Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces

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1 Persistence of regularity for solutions of the Boussines euations in Sobolev spaces Igor Kukavica, Fei Wang, an Mohamme Ziane Saturay 9 th August, 05 Department of Mathematics, University of Southern California, os Angeles, CA s: kukavica@usc.eu, wang88@usc.eu, ziane@usc.eu Abstract We aress the global regularity of solutions to the Boussines euations with zero iffusivity in two spatial imensions. Previously, the persistence in the space H +s (R ) H s (R ) for all s 0 has been obtaine. In this paper we aress the persistence in general Sobolev spaces, establishing it on a time interval which is almost inepenent of the size of the initial ata. Namely, we prove that if (u 0, ρ 0) W +s, (R ) W s, (R ) for s (0, ) an [, ), then the solution (u(t), ρ(t)) of the Boussines system stays in W +s, (R ) W s, (R ) for t [0, T ], where T epens logarithmically on the size of initial ata. If we furthermore assume that s >, then we get the global persistence in the space W +s, (R ) W s, (R ). Moreover, we prove the global persistence in the space W +s, (R ) W s, (R ) for the initial ata with compact support, as well us for ata in W +s, (T ) W s, (T ), without any restriction on s (0, ) an [, ). Mathematics Subject Classification: 35K55, 35M33, 76B03, 76D05 Keywors: Boussines euations, commutator estimate, Kato-Ponce type ineualities, global well-poseness Introuction In this paper, we aress the persistence of regularity for the D Boussines euations with zero iffusivity in general Sobolev spaces. u t + u u u + p = ρe (.) iv u = 0 (.) ρ + u ρ = 0 (.3) t Here, u is the velocity solving the D Navier-Stokes euation ([CF, DG, FMT, R, T, T3]) riven by ρ, which represents the ensity or temperature of the flui, epening on the physical context, an e = (0, ) T. This system arises in several ifferent physical scenarios. One situation is the limiting case of small iffusion Rayleigh-Bénar problem, where ρ represents the temperature. Also, the Boussines system serves as a simplifie moel for the 3D Navier-Stokes euations since it incorporates a vortex stretching effect.

2 Due to the smoothing effect, the Boussines system with viscosity an non-zero iffusion ρ + u ρ = κ ρ (.4) t replacing (.3) is easier to treat than the same system without the iffusion. The global well-poseness of the system (.) (.3) with (.3) replace by (.4), where κ > 0, was obtaine in [CD]. In 006, Chae [C] prove the global existence an uniueness of solutions to the system (.) (.3), proving that if (u 0, ρ 0 ) H m (R ) H m (R ), then the solution (u, ρ) stays in H m (R ) H m (R ) for all t > 0, where m is an integer greater than or eual to 3. The main iea in the proof was to show that T 0 ρ t <. At the same time, Hou an i [H] prove the persistence of regularity for the solutions in H m (R ) H m (R ) also for integer m greater than or eual to 3, with the key ingreient in their proof being an upper boun for u. The persistence results in the class H s (R ) H s (R ) for a larger range of s were obtaine more recently. For s =, one may refer to [DP, T], in which the global existence an uniueness results were proven for the two-imensional non-iffusive Boussines system with viscosity only in the horizontal irection. Recently, the case s (, 3) was resolve in [HKZ, HKZ]. The key step in the first paper was an estimate for B(u, v) while a borerline commutator estimate for [Λ s j, v]ρ (R ) was obtaine in the secon paper. For further results on the well-poseness an persistence, we refer the reaer to [ACW, BS, BrS, CG, CR, CN, CW, ES, HK, HK, HKR, KTW, PZ, T, W]. Note that the global existence for another extreme case when (.) is replace by is still an open problem. u t + u u + p = ρe (.5) In this paper, we consier the persistence of the system (.) (.3) in the Sobolev space W +s, (R ) W s, (R ) where s (0, ) an [, ). In comparison with the base result, we are face with the following ifficulties. First, the velocity oes not nee to belong to (R ) an thus we can not use the energy ineuality. In orer to avoi this problem, we couple the p estimate for the velocity an that of the vorticity. Secon, the available commutator estimates an the Kato-Ponce ineuality are inaeuate for our purpose. What we nee here is the estimate for [Λ s j, g]f (R ). As in [KP], we write this commutator as a Coifman-Meyer operator in the freuency regions ξ η an ξ η an use the complex interpolation in the remaining region. Also, the Brezis-Gallouet an relate Beale-Kato-Maja ineualities ([BKM]) o not apply in our situation. Instea we prove (cf. emma 4. below). Λu C ( + log( + Λ s ω r) ) +/ ( + ω + ( ω / ) / ) (.6) Since the power of the logarithm is too high, we may only obtain a local persistence result, with the existence time epening on the initial ata logarithmically. As we may see, a huge ifficulty to get a global result is the estimate of ω. If we furthermore assume that s >, then we may prove ω(t) C + t (.7) (cf. emma 6.), allowing us to prove the global existence in W +s, (R ) W s, (R ). In Theorem 6.3 below we also obtain the global persistence in the intersection space (W +s, (R ) W s, (R )) (H +s (R ) H s (R )) where s (0, ) an [, ). As a corollary of this result, we obtain

3 the global persistence in the space W +s, (R ) W s, (R ) for the initial ata with compact support. The global existence in W +s, (T ) W s, (T ) is also establishe in Theorem 7. below. On almost persistence In this section we prove that if the initial ata (u 0, ρ 0 ) belongs to W +s, (R ) W s, (R ) where s (0, ) an (, ), then (u(t), ρ(t)) W +s, (R ) W s, (R ) for t [0, T ), where T epens logarithmically on the size of the initial ata. intersection space (W +s, (R ) W s, (R )) (H +s (R ) H s (R )). Further below we prove the global persistence in the Before stating the first main theorem, we assert the global persistence in W, (R ) (R ). Proposition.. Assume that, for some, we have (u 0, ρ 0 ) W, (R ) (R ) with iv u 0 = 0. Then there exists a uniue solution (u, ρ) to the euations (.) (.3) such that u C ( [0, ), W, (R ) ) an ρ C ( [0, ), (R ) ). Moreover, with ω = u, there exists C epening on u 0 W, an ρ 0 such that u(t) C t +, t [0, ) (.) an with t 0 ( ω / ) s C( + t)/ (.) ρ(t) = ρ 0, t [0, ) (.3) hol. The following is our first main statement. Theorem.. et s (0, ) an (, ). Assume that u 0 W +s, M 0 with iv u 0 = 0 an ρ 0 W s, M, where M 0, M are constants. Then there exists a constant C epening on u 0 W, an ρ 0 such that for {( ) /3 T = min +, C( + log / ( + M 0 + M )) C( + log / ( + M 0 + M ))), } /( ), (.4) C( + log / ( + M 0 + M ))) /( ) there exists a uniue solution (u, ρ) to the euations (.) (.3) such that u C ( [0, T ), W +s, (R ) ) an ρ C ( [0, T ), W s, (R ) ). It is clear from the proof below that we can take C to be proportional to u 0 W, + ρ 0. Proof of Proposition.. Since the case = is covere in [T], we assume >. We start by multiplying the euation (.) with u u an integrating it with respect to x obtaining t u + (u u) u u x u u u x + p u u x = ρe u u x. (.5) 3

4 Due to the ivergence-free conition for u, we have (u u) u u x = 0 while integrating by parts, we obtain u u u x = j u k j u k u x + ( ) (u k j u k )(u l j u l ) u 4 x, (.6) where the summation convention on repeate inices is use throughout. Denoting the right sie of the above euality by D 0, we get from (.5) t u + D 0 = p u u x + p u ρe u u x + ρ u. (.7) Noting that D 0 0, we only nee to estimate the pressure term. We take the ivergence of the euation (.) an obtain p = iv (u u) ρ. (.8) Writing p = ( ) ( iv (u u) ρ ), (.9) we may apply the Calerón-Zygmun theorem in orer to get p C( u u + ρ ) C( u u + ρ ) C( u W, u + ρ ) C( u + u ) u + C ρ C( u + ω ) ω + C ρ, (.0) where ω = u is the vorticity an where the constant epens on (consiere fixe). In orer to get the boun for ρ, we multiply the euation (.3) by ρ ρ an integrate with respect to x to get where we use Therefore, the norm of ρ is preserve by this system, i.e., Combining (.7) with (.0) an (.3) leas to Taking the curl of the euation (.), we get t ρ = 0, (.) u ρ ρ ρ x = 0. (.) ρ(t) = ρ 0, t > 0. (.3) t u + D 0 C( u ω + ω + ) u. (.4) ω t + u ω ω = (ρe ) = ρ. (.5) We multiply both sies of the euation (.5) by ω ω an integrate it to get an estimate of ω, ω t ω ω x + u ω ω ω x ω ω ω x = ρ ω ω x. (.6) 4

5 Due to the ivergence-free conition, we have u ω ω ω x = 0. Integrating by parts the thir term on the left sie of (.6), we arrive at t ω + ( ) By Höler s ineuality an using ( ) ( ω / ) = ( ) ρ ω ω x. (.7) ω ( )/ ω = ( ω / ), (.8) we have ρ ω ω x C ρ ( ω / ) ω ( )/. (.9) We apply the ineuality (.9) to the euation (.7) an use Young s ineuality to get t ω + ( ) where, recall, the constant C epens on. ( ) ( ) ( ω / ) ( ) ( ω / ) + C ρ ω, (.0) Absorbing the first term on the right sie of the above ineuality, we get t ω + C ( ω / ) C ρ ω. (.) The ineuality (.), combine with (.3), implies as well as ω(t) C + t, t [0, ), (.) t 0 ( ω / ) s C( + t)/, (.3) where C epens on the initial ata u 0 W, an ρ 0. Using (.) in (.4) an applying the Gronwall s ineuality then conclues the proof. 3 A commutator lemma The main step in the proof of Theorem. is a commutator estimate, which is state next. et Λ = ( ) /. emma 3.. et s (0, ) an f, g S(R ). For < < an j {, }, [Λ s j, g]f (R ) C f r Λ +s g r + C Λ s f r Λg r (3.) hols, where r, r, r [, ] an r [, ) satisfy / = /r + / r = /r + / r, an where C is a constant epening on r, r, r, r, s, an. 5

6 Recall that [Λ s j, g]f = Λ s j (gf) gλ s j f. (3.) This lemma can be seen as an extension of the Kato-Ponce ineuality since we allow r =. The proof uses the ieas from [KP]. Proof. Taking the Fourier transform of [Λ s j, g]f, we get ([Λ s j, g]f)ˆ(ξ) = i ( ξ s ξ j ξ η s (ξ η) j ) ˆf(ξ η)ĝ(η) η. R (3.3) Therefore, [Λ s j, g]f =c 0 e ix ξ ( ξ s ξ j ξ η s (ξ η) j ) ˆf(ξ η)ĝ(η) η ξ =c 0 e ix (ξ+η) ( ξ + η s (ξ + η) j ξ s ξ j ) ˆf(ξ)ĝ(η) η ξ 3 =c 0 e ix (ξ+η) ( ξ + η s (ξ + η) j ξ s ξ j ) ˆf(ξ)ĝ(η)Φ k ( ξ /) η ξ (3.4) k= where c 0 = i/4π, an where Φ k : R [0, ] are C cut-off functions such that 3 Φ k = on [0, ) (3.5) k= with Denote supp Φ [ /, /], supp Φ [/4, 3], supp Φ 3 [, ). (3.6) A k (ξ, η) = ( ξ + η s (ξ + η) j ξ s ξ j ) ˆf(ξ)ĝ(η)Φ k ( ξ /), k =,, 3. (3.7) The commutator (3.) may be rewritten as 3 [Λ s j, g]f = c 0 k= e ix (ξ+η) A k (ξ, η) η ξ. (3.8) First, we write A as A (ξ, η) = ξ + η s (ξ + η) j ξ s ξ j +s Φ ( ) ξ ˆf(ξ)(Λ +s g)ˆ(η) =σ (ξ, η) ˆf(ξ)(Λ +s g)ˆ(η). (3.9) It is easy to check that σ C, (3.0) where C is a constant an where we use ξ on supp Φ. Taking the first erivative with respect to ξ l, where l =,, we obtain σ ξ l C C ξ + (3.) 6

7 while ifferentiating with respect to η l for l =, leas to σ η l C C ξ +. (3.) Continuing by inuction, we get α ξ β η σ C( α, β )( ξ + ) ( α + β ), α, β N 0. (3.3) By the Coifman-Meyer theorem, we get e ix (ξ+η) A (ξ, η) η ξ C f r Λ +s g r (3.4) where / = /r + / r. Postponing the treatment of A, we eal with A 3 next. We rewrite this symbol as ( ) ξ A 3 (ξ, η) = ξ + η s η j Φ 3 ( ) ξ + η s ξ =i ξ s Φ 3 (Λ s f)ˆ(ξ)( j g)ˆ(η) + ( ξ + η s ξ s )ξ j ξ s Φ 3 ( ) ξ ˆf(ξ)ĝ(η) + ( ξ + η s ξ s )ξ j Φ 3 ˆf(ξ)ĝ(η) ( ξ =σ 3, (ξ, η)(λ s f)ˆ(ξ)( j g)ˆ(η) + ( ξ + η s ξ s )ξ j ξ s Φ 3 ( ξ ) (Λ s f)ˆ(ξ)ĝ(η) ) (Λ s f)ˆ(ξ)ĝ(η). (3.5) Note that in the region Φ 3 > 0, we have ξ >. For σ 3 it is easy to check that α ξ β η σ 3 C( ξ + ) ( α + β ), α, β (Z + ). (3.6) In orer to estimate the secon term on the far right sie of (3.5), we write ξ j ( ξ + η s ξ s ) =ξ j which implies that the secon term may be rewritten as A irect computation shows that ( ) ξ iφ 3 (Λ s f)ˆ(ξ)( k g)ˆ(η) 0 r ( ξ + rη s ) r =ξ j s ξ + rη s (ξ + rη) η r, (3.7) 0 0 s ξ + rη s (ξ k + rη k )ξ j ξ s = σ 3 (ξ, η)(λ s f)ˆ(ξ) ( g)ˆ(η). (3.8) α ξ β η σ 3 (ξ, η) C( ξ + ) ( α + β ), α, β N 0. (3.9) r Therefore, we obtain that e ix (ξ+η) A 3 η ξ C Λ s f r g r, (3.0) 7

8 where / = /r + / r. Finally we treat the secon term of the sum on the right sie of (3.8), in which case ξ an are comparable. Note that ( ) ξ A (ξ, η) = ξ + η s (ξ + η) j Φ ( ) ξ ˆf(ξ)ĝ(η) ξ s ξ j Φ ˆf(ξ)ĝ(η) = A A. (3.) First we eal with the simpler term A, which may be written as ( ) A (ξ, η) = ξ s ξ j ξ ξ s Φ (Λ s f)ˆ(ξ)(λg)ˆ(η). (3.) Applying the Coifman-Meyer theorem we get e ix (ξ+η) A (ξ, η) η ξ C Λ s f r Λg r. (3.3) In orer to conclue the lemma we only nee to get a similar estimate for A, to which the above metho oes not apply. The main reason is that when we take the erivative, the factor of ξ + η appears in the enominator. Thus, as in [KP], we use the complex interpolation to avoi this ifficulty. First, we have e ix (ξ+η) A (ξ, η) η ξ C(b) Λ s f r Λg r, (3.4) where s = a + ib, with a a sufficiently large positive number an b R, an where C(b) is a polynomial. Next we consier the case s = ib with b R. In orer to get an estimate for e ix (ξ+η) A η ξ, we rewrite e ix (ξ+η) A (ξ, η) η ξ = = = e ix (ξ+η) ξ + η s (ξ + η) j ˆf(ξ)ĝ(η)Φ η ξ e ix (ξ+η) ξ + η s (ξ + η) j e ix ξ ξ s ˆf(ξ)(Λg)ˆ(η)Φ η ξ ξj ˆf(ξ η)(λg)ˆ(η)φ η ξ. (3.5) Using the Hörmaner-Mikhlin multiplier theorem for the symbol ξ s = ξ ib, we have e ix (ξ+η) A (ξ, η) η ξ C(b) e ix ξ ξj ˆf(ξ η)(λg)ˆ(η)φ η ξ = C(b) e ix (ξ+η) (ξ + η) j Φ ˆf(ξ)(Λg)ˆ(η) η ξ. (3.6) Since ( ( )) (ξ + ξ α η β η)j ξ Φ C( ξ + ) ( α + β ), α, β N 0, (3.7) using the Coifman-Meyer theorem, we have e ix (ξ+η) A (ξ, η) η ξ C(b) f r Λg r. (3.8) 8

9 Thus, using the complex interpolation ineuality, we get e ix (ξ+η) A (ξ, η) η ξ C(b) Λ s f r Λg r (3.9) for any s such that Re(s) (0, ). Taking s (0, ), since Im(s) = 0, we have e ix (ξ+η) A (ξ, η) η ξ C Λ s f Λg r (3.30) r an the proof is complete. 4 An boun The following statement provies bouns on the velocity an the vorticity in terms of norms of the erivatives of the vorticity. emma 4.. Assume that u S(R ) is a vector fiel satisfying iv u = 0. et 0 < s < an r be such that rs >. For <, with ω = u, the ineualities an hol, where C = C(r, s, ). Λu C ( + log( + Λ s ω r) ) +/ ( + ω + ( ω / ) / ) (4.) ω C ( + log( + Λ s ω r) ) / ( + ω + ( ω / ) / ) (4.) Proof. First we prove that f C( f b + Λ s f r), b, f S(R ) (4.3) where the constant C epens only on r an s, i.e., it is inepenent of b. Without loss of generality, we may assume f b + Λ s f r =. Using the stanar ittlewoo-paley notation, we have f S 0 f + By Bernstein s ineuality, we boun the first term on the right sie as j f. (4.4) j=0 S 0 f C S 0 f b C. (4.5) In orer to estimate the secon term on the right sie of (4.4), we write j f C j/r j f r C j/r js Λ s j f r C j(/r s), (4.6) where we use Bernstein s ineuality in the first step. Since rs >, we may sum up the terms an obtain f C, thus leaing to (4.3). By rescaling (4.3), we obtain the Gagliaro-Nirenberg ineuality f C f a b Λ s f a r (4.7) 9

10 where a = + b(s /r). (4.8) Applying the above ineuality to Λu with b to be etermine, we get Λu C Λu a Λ +s u a b r Cb ω a Λ s ω a b Cb( + ω r b) Λs ω a r, (4.9) where the parameters r, s, an are fixe, while we nee to track the epenence of the constants on b. Also, we have ω b = ω / / C b/ Using (4.0) in (4.9), we get ( )/ b ω / /b ( ω / ) /b Cb / ( ω / + ( ω / ) ) / Cb / ( ω + ( ω / ) / ). (4.0) Λu Cb +/ ( + ω + ( ω / ) / ) Λ s ω a r. (4.) Therefore, by setting b = + log( + Λ s ω r), we obtain Λu C ( + log( + Λ s ω r) ) +/ ( + ω + ( ω / ) / ). (4.) Inee, note that Λ s ω a = exp(a log r Λs ω r), which is boune by a constant by our choice of b an (4.8). Thus (4.) is establishe. Similarly to (4.9), we get Repeating the above proceure, we get which gives (4.). ω C ω a Λ s ω a b r. (4.3) ω C ( + log( + Λ s ω r) ) / ( + ω + ( ω / ) / ), (4.4) 5 The proof of Theorem. We are now reay to prove Theorem.. Since the persistence in W, (R ) (R ) has been aresse in Proposition., we focus on the estimate of the highest erivatives of u ρ, namely on Λ s ω Λ s ρ. Since the cancellation property is not available, we obtain an extra term involving Λu. Using results in Section 4, we may boun it by log +/ ( + Λ s ω r)( + ω + ( ω / ) / ), leaing to logarithmic epenence on the norms. Note that since +/ >, we o not obtain the global existence (note, however, the global results in Sections 6 an 7). Proof of Theorem.. Applying the operator Λ s to the euation (.5), multiplying it with Λ s ω Λ s ω, an integrating in x, we get t Λs ω Λ s ω Λ s ω Λ s ω x + Λ s (u ω) Λ s ω Λ s ω x = Λ s ρ Λ s ω Λ s ω x. (5.) an 0

11 Next, we integrate by parts in the secon term on the left sie of the above euation an move the thir term to the right sie in orer to get ( ) ( Λ s ω / ) t Λs ω + ( ) = Λ s ρ Λ s ω Λ s ω x Λ s (u ω) Λ s ω Λ s ω x = I + I, (5.) where we also use ( ) Λ s ω Λ s ω x = ( Λ s ω / ). (5.3) For the term I, we integrate by parts an use Höler s ineuality to obtain I = ( ) Λ s (ρe ) Λ s ω Λ s ω x Since C Λ s ρ Λ s ω Λ s ω ( )/ Λ s ω ( )/ /( ) =C Λ s ρ ( Λ s ω / ) Λ s ω ( )/. (5.4) I = (Λ s (u ω) u Λ s ω ) Λ s ω Λ s ω x, (5.5) we have by Höler s ineuality an emma 3. Using (5.4) an (5.6) in (5.), we get I Λ s (u ω) u Λ s ω Λ s ω t Λs ω + ( ) C( Λ s ω Λu + ω Λ +s u ) Λ s ω C Λ s ω ( Λu + ω ). (5.6) ( ) ( Λ s ω / ) C Λ s ρ ( Λ s ω / ) Λ s ω ( )/ + C Λs ω ( Λu + ω ) C Λ s ρ + ( Λ s ω / ) + C Λs ω + C Λ s ω ( Λu + ω ), (5.7) where we use Young s ineuality in the last step. Absorbing the secon term on the right sie into the left sie of the above ineuality, we obtain t Λs ω + ( Λs ω / ) C Λ s ρ + C Λs ω + C Λs ω ( Λu + ω ). (5.8) Aing the ineualities (.4) an (5.8) an omitting the D 0 term, we arrive at t ( u + Λs ω ) + ( Λs ω / ) C( u ω + ω + ) u + C Λs ρ + C Λs ω + C Λ s ω ( ω + Λu ). (5.9)

12 Next, we nee an appropriate estimate for Λ s ρ an ω + Λu. We apply the operator Λ s to the euation (.3), multiply it with Λ s ρ Λ s ρ, an integrate it with respect to x to get t Λs ρ = Λ s (u ρ) Λ s ρ Λ s ρ x = Λ s iv (uρ) Λ s ρ Λ s ρ x, (5.0) where we use the ivergence-free conition for u. Since (Λ Λ s iv (uρ) Λ s ρ Λ s ρ x = s iv (uρ) u Λ s ρ ) Λ s ρ Λ s ρ x (5.) we obtain by emma 3. Λ s iv (uρ) Λ s ρ Λ s ρ x Λ s iv (uρ) u Λ s ρ Λ s ρ C Λ s ρ Λu Λ s ρ + C ρ r Λ +s u s Λ s ρ, (5.) where r = /( s) an s = /s. Denoting { [, /( s)], if s < I = [, ), otherwise, (5.3) we observe that ρ 0 W s, implies ρ 0 r for r I, an thus ρ(t) r = C(M, r ), t > 0. (5.4) For Λ +s u s, by the Sobolev embeing theorem an the Gagliaro-Nirenberg ineuality, we have Λ +s u s C Λ s ω s C Λ s ω / / s / C Λ s ω / /( α) Λ s ω / α/ =C Λ s ω α Λs ω / α/, (5.5) where α = s/. Therefore, combining (5.0) an (5.) with (5.5) leas to t Λs ρ C Λu ρ W s, + C ρ r ( u W +s, + Λ s ω α Λs ω / α/ ) Λ s ρ C Λu ρ W s, + C u W +s, + C Λ s ω + C Λs ρ + Λ s ω /, (5.6) where we use Young s ineuality an (5.4). Aing the above estimate to (5.9) an setting F (t) = u + Λs ω + ρ + Λs ρ, we get F (t) + ( ) ( Λs ω / ) C ( + F (t) ) + CF (t) ω + CF (t)( Λu + ω ). (5.7)

13 Finally, we hanle the terms Λu an ω. By emma 4., we get from (5.7) F (t) + ( ) ( Λs ω / ) C ( + F (t) ) + CF (t) ω + CF (t) ( + ( ω / ) / )( + log +/ ( + Λ s ω r) ) = C ( + F (t) ) + CF (t) + t + CF (t)x(t) ( + log +/ ( + Λ s ω r) ), (5.8) where we chose r > max (/s, ) an enote X(t) = + ( ω / ) /. By (.3) t 0 X (s) s C( + t) / (5.9) hols. As for the term Λ s ω r, we have by the Gagliaro-Nirenberg an Young s ineualities, Λ s ω r = Λ s ω / / r/ ( Λ s ω / α ( Λ s ω / ) α )/ Λ s ω + ( Λ s ω / ) / Λ s ω + C + ( Λ s ω / ). (5.0) Now we conclue the proof by applying the following Gronwall type lemma to (5.8). emma 5.. Assume that F (t) is a continuously ifferentiable function in a neighborhoo of 0 an F (0) M. et X(t), Y (t) an D(t): [0, ) [0, ) be such that t 0 X(s) s C( + t) / (5.) for some an Y (t) D(t)/ + F (t) + M, where M an M are constants. Furthermore, assume that F (t) satisfies F (t) + D(t) C( + F (t)) + CF (t) + t + CF (t)x(t) ( + log( + Y (t)) ) α for some α > 0. With T = if 0 < α an {( ) /3 T = min + C( + log α, (M + )) C( + log α (M + )), } /( ) C( + log α (M + )) /( ) (5.) (5.3) for some constant C if α >, we then have where C is a constant epening only on M, M, an T. F (t) <, t [0, T ) (5.4) Proof. For the case α (0, ], as in [HKZ], we have a basic ineuality F X( + log( + Y )) α F X( + log( + Y )) F X + Y C + F X log( + CF X). (5.5) 3

14 Hence, we have F (t) + D(t) C( + F (t)) + Y (t) + CF (t)x(t) + CF (t) + t + CF (t)x(t) log( + CF (t)x(t)) C + CF (t) + D(t) + CF (t)x(t) + CF (t) + t + CF (t)x(t) log( + F (t)) + CF (t)x(t) log( + CX(t)). (5.6) The esire result is then obtaine by the usual Gronwall lemma. If on the other han α >, we get F X( + log α ( + Y )) F X + Y C + CF X logα ( + CF X). (5.7) Similarly to (5.6), we arrive at F (t) + D(t) C + CF (t) + D(t) + CF (t)x(t) + CF (t) + t + CF (t)x(t) log α ( + F (t)) + CF (t)x(t) log α ( + CX(t)) (5.8) an the lemma is establishe by the classical Gronwall s ineuality. 6 Persistence for s > an for the intersection spaces Given initial atum (u 0, ρ 0 ) W +s, W s, we o not know in general whether the solution (u, ρ) stays in the space W +s, W s, for all time. However, we can prove that this is so if s > or if the initial ata (u 0, ρ 0 ) belongs to the intersection space X = (H +s H s ) (W +s, W s, ). We consier the case s > in the next theorem, while the intersection space is aresse in Theorem 6.3 below. Theorem 6.. et s (0, ) an [, ) be such that s >. Assume that u 0 W +s, M 0 with iv u 0 = 0 an ρ 0 W s, M, where M 0, M are constants. Then there exists a uniue solution (u, ρ) of the euations (.) (.3) such that u C ( [0, ), W +s, (R ) ) an ρ C ( [0, ), W s, (R ) ). The key ingreient making the global persistence work is the following uniform boun on the vorticity. emma 6.. Assume that > an ω 0 W, (R ), an let ρ (0, ; (R ) (R )). Then the uniue solution ω of the euation (.5) with the initial ata ω 0 satisfies ω C([0, ); (R )). Moreover for any T > 0 the ineuality ω(t) C( ρ (0, ; ), ω 0 W,) + T, t [0, T ] (6.) hols. Proof. Fix T > 0. By (.) we have ω(t) C( ρ (0, ; )) + T, t [0, T ]. (6.) 4

15 For p, enote φ p = ω p. Using the estimate ρ ω p ω x C p ρ ( ω p ) ω p, (6.3) p we obtain from (.7) that p φ p + p p ( ω p ) Cp ρ ω p Cp ρ (φ p) /p (φ p ) (p )/p. (6.4) Applying the Gagliaro-Nirenberg ineuality f C f / f / to f = ω p leas to which combine with (6.4) implies Note that if p φ p + p Cp φ p Cφ p ( ω p ), (6.5) ( φp φ p ) Cp ρ (φ p) /p (φ p ) (p )/p. (6.6) φ p C p/(p+) p p/(p+) ρ p/(p+) φ(p+)/(p+) p, (6.7) for a sufficiently large constant C, the secon term on the left sie ominates the term on the right an thus φ p 0. (6.8) Denoting K p = ω (0,T ; p (R) ), (6.9) φ p may be estimate by φ p (t) max { φ p (0), C /p p /p ρ /p } K(p+)/(p+) p, t [0, T ]. (6.0) Taking the supremum in t, we obtain By inuction an (6.), we arrive at K p max { C ω 0 W s,, C /p p /p ρ /p } K(p+)/(p+) p. (6.) K n+ max{c ω 0 W s,, K }C i= /i ( i ) /i ( + ρ (0, ; )) i=0 i=0 /i C( ρ (0, ; ), ω 0 W s,) + T, t [0, T ]. (6.) Since the constant C oes not epen on n, we may pass to the limit n to conclue the proof. Next we prove Theorem 6.. 5

16 Proof of Theorem 6.. Since the persistence in H +s H s was alreay aresse in [HKZ], we only nee to consier the case >. The proof is similar to that of Theorem. except that we use a ifferent estimate for Λu an ω. Recall that F (t) + ( ) ( Λs ω / ) C( + F (t) ) + t + CF (t)( Λu + ω ), (6.3) where F (t) = u + Λs ω + ρ + Λs ρ. By emma 6., we have From (4.9), we euce ω C + t, t 0. (6.4) Λu Cb ω a Λ s ω a b r Cb ω ( a)/b ω ( /b)( a) Λs ω a r Cb( + ω + ω ) Λ s ω a r, (6.5) where rs >, b, an a = /( + b(s /r)). b = + log( + Λ s ω r) in orer to get Similarly as in the proof of emma 4., we set Λu C ( + log( + Λ s ω r) ) ( + ω + ω ). (6.6) Therefore, noting that ω(t) C + t, we obtain F (t) + ( ) ( Λs ω / ) C ( + F (t) )( + log( + Λ s ω r) ) + t. (6.7) We conclue the proof by using emma 5.. Denoting (u, ρ) X = max { u H +s, u W +s,, ρ H s, ρ W s,}, the global persistence result in the intersection space X reas as follows. Theorem 6.3. et s (0, ) an [, ). Assume that (u 0, ρ 0 ) X M where M is an arbitrary positive constant. There exists a uniue solution (u, ρ) to the euations (.) (.3) such that (u, ρ) C ( [0, ), X). Moreover, (u(t), ρ(t)) X C(M, T ), t [0, T ] (6.8) for any fixe T > 0. In the proof, we nee the following moification of emma 4.. emma 6.4. Assume that u S(R ). Also, let 0 < s < an r be such that rs >. For <, with ω = u, we have the ineualities Λu C ( + log( + Λ +s u r) ) / ( + u H ) (6.9) an ω C ( + log( + Λ s ω r) ) / ( + ω H ). (6.0) where C = C(r, s, ). 6

17 Proof of emma 6.4. By (4.9), we have Λu where b an a is given in (4.8). Also, since as in (4.0), we get C Λu a b Λ +s u a r C( + Λu b) Λ+s u a r (6.) Λu b Cb / ( Λu + Λ u ) Cb / u H (6.) Λu Cb / ( + u H ) Λ +s u a r (6.3) Then choose b = +log(+ Λ +s u r) an (6.9) follows. The ineuality (6.0) is proven analogously. Proof of Theorem 6.3. Since the persistence in H +s H s was alreay aresse in [HKZ], we only nee to consier the case >. The proof is similar to that of Theorem. except that we use a ifferent estimate for Λu an ω. Recall that we have F (t) + ( ) ( Λs ω / ) C( + F (t) ) + t + CF (t)( Λu + ω ), (6.4) where F (t) = u + Λs ω + ρ + Λs ρ. By emma 6.4 an the Calerón-Zygmun ineuality, we have an Therefore, we obtain From [HKZ] we recall that Λu C ( + log( + Λ +s u r) ) / ( + u H ) C ( + log( + Λ s ω r) ) / ( + u H ) (6.5) ω C ( + log( + Λ s ω r) ) / ( + ω H ) F (t) + ( ) ( Λs ω / ) C ( + log( + Λ s ω r) ) / ( + u H ). (6.6) C ( + F (t) ) + t + CF (t)( + u H ) ( + log( + Λ s ω r) ) /. (6.7) T 0 u(t) H t <. (6.8) Therefore, together with (5.0), we conclue the proof by using emma 5.. Corollary 6.5. et (u 0, ρ 0 ) W +s, (R ) W s, (R ) be compactly supporte an assume that s an are the same as in the above theorem. There exists a uniue solution (u, ρ) to the euations (.) (.3) such that (u, ρ) C ( [0, ), W +s, (R ) W s, (R ) ). Moreover, u(t) W +s, (R ), ρ(t) W s, (R ) C, t [0, T ] (6.9) for any fixe T > 0, where C epens on the initial ata an T. 7

18 Proof. Since (u 0, ρ 0 ) X by the assumptions, the assertion follows by applying Theorem Persistence with perioic bounary conitions Using the above theorem, we may also get the global persistence in the perioic omain. Theorem 7.. et u 0 W +s, (T ), ρ 0 W s, (T ) M where M is an arbitrary positive constant an where 0 < s < an [, ). Then there exists a uniue solution (u, ρ) to the euations (.) (.3) such that (u, ρ) C ( [0, ), W +s, (T ) W s, (T ) ). Moreover, u(t) W +s, (T ), ρ(t) W s, (T ) C(M, T ), t [0, T ] (7.) for any fixe T > 0. Proof. Since H +s (T ) W +s, (T ) an H s (T ) W s, (T ), the theorem follows by applying Theorem 6.3 provie we can prove emma 3. for the perioic omain T. In fact the Coifman-Mayer estimate still hols for the perioic omain as shown in [Wo]. Then the same argument as in the proof of emma 3. shows that [Λ s j, g]f (T ) C f r (T ) Λ +s g r (T ) + C Λ s f r (T ) Λg r (T ) (7.) hols, where f, g, s, j,, r, r, r, an r are as in the statement of emma 3., with the only ifferences replacing ξ R an η R by iscrete variables m T an n T. Acknowlegments The authors woul like to thank the referee for useful remarks. IK an FW were supporte in part by the NSF grant DMS-3943, while MZ was supporte in part by the NSF grant DMS References [ACW] D. Ahikari, C. Cao, an J. Wu, Global regularity results for the D Boussines euations with vertical issipation, J. Differential Euations 5 (0), no. 6, [BG] H. Brézis an T. Gallouet, Nonlinear Schröinger evolution euations, Nonlinear Anal. 4 (980), no. 4, [BS] [BKM].C. Berselli an S. Spirito, On the Boussines system: regularity criteria an singular limits, Methos Appl. Anal. 8 (0), no. 4, J.T. Beale, T. Kato, an A. Maja, Remarks on the breakown of smooth solutions for the 3-D Euler euations, Comm. Math. Phys. 94 (984), no.,

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