GLOBAL WELL-POSEDNESS FOR EULER-BOUSSINESQ SYSTEM ...

arXiv:0903.3747v1 [math.AP] 22 Mar 2009

GLOBAL WELL-POSEDNESS FOR EULER-BOUSSINESQ SYSTEM WITH CRITICAL DISSIPATION T. HMIDI, S. KERAANI, AND F. ROUSSET Abstract. In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global well-posedness results.

1. Introduction Boussinesq systems of the type  ∂ v + v · ∇v + ∇p = θe2 + νDv v    t ∂t θ + v · ∇θ = κDθ θ div v = 0    v|t=0 = v 0 , θ|t=0 = θ 0

are simple models widely used in the modelling of oceanic and atmospheric motions. These models also appear in many other physical problems, we refer for instance to [5, 3] for more details. Here, we focus on the two-dimensional case, the space variable x = (x1 , x2 ) is in R2 , the velocity field v is given by v = (v 1 , v 2 ) and the pressure p and the temperature θ are scalar functions. The factor θe2 in the velocity equation, the vector e2 being given by (0, 1), models the effect of gravity on the fluid motion. The operator Dv and Dθ whose form may vary are used to take into account the possible effects of diffusion and dissipation in the fluid motion, thus the constants ν ≥ 0, κ ≥ 0 can be seen as the inverse of Reynolds numbers. Mathematically, the simplest model to study is the fully viscous model when ν > 0, κ > 0 and Dv = ∆, Dθ = ∆. The properties of the system are very similar to the one of the two-dimensional Navier-Stokes equation and similar global well-posedness results can be obtained. The most difficult model for the mathematical study is the inviscid one, i.e. when ν = κ = 0. A local existence result of smooth solution can be proven as for symmetric hyperbolic quasilinear systems, nevertheless, it is not known if smooth solutions can develop singularities in finite time. Indeed, the temperature θ is the solution of a transport equation and the vorticity ω = curl v = ∂1 v 2 − ∂2 v 1 solves the equation (1.1)

∂t ω + v · ∇ω = ∂1 θ.

1991 Mathematics Subject Classification. 76D03 (35B33 35Q35 76D05). Key words and phrases. Boussinesq system, transport equations, paradifferential calculus. 1

2

T. HMIDI, S. KERAANI, AND F. ROUSSET

The main difficulty is that to get an L∞ estimate on ω which is crucial to prove global existence of smooth solutions for Euler type equation, one needs to estimate RT ∞ 0 ||∂1 θ||L and, unfortunately, no a priori estimate on ∂1 θ is known. In order to understand the coupling between the two equations in Boussinesq type systems, there have been many recent works studying Boussinesq systems with partial viscosity i.e. with a viscous term only in one equation. For κ > 0, ν = 0 and Dθ = ∆, the question of global existence is solved recently in a series of papers. In [7], Chae proved the global existence and uniqueness for initial data (v 0 , θ 0 ) ∈ H s × H s , with s > 2, see also [20]. This result was recently extended in [16] by the two first 2

+1

−1+ 2

p and θ 0 ∈ Bp,1 p ∩ Lr , r > 2. More recently, the authors to initial data v 0 ∈ Bp,1 study of global existence of Yudovich solutions for this system has been done in [14]. We also mention that in [15], Danchin and Paicu were able to construct global strong solutions (still for κ > 0, ν = 0) for a dissipative term of the form Dθ = ∂11 θ instead of ∆θ. Recently the first author and Zerguine [19] proved the global well-posedness for fractional diffusion Dθ = −|D|α for α ∈]1, 2[ where the operator |D|α is defined by F(|D|α u)(ξ) = |ξ|α (Fu)(ξ). In these works, the global existence result relies on the fact that the only smoothing effect due to the transport-(fractional) diffusion equation

∂t θ + v · ∇θ + |D|α θ = 0

governing the temperature is sufficient to counterbalance the amplification of the vorticity. However the case α = 1 is not reached by their method. The main reason is that this case can be seen as critical in the previous approaches in the sense that the smoothing effect for the temperature equation does not provide the L1T (L∞ ) bound for ∂1 θ which seems needed to control the amplification of the L∞ norm of the vorticity. Note that such an estimate is nevertheless almost true since ∂1 θ can ˜ 1 (L∞ ) which has the same scaling (see below for the be estimated in the space L T definition). The aim of this paper is the study of the well-posedness for this case, i.e., we focus on the system  ∂t v + v · ∇v + ∇p = θe2    ∂t θ + v · ∇θ + |D|θ = 0 (1.2) div v = 0    v|t=0 = v 0 , θ|t=0 = θ 0 . Note that we have taken κ = 1 which is legitimate since we study global wellposedness issues. Indeed, we can always change the coefficient κ into 1 by a change of scale. We also point out that at first sight, the system (1.2) contains the mathematical difficulties of the critical quasigeostrophic equation introduced in [10] (1.3)

∂t θ + v · ∇θ + |D|θ = 0,

v = ∇⊥ |D|−1 θ

which was much studied recently. Indeed, in (1.2) the link between v and θ is not given by the Riesz transform but by a dynamical equation, the first equation of (1.2). Nevertheless, from this velocity equation one gets that v has basically the

GLOBAL WELL-POSEDNESS FOR BOUSSINESQ SYSTEM

3

regularity of θ as in the quasigeostrophic equation. The global well-posedness for (1.3) was obtained recently by Kiselev, Nasarov and Volberg [21]. We also refer to the work [6] by Caffarelli and Vasseur about the regularity of weak solutions. Other discussions can be found in [1, 11, 12]. The main result of this paper is a global well-posedness result for the system (1.2) (see section 2 for the definitions and the basic properties of Besov spaces). 1 ˙ 1,p be a divergence-free vector field ∩W Theorem 1.1. Let p ∈]2, ∞[, v 0 ∈ B∞,1 0 ∩ Lp . Then there exists a unique global solution (v, θ) to the of R2 and θ 0 ∈ B∞,1 system (1.2) with 1 p 0 ˙ 1,p ), e1 (R+ ; B 1 ). v ∈ L∞ θ ∈ L∞ ∩L loc R+ ; B∞,1 ∩ W loc R+ ; B∞,1 ∩ L loc p,∞

A few remarks are in order.

Remark 1.2. If we take θ = 0 then the system (1.2) is reduced to the well-known 2D incompressible Euler system. It is well known that this system is globally wellposed in H s for s > 2. The main argument for globalization is the BKM criterion [2] ensuring that the development of finite time singularities for Kato’s solutions is related to the blowup of the L∞ norm of the vorticity near the maximal time existence. In [26] Vishik has extended the global existence of strong solutions result 1+2/p to initial data lying on the spaces Bp,1 . Notice that these spaces have the same scaling as Lipschitz functions (the space which is relevant for the hyperbolic theory) and in this sense they are called critical. We emphasize that the application of the BKM criterion requires a super-lipschitzian regularity (H s with s > 2 for example). 1+2/p For that reason the question of global existence in the critical spaces Bp,1 is hard to deal with because these spaces have only a lipschitzian regularity and the BKM criterion cannot be used. 1+2/p 1 ˙ 1,r for all p ∈ [1, +∞[ and r > max{p, 2}, then the space Since Bp,1 ֒→ B∞,1 ∩W 1+2/p

of initial velocity in our theorem contains all the critical spaces Bp,1 except the 1 biggest one, that is B∞,1 . For the limiting case we have been able to prove the global existence only up to the extra assumption ∇v 0 ∈ Lp for some p ∈]2, ∞[. The reason behind this extra assumption is the fact that to obtain a global L∞ bound for the vorticity we need before to establish an Lp estimate for some p ∈]2, ∞[ and it is not clear how to get rid of this condition.

Remark 1.3. Since ∇v, ∇θ ∈ L1loc (R+ ; L∞ ) (see Remark 5.6 below for θ) then we 1+2/p can easily propagate all the higher regularities: critical (i.e. v0 ∈ Bp,1 with p finite) and sub-critical (for example v0 ∈ H s , for s > 2). The main idea in the proof of Theorem 1.1 is to really used the structural properties of the system solved by (ω, θ), ω = curl v = ∂1 v 2 − ∂2 v 1 . Indeed, if we neglect the nonlinear terms for the moment, one gets the system ∂t ω = ∂1 θ,

∂t θ = −|D|θ

4


and we notice that its symbol given by A(ξ) =

0 iξ1 0 −|ξ|

is diagonalizable for ξ 6= 0 with two real distinct eigenvalues which are 0 and −|ξ|. By using the Riesz transform R = ∂1 /|D|, one gets that the diagonal form of the system is given by ∂t Rθ = |D|Rθ, ∂t ω + Rθ = 0.

This last form of the system is much more convenient in order to perform a priori estimates. To prove Theorem 1.1, we shall use the same idea, we shall diagonalize the linear part of the system and then get a priori estimates from the study of the new system. The main technical difficulty in this program when one takes the nonlinear terms into account is to evaluate in a sufficiently sharp way the commutator [R, v · ∇] between the Riesz transform and the convection operator. Such commutator estimates are stated and proven in section 3 of the paper. The diagonalization approach used in this paper also allows to prove global wellposedness in different spaces for a ”Boussinesq-Navier-Stokes” system i.e. the system which corresponds to Dθ = 0 and Dv = −|D|. This is discussed in a companion paper [18]. The remaining of the paper is organized as follows. In section 2 we recall some functional spaces and we give some of their useful properties. Section 3 is devoted to the study of some commutators involving the Riesz transform. In section 4 we study a linear transport-(fractional) diffusion equation. Especially, we establish some smoothing effects and a logarithmic estimate type. In section 5 we give the proof of Theorem 1.1 which is It is splitted into three parts. We first establish some a priori estimates, then we prove the uniqueness and finally we briefly explain how one can easily combine a procedure of smoothing out of the initial data with the a priori estimates to get the existence part of the theorem. An appendix is devoted to the proof of a technical commutator lemma. 2. Notations and preliminaries 2.1. Notations. Throughout this work we will use the following notations. • For any positive A and B the notation A . B means that there exist a positive harmless constant C such that A ≤ CB. • For any tempered distribution u both u ˆ and Fu denote the Fourier transform of u. • Pour every p ∈ [1, ∞], k · kLp denotes the norm in the Lebesgue space Lp . • The norm in the mixed space time Lebesgue space Lp ([0, T ], Lr (Rd ) is denoted by k · kLpT Lr (with the obvious generalization to k · kLpT X for any normed space X ). • For any pair of operators P and Q on some Banach space X , the commutator [P, Q] is given by P Q − QP . ˙ 1,p the space of distributions u such that ∇u ∈ Lp . • For p ∈ [1, ∞], we denote by W


5

2.2. Functional spaces. Let us introduce the so-called Littlewood-Paley decomposition and the corresponding cut-off operators. There exists two radial positive functions χ ∈ D(Rd ) and ϕ ∈ D(Rd \{0}) such that X i) χ(ξ) + ϕ(2−q ξ) = 1; ∀ q ≥ 1, supp χ ∩ supp ϕ(2−q ) = ∅ q≥0 ϕ(2−j ·)

ii) supp ∩ supp ϕ(2−k ·) = ∅, if |j − k| ≥ 2. For every v ∈ S ′ (Rd ) we set −q

∆−1 v = χ(D)v ; ∀q ∈ N, ∆q v = ϕ(2 The homogeneous operators are defined by ˙ q v = ϕ(2−q D)v, ∆

S˙ q v =

D)v

and Sq =

q−1 X

∆j .

j=−1

X

˙ j v, ∆

j≤q−1

∀q ∈ Z.

From [4] we split the product uv into three parts: uv = Tu v + Tv u + R(u, v), with Tu v =

X

Sq−1 u∆q v,

R(u, v) =

q

X

˜ qv ∆ q u∆

˜q = and ∆

q

1 X

∆q+i .

i=−1

s as For (p, r) ∈ [1, +∞]2 and s ∈ R we define the inhomogeneous Besov space Bp,r the set of tempered distributions u such that s kukBp,r := 2qs k∆q ukLp r < +∞. ℓ

s B˙ p,r

The homogeneous Besov space is defined as the set of u ∈ S ′ (Rd ) up to polynomials such that ˙ q ukLp kukB˙ s := 2qs k∆ < +∞. r p,r

ℓ (Z)

s LρT Bp,r

the space of distributions u such that Let T > 0 and ρ ≥ 1, we denote by

:= 2qs k∆q ukLp r ρ < +∞. kukLρ Bp,r s T LT

ℓ

eρ B s L T p,r

if We say that u belongs to the space kukLe ρ B s := 2qs k∆q ukLρ Lp T

p,r

T

ℓr

< +∞.

By a direct application of the Minkowski inequality, we have the following links between these spaces. Let ε > 0, then s e ρ B s ֒→ Lρ B s−ε , if LρT Bp,r ֒→ L T p,r T p,r

s+ε s s e ρ Bp,r LρT Bp,r ֒→ L ֒→ LρT Bp,r , if T

r ≥ ρ, ρ ≥ r.

We will make continuous use of Bernstein inequalities (see [8] for instance).

6


Lemma 2.1. There exists a constant C such that for q, k ∈ N, 1 ≤ a ≤ b and for f ∈ La (Rd ), 1

1

sup k∂ α Sq f kLb ≤ C k 2q(k+d( a − b )) kSq f kLa ,

|α|=k

C −k 2qk k∆q f kLa

≤

sup k∂ α ∆q f kLa ≤ C k 2qk k∆q f kLa .

|α|=k

3. Riesz transform and commutators In the next proposition we gather some properties of the Riez operator R = ∂1 /|D|. Proposition 3.1. Let R be the Riez operator R = ∂1 /|D|. Then the following hold true. (1) For every p ∈]1, +∞[, kRkL(Lp ) . 1. (2) Let χ ∈ D(Rd ). Then, there exists C > 0 such that k|D|s χ(2−q |D|)RkL(Lp ) ≤ C2qs ,

for every (p, s, q) ∈ [1, ∞]×]0, +∞[×N. We notice that the previous results hold true if we change |D|s by ∇s with s ∈ N. (3) Let C be a fixed ring. Then, there exists ψ ∈ S whose spectum does not meet the origin such that Rf = 2qd ψ(2q ·) ⋆ f for every f with Fourier transform supported in 2q C. Proof. (1) It is a classical Calderón-Zygmund theorem (see [24] for instance). ξi . K is a tempered distribution such (2) Let K ∈ S ′ such that FK(ξ) = |ξ|s χ(|ξ|) |ξ|

that its Fourier transform FK is C ∞ (Rd \ {0}) and satisfies for every α ∈ Nd |∂ξα FK(ξ)| ≤ Cα |ξ|s−|α| .

According to Mikhlin-H¨ ormander Theorem (see [24] for instance) we have for every d d α ∈ N and x ∈ R \ {0}, Then, it ensues that

|∂xα K(x)| ≤ Cα′ |x|−s−d−|α| .

|K(x)| ≤ C|x|−d−s ∀x 6= 0. 1 Since FK is obviously in L , we also have that K ∈ C0 (Rd ). This removes the singularity at the origin and gives |K(x)| ≤ C(1 + |x|)−d−s

∀x ∈ Rd .

Therefore we get that the kernel K ∈ L1 . Now for every u ∈ Lp we have |D|s Rχ(|D|)u = K ⋆ u. We can now conclude the case q = 0 by using the classical Young inequality for convolution products. The case q ≥ 1 can be derived from q = 0 via an obvious argument of homogeneity.


7

(3) This is can be done easily by introducing a judiciously chosen cut-off function. The following lemma will be useful in the proof of many commutator estimates. Lemma 3.2. Given (p, m) ∈ [1, ∞]2 such that p ≥ m′ with m′ the conjugate exponent of m. Let f, g and h be three functions such that ∇f ∈ Lp , g ∈ Lm and ′ xh ∈ Lm . Then, kh ⋆ (f g) − f (h ⋆ g)kLp ≤ kxhkLm′ k∇f kLp kgkLm . Proof. We have by definition and Taylor formula Z h(x − y)g(y) f (y) − f (x) dy h ⋆ (f g)(x) − f (h ⋆ g)(x) = Rd Z 1Z h i g(y)h(x − y) (y − x) · ∇f (x + t(y − x)) dydt. = 0

Rd

Using Hölder inequalities and making a change of variables z = t(x − y) we get |h ⋆ (f g)(x) − f (h ⋆ g)(x)| ≤ kgkLm

Z

0

1Z

′

Rd

′

−1 m t−d hm 1 (t z)|∇f | (x − z)dz

1 m′

,

where we set h1 (z) = |z||h(z)|. Using Convolution inequalities we obtain since p ≥ m′ kh ⋆ (f g) − f (h ⋆ g)kLp

1

′

′

1

m m′ m′ ≤ kgkLm khm 1 kL1 k|∇f | k p

L m′

≤ kgk

Lm

kh1 k

Lm′

k∇f k . Lp

As explained in the introduction, the control of the commutator between R and the convection operator v · ∇ is a crucial ingredient in the proof of Theorem 1.1. Theorem 3.3. Let v be is a smooth divergence-free vector field. (1) For every (p, r) ∈ [2, ∞[×[1, ∞] there exists a constant C = C(p, r) such that k[R, v · ∇]θkBp,r 0 ≤ Ck∇vkLp kθkB∞,r 0 + kθkLp ,

for every smooth scalar function θ. (2) For every (r, ρ) ∈ [1, ∞]×]1, ∞[ and ǫ > 0 there exists a constant C = C(r, ρ, ε) such that ǫ k[R, v · ∇]θkB∞,r ≤ C(kωkL∞ + kωkLρ ) kθkB∞,r + kθkLρ , 0

for every smooth scalar function θ.

8


Proof. We split the commutator into three parts, according to Bony’s decomposition, X X [R, v · ∇]θ = [R, Sq−1 v · ∇]∆q θ + [R, ∆q v · ∇]Sq−1 θ q∈N

X

+

q∈N

e qθ [R, ∆q v · ∇]∆

q≥−1

=

X

Iq +

q∈N

X

IIq +

q∈N

= I + II + III.

X

IIIq

q≥−1

We start with the estimate of the first term I. According to the point (3) of Proposition 3.1 there exists h ∈ S whose spectum does not meet the origin such that Iq (x) = hq ⋆ (Sq−1 v · ∇∆q θ) − Sq−1 v · (hq ⋆ ∇∆q θ), dq where hq (x) = 2 h(2q x). Applying Lemma 3.2 with m = ∞ we get . kxhq kL1 k∇Sq−1 vkLp k∆q ∇θkL∞ . k∇vkLp k∆q θkL∞ .

kIq kLp

(3.1)

In the last line we’ve used Bernstein inequality and kxhq kL1 = 2−q kxhkL1 . Combined with the trivial fact X X Iq = Iq ∆j q

|j−q|≤4

this yields

kIkBp,r 0

.

X

q≥−1

kIq krLp

1 r

. k∇vkLp kθkB∞,r . 0 Let us move to the second term II. As before one writes IIq (x) = hq ⋆ (∆q v · ∇Sq−1 θ) − ∆q v · (hq ⋆ ∇Sq−1 θ),

and then we obtain the estimate kIIq kLp

. 2−q k∆q ∇vkLp kSq−1 ∇θkL∞ X 2j−q k∆j θkL∞ . . k∇vkLp j≤q−2

Combined with convolution inequalities this yields

kIIkBp,r . k∇vkLp kθkB∞,r . 0 0 Let us now deal with the third term III. Using that the divergence of ∆q v vanishes, we rewrite III as X X X e q θ) − e q θ) + e qθ III = Rdiv(∆q v ∆ div(∆q v R∆ [R, ∆q v · ∇]∆ q≥2

= J1 + J2 + J3 .

q≥2

q≤1


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Using Proposition 3.1-(2), we get

∆j Rdiv(∆q v ∆ e q θ) p . 2j k∆q vkLp k∆ e q θkL∞ . L

e q θ is supported away from zero for q ≥ 2 then Proposition 3.1 (3) yields Also, since ∆

e q θ) p . 2j k∆q vkLp kR∆ e q θkL∞

∆j div(∆q v R∆ L e q θkL∞ . . 2j k∆q vkLp k∆

Therefore we get k∆j (J1 + J2 )kLp

.

X

q∈N q≥j−4

e q θkL∞ 2j k∆q vkLp k∆

. k∇vkLp

X

q∈N q≥j−4

2j−q k∆q θkL∞ ,

where we have again used Bernstein inequality to get the last line. It suffices now to use convolution inequalities to get . . k∇vkLp kθkB∞,r kJ1 + J2 kBp,r 0 0 For the last term J3 we can write X X e q θ(x) = e q θ(x), [R, ∆q v · ∇]∆ [div χ e(D)R, ∆q v]∆ −1≤q≤1

q≤1

where χ e belongs to D(Rd ). Proposition 3.1 ensures that div χ e(D)R is a convolution ˜ satisfying operator with a kernel h ˜ |h(x)| . (1 + |x|)−d−1 .

Thus J3 =

X q≤1

˜ ⋆ (∆q v · ∆ ˜ ⋆∆ ˜ q θ) − ∆q v · (h ˜ q θ). h

First of all we point out that ∆j J3 = 0 for j ≥ 6, thus we just need to estimate the ˜ belongs to Lp′ for p′ > 1 then using Lemma low frequencies of J3 . Noticing that xh 3.2 with m = p ≥ 2 we obtain X ˜ p′ k∆q ∇vkLp k∆ e q θkLp k∆j J3 kL∞ . kxhk L q≤1

. k∇vkLp

X

−1≤q≤1

k∆q θkLp .

This yields finally kJ3 kBp,r . k∇vkLp kθkLp . 0 This completes the proof of the first part of Theorem 3.3. The second part can be done in the same way so that we will only give here a shorten proof. To estimate

10


the terms I and II we use two facts: the first one is k∆q ∇ukL∞ ≈ k∆q ωkL∞ for all q ∈ N. The second one is k∇Sq−1 vk

L∞

. k∇∆−1 vk

L∞

. kωk

Lρ

+

q−2 X j=0

+ qkωk

L∞

k∆j ∇vkL∞

.

For the remainder term we do strictly the same analysis as before except for J3 : we apply Lemma 3.2 with p = ∞ and m = ρ leading to X ˜ ρ′ k∆q ∇vkL∞ k∆ e q θkLρ k∆j J3 kLp . kxhk L q≤1

. k∇vkLρ

X

−1≤q≤1

k∆q θkLρ

. kωkLρ kθkLρ . This ends the proof of the theorem.

4. Transport-Diffusion equation In this section we will give some useful estimates for any smooth solution of a linear transport-diffusion model given by

(TD)

∂t θ + v · ∇θ + |D|θ = f θ|t=0 = θ 0 .

We will discuss three kinds of estimates: Lp estimates, smoothing effects and logarithmic estimates. The proof of the following Lp estimates can be found in [13]. Proposition 4.1. Let v be a smooth divergence-free vector field of Rd and θ be a smooth solution of (TD). Then we have for every p ∈ [1, ∞] Z t 0 kf (τ )kLp dτ. kθ(t)kLp ≤ kθ kLp + 0

We intend to prove the following smoothing effect. Theorem 4.2. Let v be a smooth divergence-free vector field of Rd with vorticity ω. Then, for every p ∈ [1, ∞[ there exists a constant C such that sup 2q k∆q θkL1t Lp ≤ Ckθ 0 kLp + Ckθ 0 kL∞ kωkL1t Lp , q∈N

for every smooth solution θ of (TD) with f ≡ 0. Proof. We start with localizing in frequencies the equation: for q ≥ −1 we set θq := ∆q θ. Then ∂t θq + v · ∇θq + |D|θq = −[∆q , v · ∇]θ.


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Recall that θq is real function since the functions involved in the dyadic partition of the unity are radial. Then multiplying the above equation by |θq |p−2 θq , integrating by parts and using Hölder inequalities we get Z 1 d (|D|θq )|θq |p−2 θq dx ≤ kθq kp−1 kθq kpLp + Lp k[∆q , v · ∇]θkLp . p dt 2 R Recall from [9] the following generalized Bernstein inequality Z (|D|θq )|θq |p−2 θq dx, c2q kθq kpLp ≤ R2

where c depends on p. Inserting this estimate in the previous one we obtain 1 d p kθq kpLp + c2q kθq kpLp . kθq kp−1 Lp k[∆q , v · ∇]θkL . p dt Thus we find d (4.1) kθq kLp + c2q kθq kLp . k[∆q , v · ∇]θkLp . dt To estimate the right hand-side, we shall use the following lemma (see the appendix for the proof of this lemma).

Lemma 4.3. Let v be a smooth divergence-free vector field and θ be a smooth scalar function. Then, for all p ∈ [1, ∞] and q ≥ −1, 0 . k[∆q , v · ∇]θkLp . k∇vkLp kθkB∞,∞

Combined with (4.1) this lemma yields d ct2q q 0 e kθq (t)kLp . ect2 k∇v(t)kLp kθ(t)kB∞,∞ dt q . ect2 kω(t)kLp kθ 0 kL∞ .

To get the last line, we have used the conservation of the L∞ norm of θ and the classical fact k∇vkLp . kωkLp ∀p ∈]1, +∞[. Integrating the differential inequality we get Z t q q 0 −ct2 0 e−c(t−τ )2 kω(τ )kLp dτ. kθq (t)kLp . kθq kLp e + kθ kL∞ 0

Integrating in time yields finally q

2 kθq kL1t Lp

.

kθq0 kLp

0

+ kθ kL∞

0

0

. kθ kLp + kθ kL∞ which is the desired result.

Z

Z

t

kω(τ )kLp dτ

0 t 0

kω(τ )kLp dτ,

Let us now move to the last part of this section which deals with some logarithmic estimates generalizing the results of [26, 16]. First we recall the following result of propagation of Besov regularities.

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Proposition 4.4. Let (p, r) ∈ [1, ∞]2 , s ∈] − 1, 1[ and θ a smooth solution of (TD). Then we have Z t 0 CV (t) s dτ , s e−CV (τ ) kf (τ )kBp,r kθ kBp,r + kθkLe ∞ B s . e p,r

t

0

where V (t) = k∇vkL1t L∞ .

The proof of this result is omitted here and it can be done similarly to the inviscid case [8], using especially Proposition 4.1. Now we will show that for the index regularity s = 0 we can obtain a better estimate with a linear growth on Lipschitz norm of the velocity. Theorem 4.5. There exists C > 0 such that if κ ≥ 0, p ∈ [1, ∞] and θ a solution of (∂t + v · ∇ + κ|D|)θ = f, then we have Z t ∞ k∇v(τ )k dτ . 1 + + kf k kθkLe ∞ B 0 ≤ C kθ 0 kBp,1 0 0 L L1t Bp,1 t

p,1

0

Proof. We mention that the result is first proved in [26] for the case κ = 0 by using the special structure of the transport equation. In [17] the first two authors generalized Vishik’s result for a transport-diffusion equation where the dissipation term has the form −κ∆θ. The method described in [17] can be easily adapted here for our model. Let q ∈ N ∪ {−1} and denote by θq the unique global solution of the initial value problem ( ∂t θq + v · ∇θq + |D|θ q = ∆q f, (4.2) θ q |t=0 = ∆q θ 0 . Using Proposition 4.4 with s = ± 21 we get kθ q k

1

±2 e ∞ Bp,∞ L t

. k∆q θ 0 k

1 ±2 Bp,∞

+ k∆q f k

±1 2 L1t Bp,∞

eCV (t) ,

where V (t) = k∇vkL1t L∞ . Combined with the definition of Besov spaces this yields for j, q ≥ −1 CV (t) 0 p − 1 |j−q| p . 2 2 k∆ θ k + k∆ f k . (4.3) k∆j θq kL∞ 1 Lp e q L q L L t t

By linearity and again the definition of Besov spaces we have X X p + p, k∆j θq kL∞ kθkLe ∞ B 0 ≤ (4.4) k∆j θq kL∞ t L t L t

p,1

|j−q|≥N

|j−q|