χyθy + hdy, 76 and ρ e 4 x = c d x H D x, y, dy. 77 We used here he fac ha χθ and χθ + h are compacly suppored in and hence x H D x, y h, χyθydy = x H D x, y, χy + hθy + hdy. The following elemenary lemma will be used in several insances: LEMMA 4. Le ρ >, p >. Then if m, j, m + j >, and ρ ρ m p j e p K d CK,m,j p m 78 e p Kρ K d CK + log + p if m = and j =, wih consans C K,m,j and C K independen of ρ and p. Noe ha when m + j >, ρ = is allowed. We sar esimaing he erms in 73. For e we use he inequaliy 36, and i hen follows from Lemma 4 wih m = d + ha e x C h θ L dλ χydy x + λh y d+ and herefore we have from 44 ha h e x C θ L dx holds for dx l. Concerning e 3 we use Lemma 4 wih m = d and j =, in conjuncion wih 3 and obain e 3 x C h θ L χy x y d dy and herefore we obain from 45 h e 3 x C θ L 8 dx holds for dx l. Regarding e 4 we can spli i ino wih e 5 x = e 4 x = e 5 x + e 6 x ρ x H D x, y, χydy 79 8

15 and e 6 x = ρ x H D x, y, χydy. Now e 6 is bounded using he Lemma 4 wih m = d and j =, in conjuncion wih 3 and 44 and obain χy e 6 x C dy C 8 x y d for dx l, wih a consan independen of l. For e 5 we use he fac ha χ is a fixed smooh funcion which vanishes a he boundary. In order o bound he erms e and e 5 we need o use addiional informaion, namely he inequaliies 37 and 38. For e 5 we wrie e 5 x = ρ d x H D x, y, + y H D x, y, χydy + ρ d H Dx, y, y χydy, and using 37 and Lemma 4 wih m =, j = and 45 we obain he bound e 5 x C + log + ρ dx and herefore, in view of 45 and ρ dx we have + Cρ for dx l, wih C depending on bu no on l. Consequenly, we have χy x y d dy e 5 x C 83 e 4 x C 84 for dx l, wih a consan C depending on only. In order o esimae e we wrie H D x + h, y, H D x, y h, = h x + y H D x + λh, y + λ h, dλ 85 and use 38 and Lemma 4 wih m =, j = o obain e x h θ L dλ ρ d x x + y H D x + λh, y + λ h χy dy C h θ L dλ ρ 3 e dx 4K d and hus h e x C θ L dx holds for dx l. Summarizing, we have ha 5 86 h v h x C θ L dx + C δ hθx 87 for dx l. We now esimae u h using 3 and a Schwarz inequaliy u h x c ρ + x y H D x, y, χδ h θy χδ h θxdy { ρ ρ 3 d } H Dx, y, χδ h θy χδ h θ dy We have herefore u h x C ρdfx. 88 where f = χδ h θ and Df is given in Theorem 3. Regarding δ h u ou we have δ h u ou h x C θ L ρ in view of 36. Puing ogeher he esimaes 87, 88 and 89 we have 89

16 6 PETER CONSTANTIN AND MIHAELA IGNATOVA PROPOSITION. Le χ be a good cuoff, and le u be defined by 8. Then ρdfx h δ h ux C + θ L dx + h + δ h θx ρ holds for dx l, ρ cdx, f = χδ h θ and wih C a consan depending on. Now we will obain similar esimaes for u. We sar wih he represenaion 9 ux = u in x + u ou x 9 where ρ u in x = c x x H D x, y, θydy 9 and ρ = ρx cdx. In view of 36 we have We spli now where and wih and and Now u ou x C ρ θ L 93 u in x = gx + g x + g x + g 3 x + g 4 xfx 94 fx = χx θx 95 ρ gx = c x H D x, y, fy fxdy, 96 ρ g x = c g x = c g 3 x = c ρ ρ x x H D x, y, χyθydy, 97 x x + y H D x, y, χyθydy, 98 x H D x, y, y χyθydy, 99 ρ g 4 x = c x H D x, y, dy. g x C dx θ L holds for dx l because of 36, ime inegraion using Lemma 4 and hen use of 44. For g x we use 38 and hen Lemma 4 o obain g x C dx θ L for dx l. Now g 3 x C dx θ L 3 holds because of 3, Lemma 4 and hen use of 45. Regarding g 4, in view of y H D x, y, dy = 4 we have ρ g 4 x = c x + y H D x, y, dy

17 and, we hus obain from 37 and from Lemma 4 wih m = j = because ρ cdx. Finally we have using a Schwarz inequaliy like for 88 Gahering he bounds we have proved g 4 x C 5 gx C ρdf. 6 PROPOSITION 3. Le χ be a good cuoff wih scale l and le u be given by 8. Then ρdf ux C + θ L dx + + θx ρ holds for dx l, ρ cdx and f = χ θ wih a consan C depending on. We consider he finie difference 5. Commuaors 7 7 δ h Λ D θx = Λ D θx + h Λ D θx 8 wih dx l and h l 6. We use a good cuoff χ again and denoe fx = χxδ h θx. 9 We sar by compuing δ h Λ D θx = Λ D fx + c 3 d H Dx, y, H D x + h, y, χyθydy c 3 d H Dx + h, y, H D x, y h, χyθydy c 3 d H Dx, y, δ h χyθy + hdy = Λ D fx + E x + E x + E 3 x. obeys LEMMA 5. There exiss a consan Γ such ha he commuaor C h θ = δ h Λ D θ Λ D χδ h θ h C h θx Γ dx θ L for dx l, h l 6, f = χδ hθ and θ H L. Proof. We use. For E x we use a similar argumen as for e leading o 8, namely he inequaliy 3 and Lemma 4 wih m = d +, j =, and 44 o obain E x C h dx θ L. For E we proceed in a manner analagous o he one leading o he bound 86, by using 85, 37, Lemma 4 wih m = d +, j =, and 45 o obain For E 3 we use E 3 x h θ L E x C h dx θ L. 3 d and using Lemma 4 wih m = d +, j = and 45 we obain E 3 x C h dx θ L, H D x, y, χy dy

18 8 PETER CONSTANTIN AND MIHAELA IGNATOVA concluding he proof. We consider now he commuaor [, Λ D ]. LEMMA 6. There exiss a consan Γ 3 depending on such ha for any smooh funcion f vanishing a and any x we have [, Λ D ]fx Γ 3 dx f L. 3 If χ is a good cuoff wih scale l and if θ is a smooh bounded funcion in D Λ D, hen obeys C χ θ = Λ D θ Λ D χ θ 4 C χ θx = Λ D θ Λ D χ θx Γ 3 dx θ L 5 for dx l, wih a consan Γ 3 independen of l. Proof. We noe ha and herefore [, Λ D ]fx = c 3 [, Λ D ]fx = c x H D x, y, fy H D x, y, y fy dy 6 3 x + y H D x, y, fydy. 7 The inequaliy 3 follows from 37 and Lemma 4. For he inequaliy 5 we need also o esimae Cx = c s 3 H D x, y, χyθydy by he righ hand side of 5, and his follows from 45 in view of SQG: Hölder bounds We consider he equaion 7 wih u given by 8 and wih smooh iniial daa θ compacly suppored in. We noe ha by he Córdoba-Córdoba inequaliy we have We prove he following uniform inerior Hölder bound: θ L θ L. 8 THEOREM 5. Le θx, be a smooh soluion of 7 in he smooh bounded domain. There exiss a consan < α < depending only on θ L, and a consan Γ > depending on he domain such ha, for any l > sufficienly small holds. θx + h, θx, sup dx l, h l 6, h α θ C α + Γl α θ L 9 Proof. We ake a good cuoff χ used above, h l 6 and observe ha, from he SQG equaion we obain he equaion + u + δ h u h δ h θ + Λ D χδ h θ + C h θ = where C h θ is he commuaor given above in. Denoing as before in 9 f = χδ h θ we have afer muliplying by δ h θ and using he fac ha χx = for dx l, L χ δ h θ + Df + δ h θc h θ = where L χ g = g + u x g + δ h u h g + Λ D χ g and Df is given in Theorem 3.

19 Muliplying by h α where α > will be chosen small o be small enough we obain L δh θx χ h α + h α Df α δ hu δh θx h h α + C h θ δ h θ h α. 3 The facor α comes from he differeniaion δ h u h h α and is smallness will be crucial below. Le us record here he inequaliy 47 in he presen case: valid poinwise, when h l 6 and dx l, where Df γ h θ L δ h θ d 3 + γ dx δ h θ, 4 δ h θ d = δ h θ, h if δ h θx M θ L dx, and δ h θ d = oherwise. We use now he esimaes 9, and a Young inequaliy for he erm involving ρdf o obain L χ δh θx h α +C α θ L dx + ρ +Γ h dx θ L δ h θ h α + h α Df C α h α ρ δ h θ 4 h α δ h θ + C α δ h θ h α δ h θ for dx l, h l 6. Le us choose ρ now. We se { δh θx h θ L, if δ h θx M θ h L dx ρ =, dx, if δ h θx M θ h L dx, 6 where we pu M = M Γ γ +, 7 where M is he consan from Theorem 3, Γ is he consan from and γ is he consan from 4. This choice was made in order o use he lower bound on Df o esimae he conribuion due o he inner piece u h see 7 of δ h u and he conribuion from he commuaor C h θ. We disinguish wo cases. The firs case is when δ h θx M θ h L. Then we have L χ δh θx h α +Γ h dx θ L δ h θ h α. dx + h α Df C [α θ L + + M α θ L The choice of M was such ha, in his case Γ We choose now α by requiring o saisfy and obain from 8 h dx θ L δ hθx h α γ 8 δ hθx 3 h α θ L χ ] δ h θ 3 h α θ L L. 8 ɛ = α θ L 9 C M ɛ + + M ɛ γ 8 δh θx h α for dx l, h l 6, in he case f M θ L h dx h α Df 3

20 PETER CONSTANTIN AND MIHAELA IGNATOVA The second case is when he opposie inequaliy holds, i.e, when δ h θx M θ L dx. Then, using ρ = dx we obain from 5 L χ δh θx h α + h α Df C M ɛ + M + ɛ dx δ hθx h α +Γ dx θ L δ h θ h α γ 8dx δh θx h α + Γ M θ L dx 3 h α. Summarizing, in view of he inequaliies 3 and 3, he damping erm γ dx δ hθx in 4 and he choice of small ɛ in 3, we have ha L χ δh θx h α holds for dx l and h l 6 where + γ 4dx h 3 δh θx h α B 33 B = 6 +α Γ M θ L dx α = Γ γ 4 θ L dx α 34 wih Γ depending on. Wihou loss of generaliy we may ake Γ > 46 α so ha when h l 6. We noe ha L χ δh θx h α δ h θ h α < Γ l α θ L holds for any, x wih dx l and h l 6. We ake δ >, T >. We claim ha, for any δ > and any T > + γ δh θx 4dx h α Γ l α θ L 35 δ h θx sup dx l, h l 6, T h α + δ [ θ C α + Γ l α θ ] L holds. The res of he proof is done by conradicion. Indeed, assume by conradicion ha here exiss T, x and h wih d x l and h l 6 such ha θ x + h, θ x, h α holds. Because he soluion is smooh, we have > + δ [ θ C α + Γ l α θ L ] = R δ h θx, h α + δ θ C α for a shor ime. Noe ha his is no a saemen abou well-posedness in his norm: may depend on higher norms. Also, because he soluion is smooh, i is bounded in C, and δ h θx sup dx l, h l h 6 C on he ime inerval [, T ]. I follows ha here exiss δ > such ha δ h θx sup dx l, h δ h α Cδ α R.

21 In view of hese consideraions, we mus have >, h δ. Moreover, he supremum is aained: here exiss x wih d x l and h such ha δ h l 6 such ha Because of 35 we have ha θ x + h, θ x, h α = s = sup dx l, h l 6 d θ x + h, θ x, d h α < = δ h θ h α > R. and herefore here exiss < such ha s > s. This implies ha inf{ > s > R} = which is absurd because we made sure ha s < R. Now δ and T are arbirary, so we have proved δ h θx sup dx l, h l 6, h α [ θ C α + Γ l α θ ] L 36 which finishes he proof of he heorem. Proof of Theorem. The proof follows from 36 because Γ does no depend on l. For any fixed x we may ake l such ha l dx l. Then 36 implies We ake he gradien of 7. We obain dx α δ hθx, h α [ θ C α + Γ α θ L ] Gradien bounds + u θ + u θ + Λ D θ = where u is he ransposed marix. Le us ake a good cuoff χ. Then g = θ obeys everywhere g + u g + Λ D χg + C χ θ + u g = 38 wih C χ given in 4. We muliply by g and, using he fac ha χx = when dx l we obain L χg + Df + gc χ θ + g u g = 39 when dx l, where L χ is similar o he one defined in : and f = χg. Recall ha Df = fλ D f Λ D f L χg + Df Γ 3 dx g θ L + C for dx l. Using a Young inequaliy we deduce L χ φ = φ + u φ + Λ D χ φ 4. Then, using 5 and 7 we deduce ρdf + θ L dx + + θx g 4 ρ L χ g + Df Γ 3 dx θ L g + C 4 ρg 4 + C 4 θ L dx + g + C 4 g 3 4 ρ for dx l. Now g = f when dx l. If gx M θ L dx hen, in view of 6 Df γ θ α C α g 3+ α α dx α α + γ dx g 43 which is a super-cubic lower bound. We choose in his case ρ = C 5 gx, 44

22 PETER CONSTANTIN AND MIHAELA IGNATOVA and he righ hand side of 4 becomes a mos cubic in g: [ L χ g + Df g 3 Γ 3 M + C 4 + ] θ L C 5 M + C 5 θ L + = K g In view of 43 we see ha L χ g + g 3 γ θ α α C α gx dx α K 46 holds for dx l, if g M θ L dx. In he opposie case, gx M θ L dx we choose and obain from 4 ρx = dx 47 L χ g + Df ] [C dx 3 4 M 4 θ 4 L + C 4M 3 θ 3 L + C 4M θ 3 L + MΓ 3 θ L = K 48 dx 3 and using he convex damping inequaliy 6 Df γ g we obain in his case L χ g + γ g x K dx dx. 49 Puing ogeher 46 and 49 and 9 we obain THEOREM 6. Le θ be a smooh soluion of 7. Then sup θx, C dx l where P θ L is a polynome of degree four. dx [ θ L + P θ L l ] 5 Proof of Theorem. The proof follows by choosing l depending on x, because he consans in 5 do no depend on l. 8. Example: Half Space The case of he half space is ineresing because global smooh soluions of 7 are easily obained by reflecion: If he iniial daa θ is smooh and compacly suppored in = R d + and if we consider is odd reflecion { θ x θ x =,... x d, if x d >, 5 θ x,..., x d if x d < hen he soluion of he criical SQG equaion in he whole space, wih iniial daa θ is globally smooh and is resricion o solves 7 here. This follows because of reflecion properies of he hea kernel and of he Dirichle Laplacian. The hea kernel wih Dirichle boundary condiions in = R d + is Hx, y, = c d where ỹ = y,..., y d, y d. More precisely, e x y 4 e x ey 4 Hx, y, = G d x y [G x d y d G x d + y d ] 5

23 wih x = x,..., x d, and Le us noe ha d G d x = e x π G ξ = x H = H e ξ π x y + y d e x d y d x d y d e x d y d 3 55 We check ha 3 is obeyed. Indeed, because e p p when p i follows ha y d x d y d e e x d y d y d x d y d if x dy d, and if p = x dy d hen In his case, if x d hen y d and hus we obain: We check 37: Firs we have x + y H = and hen y d x d y d e e x d y d e e y d x d y d e. p and pe p is bounded; if x d we wrie y d [ x y x H CH + + ] x d x d +y d G x d + y d G d x y = y d x d + x d x + y Hx, y, dy C e x d Indeed, he only nonzero componen occurs when he differeniaion is wih respec o he normal direcion, and hen xd + yd Hx, y, = c d e x y xd + y d 4 e x d +y d 4 59 Therefore We check 38: firs x + y Hx, y, dy C = C xd ξe ξ 4 dξ = C e x d 4. xd +y d e x d +y d 4 dy d G d x x + y H = x d+y d G x d + y d x y x y xd x + y H = x d+y d G x d + y d G d x y Consequenly x x + y Hx, y, C d + x y + x d + y d e x y 4 e x d +y d 4 6 and 38 follows: x x + y Hx, y, dy C + z e z x dz. d 6 6

24 4 PETER CONSTANTIN AND MIHAELA IGNATOVA We compue Θ and Λ D : Θx, = e x = Hx, y, dy = xd π e x d ξ dξ 63 and herefore 3 e d = 3 d π e x d ξ dξ = 4 x d π. REMARK 3. We noe here ha Λ s D = C sy s d is calculaed by dualiy: Λ s D, φ =, Λs D φ = c s dx s d [ φx Hx, y, φydy] = c s s d [ φxdx Θy d, φydy ] = c s s d Θy d, φydy = cs π φy s d yd e ξ dξ = C s φydy y s d where we used he symmery of he kernel H and 63. We observe ha if we consider horizonal finie differences, i.e. h d = hen C h θ vanishes, and we deduce ha sup h α θx + h, x d, θx, x d, C,α 64 x,h, wih C,α he parial C α norm of he iniial daa. This inequaliy can be used o prove ha u is bounded when d =. Indeed u x, = c x y 3 x ỹ 3 x y θy, dy 65 and he bound is obained using he parial Hölder bound on θ 64 and he uniform bounds θ L p for p =,. The ouline of he proof is as follows: we spli he inegral wih u in x = c and x y δ, x y δ u ou x = c max{ x y, x y } δ u = u in + u ou 66 x y 3 x ỹ 3 x y θy, y, θx, y, dy 67 x y 3 x ỹ 3 x y θy, y, dy 68 where in 67 we used he fac ha he kernel is odd in he firs variable. Then, for u in we use he bound 64 o derive u in δ x C,α C ρ +α dρ = CC,α δ α 69 and for u ou, if we have no oher informaion on θ we jus bound wih some L δ. Boh δ and L are arbirary. L u ou x C log θ L + CL θ δ L 7

25 Finally, le us noe ha even if θ C, he angenial componen of he velociy need no vanish a he boundary because i is given by he inegral y u x,, = c x y + y 3 θy, dy. R Appendix We skech here he proofs of and 38. We ake a poin x, a poin y and disinguish beween wo cases, if d x < x y 4 and if d x x y 4. In he firs case we ake a ball B of radius δ = d x 8 cenered a x and in he second case we ake also a ball B cenered a x bu wih radius δ = d x. We noe ha in boh cases he radius δ is proporional o d x. We ake x B x, δ, we fix y, ake he funcion hz, = H D z, y,, and apply Green s ideniy in he domain U = B,. We obain and hus = U [ s z hz, sg s x z + hz, s s + z G s x z] dzds = hx, G x y + [ ] G s x z B n hz, s hz,s n G sx z H D x, y, = G x y B [ G s x z hz, s n ] hz, s n G sx z We noe ha he x dependence is only via G, and x z is bounded away from zero. We differeniae wice under he inegral sign, and use he upper bounds 3, 3. We have x x H D x, y, x x G x y C B + min{;d y} + min{;d y} d+3 s p 3 x z s e x z 4 s s d e y z Ks dzds d+ B s p x z s e x y 4 s s d+ p y z s e y z Ks d+ B s p x z s e x y 4 s s d dy w y y z dzds y z e Ks dzds where p k ξ are polynomials of degree k. The inegrals are no singular. In boh cases x z δ, and any negaive power s k leaving e x z 8 s can be absorbed by e x z 8 s a he price x z k Cδ k, sill available. Similarly, in he firs case y z x y δ δ and in he second case y z x z x y δ. Any power s k can be absorbed by e y z Ks a he price y z k Cδ k sill leaving e y z Ks available. We noe ha if dy < dx so ha dy < is possible, hen, in view of w 3 we have y y z dy Cδ. We also noe ha view of he fac ha s x z x y + s y z s s we have a bound e x z 8 s y z Ks e x y K wih K = 6 + 4K. Pulling his exponenial ou and esimaing all he res in erms of δ we obain, in boh cases, all he inegrals bounded by Cδ d 4 and herefore we have, in boh cases, x x H D x, y, x x G x y Ce x y K δ d 4 C d e x y K because cδ. This proves 36.

26 6 PETER CONSTANTIN AND MIHAELA IGNATOVA and For 37 and 38 we sar by noicing ha i is enough o prove he esimaes x + y H D x, y, dy C e dx K 7 Bx, dx 4 Bx, dx 4 x x + y H D x, y, dy C e dx K 7 for < cd x. Indeed, if x y dx 4, individual Gaussian upper bounds for up o wo derivaives of H D suffice here is no need for cancellaions. In order o prove 7 and 7 we use a good cuoff χ wih a scale l = dx solves. We ake y Bx, dx 4. Boh x and y are fixed for now. We noe ha he funcion z hz = χzg z y hz, = [ χzg z y + χz G z y] = F z, y,, vanishes for z, and has iniial daum h = χzδz y, so, by Duhamel hz, = e h + e s F sds, which, in view of e fz = H Dz, w, fwdw yields χzg z y = χyh D z, y, + H D z, w, sf w, sdwds for all z, and recalling ha χx = χy =, and reading a z = x we have H D x, y, = G x y + H D x, w, s [ χwg s w y + χw G s w y] dwds. 73 The righ hand side inegral is no singular and can be differeniaed because he suppor of χ is far from he ball Bx, dx 4. Differeniaion x + y cancels he Gaussian G x y. The esimaes of he righ hand side and x + y x x + y H D x, w, sf w, y, sdwds H D x, w, sf w, y, sdwds d+ C e dx K d+ C e dx K for < cd x follow from Gaussian upper bounds. Inegraion dy on he ball B dx 4 picks up he volume of he ball, and hus 7 and 7 are verified.. Appendix We skech here he proof of local wellposedness of he equaion 7. We sar by defining a Galerkin approximaion. We consider he projecors P n n P n f = f j w j 74 wih f j = fxw jxdx. We consider for fixed n he approximae sysem where j= θ n + P n u n θ n + Λ D θ n = 75 u n = Λ D θ n = R Dθ n 76

27 wih P n θ n x, = θ n x, = 7 n θ n,j w j x 77 and wih iniial daa θ n = P n θ where θ is a fixed smooh funcion belonging o H H. Alhough i was wrien as a PDE, he sysem 75 is a sysem of ODEs for he coefficiens θ n,j = θ nw j dx. Le us noe ha P n does no commue wih bu does commue wih and funcions of i. The funcion u n is divergence-free and i is a finie sum of divergence-free funcions, u n x = n j= j= λ j θ n,j w j x. 78 Noe however ha u n / P n L. The normal componen of u n vanishes a he boundary because w j ν =. Moreover, because P n u n θ n θ n dx = u n θ n θ n dx = i follows ha θ n L is bounded in ime and herefore he soluion exiss for all ime. The following upper bound for higher norms is uniform only for shor ime, and i is he bound ha is used for local exisence of smooh soluions. We apply Λ D = o 75 and use he fac ha i is a local operaor, i commues wih P n and wih derivaives: Λ Dθ n + P n un Λ Dθ n u n θ n + Λ Du n θ n + Λ 3 D θ n = 79 We ake he scalar produc wih Λ D θ n. Because his is finie linear combinaions of eigenfuncions, i vansihes a and inegraion by pars is allowed. We obain d d Λ D θ n L + Λ 5 D θ n L Λ D u n L Λ D θ n L θ n L + u n L θ L Λ D θ L We noe now ha Λ Du n = n j= 8 θ n,j λ j w j = Λ D Λ Dθ n = RDΛ Dθ n. 8 Now R D is bounded in L I is in fac an isomery on componens; his follows from 7, herefore The fac ha R D is bounded in L 4 is also rue [9]. Then Moreover, i is known see for insance [] ha in d = we have Λ Du n L Λ Dθ n L. 8 Λ Du n L 4 Λ Dθ n L f L 4 C Λ D f L and herefore θ n L 4 Λ 5 D θ n L. 84 and u n L 4 C Λ 5 D θ n L. 85 Now we use he Sobolev embedding φ L C φ L 4 + φ L 86 and deduce, using also a Poincaré inequaliy d d Λ Dθ n L + Λ 5 D θ n L C Λ Dθ n L Λ 5 D θ L. 87

28 8 PETER CONSTANTIN AND MIHAELA IGNATOVA Thus, afer a Young inequaliy we deduce ha T sup Λ Dθ n L + Λ 5 D θ n L d C Λ Dθ L 88 T holds for T depending only on Λ D θ L, wih a consan independen of n. The following resul can now be obained by assing o he limi in a subsequence and using a Aubin-Lions lemma []: PROPOSITION 4. Le θ H H in d =. There exiss T > a unique soluion of 7 wih iniial daum θ saisfying θ L, T ; H H L, T ; D ΛD Higher regulariy can be obained as well. Because he proof uses L - based Sobolev spaces and Sobolev embedding, i is dimension dependen. A proof in higher dimensions is also possible, bu i requires using higher powers of, and will no be pursued here. Acknowledgmen. The work of PC was parially suppored by NSF gran DMS-9394 References [] L.A. Caffarelli and A. Vasseur, Drif diffusion equaions wih fracional diffusion and he quasi-geosrophic equaion, Ann. of Mah., 73, [] X. Cabre, J. Tan, Posiive soluions of nonlinear problems involving he square roo of he Laplacian, Adv. Mah. 4, no. 5, [3] A. Casro, D. Córdoba, J. Gomez-Serrano and A. Marin, Remarks on geomeric properies of he SQG sharp fron and he alpha-paches, Discree Coninuos Dynamical Sysems 34, [4] A. Casro, D. Córdoba, J. Gomez-Serrano, Global smooh soluions for he inviscid SQG equaion wih. Preprin arxiv: [5] P. Consanin, Geomeric saisics in urbulence, SIAM Review 36, [6] P. Consanin, D. Córdoba, and J. Wu. On he criical dissipaive quasi-geosrophic equaion, Indiana Univ. Mah. J., 5Special Issue:Dedicaed o Professors Ciprian Foias and Roger Temam Bloomingon,IN,., [7] P. Consanin, M. Ignaova, Remarks on he fracional Laplacian wih Dirichle boundary condiions and applicaions, IMRN, 6. [8] P. Consanin, M.-C. Lai, R. Sharma, Y.-H. Tseng, and J. Wu, New numerical resuls for he surface quasi-geosrophic equaion, J. Sci. Compu., 5, 8. [9] P. Consanin, A.J. Majda, and E. Tabak, Formaion of srong frons in he -D quasigeosrophic hermal acive scalar, Nonlineariy, , [] P. Consanin, V. Vicol, Nonlinear maximum principles for dissipaive linear nonlocal operaors and applicaions, GAFA [] P. Consanin, A. Tarfulea, V. Vicol, Long ime dynamics of forced criical SQG, Communicaions in Mahemaical Physics 335 5, no., [] P. Consanin and J. Wu, Regulariy of Hölder coninuous soluions of he supercriical quasi-geosrophic equaion, Ann. Ins. H. Poincaré Anal. Non Linéaire, [3] D. Córdoba, Nonexisence of simple hyperbolic blow-up for he quasi-geosrophic equaion, Ann. of Mah., [4] A. Córdoba, D. Córdoba, A maximum principle applied o quasi-geosrophic equaions. Comm. Mah. Phys. 49 4, [5] D. Córdoba, M.A. Fonelos, A.M. Mancho, and J.L. Rodrigo, Evidence of singulariies for a family of conour dynamics equaions, Proc. Nal. Acad. Sci. USA, 7 5, [6] E.B. Davies, Explici consans for Gaussian upper bounds on hea kernels, Am. J. Mah [7] C. Fefferman and J.L. Rodrigo. Analyic sharp frons for he surface quasi-geosrophic equaion. Comm. Mah. Phys., 33, [8] I.M. Held, R.T. Pierrehumber, S.T. Garner, and K.L. Swanson, Surface quasi-geosrophic dynamics, J. Fluid Mech., 8 995,. [9] D. Jerison, C. Kenig, The inhomogeneous Dirichle problem in Lipschiz domains, J. Func. Analysis 3 995, 6-. [] A. Kiselev, F. Nazarov, and A. Volberg, Global well-posedness for he criical D dissipaive quasi-geosrophic equaion, Inven. Mah., 673 7, [] J.L. Lions, Quelque mehodes de résoluion des problèmes aux limies non linéaires, Paris, Dunod 969. [] S.G. Resnick, Dynamical problems in non-linear advecive parial differenial equaions, ProQues LLC, Ann Arbor, MI, 995, Thesis Ph.D. The Universiy of Chicago.

29 9 [3] L. Silvesre, V. Vicol, and A. Zlaoš, On he Loss of Coninuiy for Super-Criical Drif-Diffusion Equaions, Arch. Raion. Mech. Anal., [4] Q. S. Zhang, The boundary behavior of hea kernels of Dirichle Laplacians, J. Diff. Eqn 8, [5] Q. S. Zhang, Some gradien esimaes for he hea equaion on domains and for an equaion by Perelman, IMRN 6, aricle ID934, -39. DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ address: cons@mah.princeon.edu DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ address: ignaova@mah.princeon.edu