E quations aux

E quations aux

D ´e r i v ´e e s

P a r t i e l l e s

2007-2008

Taoufik Hmidi

On the global well-posedness of the Boussinesq system with zero viscosity Séminaire É. D. P.(2007-2008), Exposé n^oXXIV, 15 p.

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ON THE GLOBAL WELL-POSEDNESS OF THE BOUSSINESQ SYSTEM WITH ZERO VISCOSITY

TAOUFIK HMIDI

(JOINT WORK WITH S. KERAANI)

ABSTRACT. In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with zero viscosity for rough initial data.

1. INTRODUCTION

This paper deals with the global well-posedness for the two-dimensional Boussinesq sys- tem,

(Bν,κ)











∂_tv+_v· ∇v−ν∆v+∇π=θe2,

∂_tθ+v· ∇θ−_κ∆θ=0, divv=0,

v_|t=0 =v⁰, θ_|t=0 =θ⁰.

Here,e₂ denotes the vector(_{0, 1})_,v = (v₁,v₂)is the velocity field,πthe scalar pressure andθthe temperature. The coefficientsνandκare assumed to be positive;νis called the kinematic viscosity andκthe molecular conductivity.

In the case of strictly positive coefficients ν and κ both velocity and temperature have sufficiently smoothing effects leading to the global well-posedness of smooth solutions.

This was proved by numerous authors in various function spaces (see [4, 9, 15] and the references therein).

Forν>_{0 and}κ=0 the problem of global well-posedness is well understood. In [5], Chae proved global well-posedness for initial data (v⁰,θ⁰)lying in Sobolev spaces H^s×H^s, withs>2 ( see also [14]). This result has been recently improved by Keraani and the author [12] to initial data inH^s×H^s, withs >0. However we give only a global existence result without uniqueness in the energy spaceL²×L². In a joint work with Abidi [1], we prove the uniqueness for data belonging toL²∩_B∞⁻¹_,1×_B2,1⁰ . More recently Danchin and Paicu [8] have proved the uniqueness in the energy space.

Our goal here is to study the global well-posedness of the system(B_0,κ), withκ>0. First of all, let us recall that the two-dimensional incompressible Euler system, corresponding to θ⁰ = 0, is globally well-posed in the Sobolev spaceH^s, with s > 2. This is due to the advection of the vorticity by the flow: there is no accumulation of the vorticity and thus there is no finite-time singularities according to B-K-M criterion [3]. In critical spaces likeB_p,1^2p+1the situation is more complicate because we do not know if the B-K-M criterion works or not. In [16], Vishik proved that Euler system is globally well-posed in these critical Besov spaces. He used for the proof a new logarithmic estimate taking advantage of the particular structure of the vorticity equation in dimension two. For the Boussinesq

system(B0,κ), Chae proved in [5] the global well-posedness for initial data v⁰,θ⁰ lying in Soboloev spaceH^s, withs>2. We intend here to improve this result for rough initial data. Our results reads as follows.

Theorem 1.1. Let p ∈ [1,∞[, v⁰ ∈ _B¹⁺

2 p

p,1 be a divergence-free vector field of R² andθ⁰ ∈ B⁻_p,1^1+2p ∩L^r,with2 < r < _∞. There exists a unique global solution(v,θ)to the Boussinesq system(B_0,κ),κ>0such that

v∈ _C(_R+;B¹⁺

2 p

p,1 ) and θ ∈ L^∞_loc(_R+;B⁻¹⁺

2 p

p,1 ∩L^r)∩eL¹_loc(_R+;B¹⁺

2 p

p,1 ∩_B²_r,∞)_. The situation in the case p = +_∞ is more subtle since Leray’s projector is not continu- ous on L^∞ and we overcome this by working in homogeneous Besov spaces leading to more technical difficulties. Before stating our result we introduce the following sub-space ofL^∞ :

u∈_B^∞ ⇔ kuk_B∞ :=kuk_L∞+k_∆−1uk_B˙0

∞,1 <_∞.

We notice thatB^∞is a Banach space and independent of the choice of the dyadic partition of unity. For the definition of Besov spaces and the frequency localization operator∆−1

we can see next section. Our second main result is the following:

Theorem 1.2. Let v⁰ ∈ _B∞,1¹ ,with zero divergence andθ⁰ ∈ _B^∞ . Then there exists a unique global solution(v,θ)to the Boussinesq system(B_0,κ),κ>0such that

v∈_C(_R+;B¹_∞,1) and θ ∈ L^∞_loc(_R+;B^∞)∩eL¹_loc(_R+;B²_∞,∞). For the proofs it suffices to estimate the quantities k∇v(t)k_L∞ and k∇θk_L1

tL^∞. This will be done by using some logarithmic estimates combined with some smoothing effects.

Theorem 3.1 is very crucial and it describes new smoothing effects for the transport- diffusion equation governed by a vector field which is not necessary Lipschitz but only quasi-lipschitz.

The rest of this paper is organized as follows. In section 2, we recall some preliminary results on Besov spaces. Section 3 is devoted to the proof of smoothing effects. In section 4 and 5 we give respectively the proof of Theorem 1.1 and 1.2. We give in the appendix a commutator lemma.

2. NOTATION AND PRELIMINARIES

Throughout this paper we shall denote by Csome real positive constant which may be different in each occurrence and by C₀ a real positive constant depending on the norms of the initial data.

Let us introduce the so-called Littlewood-Paley decomposition and the correspond- ing cut-off operators. There exist two radial positive functions χ ∈ _D(_R^d) and ϕ ∈ D(_R^d\{0})such that

i) χ(ξ) +

∑

q≥0

ϕ(2^−qξ) =1; ∀ q≥1, suppχ∩supp ϕ(2^−q) =∅ ii) suppϕ(₂^−p·)∩_supp ϕ(₂^−q·) =∅, if|p−q| ≥_2.

For everyv∈_S⁰(_R^d)we set

∆−1v=χ(D)v;∀q∈_N, ∆qv= ϕ(2^−qD)v and S_q=

∑

−1≤p≤q−1

∆p. The homogeneous operators are defined by

∆˙qv= ϕ(2^−qD)v, S˙_qv=

∑

j≤q−1

∆˙jv, ∀q∈_Z.

¿From [2] we split the productuvinto three parts:

uv= Tuv+Tvu+R(u,v), with

T_uv=

∑

q

S_q−1u∆qv and R(u,v) =

∑

|q⁰−q|≤1

∆qu∆q⁰v.

Let us now define inhomogeneous and homogeneous Besov spaces. For(p,r)∈[1,+_∞]² ands ∈ _Rwe define the inhomogeneous Besov spaceB^s_p,ras the set of tempered distri- butionsusuch that

kuk_B^s_p,r :=2^qsk_∆quk_L^p

`^r <+_∞.

The homogeneous Besov space ˙B^s_p,ris defined as the set ofu∈_S⁰(_R^d)up to polynomials such that

kuk_B˙s

p,r :=2^qsk_∆^˙_quk_L^p

`^r(_Z) <+_∞.

LetT>0 andρ ≥1, we denote byL^ρ_TB^s_p,rthe space of distributionsusuch that kuk_L^ρ

TB^sp,r :=2^qsk_∆quk_Lp

`^r

L^ρ_T <+_∞. We say thatubelongs to the spaceeL^ρ_TB^s_p,rif

kuk

eL^ρ_TB^sp,r :=2^qsk_∆quk_L^ρ

TL^p

`^r <+_∞.

The following result is due to Vishik [16].

Lemma 2.1. Let d ≥ 2, there exists a positive constant C such that for any smooth function f and for any diffeomorphismψofR^d preserving Lebesgue measure, we have for all p ∈ [1,+_∞] and for all j,q∈_Z,

k_∆^˙_j(_∆^˙_qf ◦ψ)k_L^p ≤ C2^−|j−q|k∇ψ^η(j,q)k_L∞k_∆^˙_qfk_L^p, with

η(j,q) =sign(j−q)_. Let us now recall the following result proven in [8, 10].

Proposition 2.2. Letν ≥ 0,(p,r) ∈ [1,∞]², s ∈]−1, 1[, v ∈ L¹_loc(_R+; Lip(_R^d))with zero divergence and f be a smooth function. Let a be any smooth solution of the transport-diffusion equation

∂_ta+v· ∇a−ν∆a= f.

Then there is a constant C :=C(s,d)such that for every t∈_R+

kak

eL^∞_tB^sp,r+ν

1

mka−_∆−1ak

eL^m_tB^sp,r^+m2

≤Ce^CV(t) ka⁰k_Bs

p,r +

Z _t

0

kf(τ)k_Bs

p,rdτ

, where V(t):=

Z _t

0

k∇v(τ)k_L∞dτ.

We shall now give a logarithmic estimate which is an extension of Vishik’s one [16]. For the proof we refer to [13].

Proposition 2.3. Let p,r ∈ [1,+_∞], v be a divergence-free vector field belonging to the space L¹_loc(_R+; Lip(_R^d))and let a be a smooth solution of the following transport-diffusion equation (withν≥0),

∂_ta+v· ∇a−_ν∆a= f a_|t=0 =a⁰.

If the initial data a⁰∈ B⁰_p,r,then we have for all t∈_R+

kak

eL^∞_t B⁰_p,r ≤C ka⁰k_B0

p,r +kfk

eL¹_tB⁰_p,r

1+

Z _t

0

k∇v(τ)k_L∞dτ , where C depends only on the dimension d but not on the viscosityν.

3. SMOOTHING EFFECTS

This section is devoted to the proof of a smoothing effect for a transport-diffusion equa- tion with respect to a vector field which is not necessary Lipschitz. This problem was studied by the author [11] in the case of singular vortex patches for two dimensional Navier-Stokes equations. The estimate given below is more precise than [11].

Theorem 3.1. Let v be a smooth divergence-free vector field ofR^dwith vorticityω := curlv.

Let a be a smooth solution of the transport-diffusion equation

∂_ta+v· ∇a−_∆a=0; a_|t=0= a⁰. Then we have for q∈_N∪ {−1}and t ≥0

2^2q Z _t

0

k_∆qa(τ)k_L∞dτ . ka⁰k_L∞

1+t+ (q+2)kωk_L1

tL^∞+k∇_∆−1vk_L1 tL^∞

. Remark 1. In [10], the author proved in the case of Lipschitz velocity the following esti- mate

(1) 2^2q

Z _t

0

k_∆qa(τ)k_L∞dτ.ka⁰k_L∞

1+t+

Z _t

0

k∇v(τ)k_L∞dτ .

But this is not useful in our case. We emphasize that the above theorem is also true when we changeL^∞ byL^p, withp∈ [1,∞].

Proof. The idea of the proof is the same as in [10]. We use Lagrangian formulation com- bined with intensive use of paradifferential calculus.

Letq∈_N^∗, then the Fourier localized functiona_q:=_∆qasatisfies

(2) ∂taq+S_q−1v· ∇aq−_∆aq= (S_q−1−Id)v· ∇aq−[_∆q,v· ∇]a := gq. Letψ_qdenote the flow of the regularized velocityS_q−1v:

ψq(t,x) =x+

Z _t

0 S_q−1v τ,ψq(τ,x)dτ.

We set

¯

aq(t,x) =aq(t,ψq(t,x)) and g¯q(t,x) =gq(t,ψq(t,x)).

¿From Leibniz formula we deduce the following identity

∆a¯_q(t,x) =

∑

D

H_q·(∂ⁱψ_q)(t,x),(∂ⁱψ_q)(t,x)^E+ (∇a_q)(t,ψ_q(t,x))·_∆ψ_q(t,x), whereHq(t,x):= (∇²aq)(t,ψq(t,x))is the Hessian matrix.

Straightforward computations based on the definition of the flow and Gronwall’s in- equality yield

∂ⁱψ_q(t,x) =e_i+hⁱ_q(t,x),

where(e_i)_i^d₌₁is the canonical basis ofR^dand the functionhⁱ_qis estimated as follows (3) khⁱ_q(t)k_L∞ .^Vq(t)e^CVq(t), with Vq(t):=

Z _t

0

k∇S_q−1v(τ)k_L∞dτ.

Applying Leibniz formula and Bernstein inequality we find (4) k_∆ψq(t)k_L∞ .^2qVq(t)e^CVq(t). The outcome is

(5) ∆a¯_q(t,x) = (_∆aq)(t,ψ_q(t,x))− R_q(t,x), with

kR_q(t)k_L∞ . k∇a_q(t)k_L∞k_∆ψ_q(t)k_L∞

+ k∇²aq(t)k_L∞sup

i

khⁱ_q(t)k_L∞+khⁱ_q(t)k²_L∞ . ^22qVq(t)e^CVq(t)ka_q(t)k_L∞.

(6)

In the last line we have used Bernstein inequality.

¿From (2) and (5) we see that ¯a_qsatisfies

(∂_t−_∆)a¯_q(t,x) =R_q(t,x) +g¯_q(t,x).

Now, we will again localize in frequency this equation through the operator∆j. So we write from Duhamel formula,

∆ja¯_q(t,x) = e^t∆∆ja_q(0) +

Z _t

0 e^(t−τ)∆∆jR_q(τ,x)dτ +

Z _t

0 e^(t−τ)∆∆jg¯_q(τ,x)dτ.

(7)

At this stage we need the following lemma (see for instance [7]).

Lemma 3.2. For u∈ L^∞and j∈_N,

(8) ke^t∆∆juk_L∞ ≤Ce^−ct22jk_∆juk_L∞, where the constants C and c depend only on the dimension d.

Combined with (6) this lemma yields, for everyj∈_N,

(9) ke^(t−τ)∆∆jR_q(τ)k_L∞ .^22qVq(τ)e^CVq(τ)e^{−c(t−τ)22j}ka_q(τ)k_L∞. Since the flow is an homeomorphism then we get again in view of Lemma 8

ke^(t−τ)∆∆jg¯q(τ)k_L∞ . ^{e−c(t−τ)22j}k[_∆q,v.∇]a(τ)k_L∞

+ k(S_q−1v−v)· ∇aqk_L∞

. (10)

From Proposition 6.1 we have

k[_∆q,v· ∇]a(t)k_L∞ . ka(t)k_L∞

k∇_∆−1v(t)k_L∞+ (q+2)kω(t)k_L∞

. ka⁰k_L∞

k∇_∆−1v(t)k_L∞+ (q+2)kω(t)k_L∞

. (11)

We have used in the last line the maximum principle: ka(t)k_L∞ ≤ ka⁰k_L∞. On the other hand sinceq∈_N^∗, we can easily obtain

k(S_q−1v−v)· ∇a_qk_L∞ . ka_qk_L∞2^q

∑

j≥q−1

2^−jk_∆jωk_L∞

. ka⁰k_L∞kωk_L∞. (12)

Putting together (7), (9), (10), (11) and (12) we find k_∆ja¯q(t)k_L∞ . ^e−ct22jk_∆ja⁰_qk_L∞

+ Vq(t)e^CVq(t)2^2q Z _t

0 e^{−c(t−τ)22j}kaq(τ)k_L∞dτ + (q+2)ka⁰k_L∞

Z _t

0 e^{−c(t−τ)22j}kω(τ)k_L∞dτ + ka⁰k_L∞

Z _t

0 e^{−c(t−τ)22j}k∇_∆−1v(τ)k_L∞dτ.

Integrating in time and using Young inequalities, we obtain for allj∈_N k_∆ja¯qk_L1

tL^∞ . (2^2j)⁻¹k_∆ja⁰_qk_L∞+ (q+2)ka⁰k_L∞kωk_L1

tL^∞+

ka⁰k_L∞k∇_∆−1vk_L1 tL^∞

+Vq(t)e^CVq(t)2^2(q−j)kaqk_L1 tL^∞.

Let Nbe a large integer that will be chosen later. Since the flow is an homeomorphism, then we can write

2^2qka_qk_L1

tL^∞ = 2^2qka¯_qk_L1 tL^∞

≤ 2^2q

|j−

∑

k_∆ja¯_qk_L1

tL^∞+

∑

|j−q|≥N

k_∆ja¯_qk_L1 tL^∞

.

Hence, for allq> N, one has 2^2qkaqk_L1

tL^∞ . ka⁰k_L∞+2^2Nka⁰k_L∞

(q+2)kωk_L1

tL^∞+k∇_∆−1vk_L1 tL^∞

+ V_q(t)e^CVq(t)2^2N2^2qka_qk_L1

tL^∞+2^2q

∑

|j−q|≥N

k_∆ja¯_qk_L1 tL^∞. According to Lemma 2.1, we have

k_∆ja¯_q(t)k_L∞ .^{2−|q−j|eCVq(t)}ka_q(t)k_L∞. Thus, we infer

2^2qka_qk_L1

tL^∞ . ka⁰k_L∞+2^2Nka⁰k_L∞

(q+2)kωk_L1

tL^∞+k∇_∆−1vk_L1 tL^∞

+ V_q(t)e^CVq(t)2^2N2^2qka_qk_L1

tL^∞+2^−Ne^CVq(t)2^2qka_qk_L1 tL^∞. For low frequencies,q≤ N, we write

2^2qka_qk_L1

tL^∞.^22Nkak_L1 tL^∞. Therefore we get forq∈_N∪ {−1},

2^2qka_qk_L1

tL^∞ . ka⁰k_L∞+₂^2Nkak_L1

tL^∞

+ 2^2Nka⁰k_L∞

(q+2)kωk_L1

tL^∞+k∇_∆−1vk_L1 tL^∞

+ Vq(t)e^CVq(t)2^2N+2^−Ne^CVq(t)

2^2qkaqk_L1 tL^∞. ChoosingNandtsuch that

Vq(t)e^CVq(t)2^2N+e^CVq(t)2^−N .^e, wheree<<1. This is possible for small timetsuch that

V_q(t)≤C₁, whereC₁is a small absolute constant.

Under this assumption, one obtains forq≥ −1 2^2qkaqk_L1

tL^∞ . kak_L1

tL^∞+ka⁰k_L∞

1+ (q+2)kωk_L1

tL^∞+k∇_∆−1vk_L1 tL^∞

.

Let us now see how to extend this for arbitrarly large time T. We take a partition(T_i)^M_i=1 of[0,T]such that

Z _Ti+1

Ti

k∇S_q−1v(t)k_L∞dt'C₁.

Reproducing the same arguments as above we find in view ofka(T_i)k_L∞ ≤ ka⁰k_L∞,

2^2q Z _Ti+1

Ti

kaq(t)k_L∞dt .

Z _Ti+1

Ti

ka(t)k_L∞dt+ka⁰k_L∞

+ ka⁰k_L∞

(q+₂)

Z _Ti+1

T_i

kω(t)k_L∞dt

+

Z _Ti+1

T_i

k∇_∆−1v(t)k_L∞dt . Summing these estimates we get

2^2qkaqk_L1

TL^∞ . kak_L1

TL^∞+ (M+1)ka⁰k_L∞+

+ ka⁰k_L∞

(q+₂)kωk_L1

TL^∞+k∇_∆−1vk_L1 TL^∞

. AsM≈Vq(T), then

2^2qka_qk_L1

TL^∞ . kak_L1

TL^∞+ (V_q(T) +1)ka⁰k_L∞+ + ka⁰k_L∞

(q+2)kωk_L1

TL^∞+k∇_∆−1vk_L1 TL^∞

. Since

k∇S_q−1vk_L∞ ≤ k∇_∆−1vk_L∞+ (q+2)kωk_L∞, then inserting this estimate into the previous one

2^2qkaqk_L1

TL^∞ . ka⁰k_L∞

(1+T) + (q+2)kωk_L1

TL^∞+k∇_∆−1vk_L1 TL^∞

.

This is the desired result.

4. PROOF OFTHEOREM1.1

We restrict ourselves to thea prioriestimates. The existence and uniqueness parts can be done in classical manner.

Proposition 4.1. For v⁰∈ _B¹⁺

2p

p,1 andθ⁰ ∈ L^r,with2<r< _∞,we have for t∈_R+ 1)

kθ(t)k_L^r ≤ kθ⁰k_L^r. 2)

kθk

L¹_tB_r,1^1+2r +kω(t)k_L∞+kωk

eL^∞_tB_∞,1⁰ ≤C₀e^eC0t. 3)

kθk

eL¹_tB²_r,∞+kθk

L¹_tB¹_p,1^+2p +kvk

eL^∞_t B¹_p,1^+2p ≤C₀e^eeC0

t

, where the constant C₀depends on the initial data.

Proof. The first estimate can be easily obtained from L^renergy estimate. For the second one, we recall that the vorticityω =∂₁v²−∂₂v¹satisifies the equation

(13) ∂_tω+v· ∇ω =∂₁θ.

Taking theL^∞ norm we get

(14) kω(t)k_L∞ ≤ kω⁰k_L∞+k∇θk_L1 tL^∞. Using the embeddingB¹_r,1^+2r ,→Lip(_R²)we obtain

kω(t)k_L∞ . kω⁰k_L∞+kθk

L¹_tB_r,1^1+2r. (15)

¿From Theorem 3.1 applied to the temperature equation and by Bernstein inequalities we deduce fore>0,

kθk_L1

tB²r,∞^−e . kθ⁰k_L^r1+t+kωk_L1

tL^∞+k_∆−1∇vk_L1 tL^∞

. kθ⁰k_L^r1+t+kωk_L1

tL^∞+k∇vk_L1 tL^p¯

, with ¯p:=max{p,r}. This leads forr>2 to the inequality

kθk

L¹_tB¹r,∞^+2r .kθ⁰k_L^r1+t+kωk_L1

tL^∞+k∇vk_L1 tL^p¯

.

On the other hand we have the classical resultk∇vk_Lp¯ ≈ kωk_Lp¯. Thus we get

(16) kθk

L¹_tB_r,1^1+2r .kθ⁰k_L^r1+t+kωk_L1

tL^∞+kωk_L1 tL^p¯

. The estimate of theL^p¯norm of the vorticity can be done as itsL^∞ norm

kω(t)k_Lp¯ ≤ kω⁰k_Lp¯ +k∇θk_L1 tL^p¯. . kω⁰k_Lp¯ +kθk

L¹_tB¹_r,1^+2r. (17)

Set f(t):= kθk

L¹_tB¹_r,1^+2r, then combining (15), (16) and (17) leads f(t).kθ⁰k_Lr(1+t+tkω⁰k_L∞∩L^p¯) +kθ⁰k_Lr

Z _t

0 f(τ)dτ.

According to Gronwall’s inequality, one has

(18) kθk

L¹_tB_r,1^1+2r ≤C₀e^C0t,

whereC₀is a constant depending on the initial data. From (15) and (16) we deduce kω⁰k_L∞∩L^p¯ ≤C₀e^C0t.

Let us now turn to the estimate ofkω(t)k_B0

∞,1. Applying Proposition 2.3 to the vorticity equation and using Besov embeddings,

kωk

eL^∞_tB_∞,1⁰ . kω⁰k_L∞+kθk_L1 tB_∞,1¹

1+k∇vk_L1 tL^∞

. kω⁰k_L∞+kθk

L¹_tB_r,1^1+2r

1+k∇vk_L1 tL^∞

. (19)

On the other hand we have

k∇v(t)k_L∞ ≤ k∇_∆−1v(t)k_L∞+

∑

q∈_N

k_∆q∇v(t)k_L∞

. k∇_∆−1v(t)k_Lp¯+kω(t)k_B0

∞,1

. kω(t)k_Lp¯ +kωk

eL^∞_t B⁰_∞,1. (20)

Putting together (18), (19) and (20) and using Gronwall’s inequality gives

(21) k∇vk_L∞+kωk

eL^∞_t B⁰_∞,1 ≤C₀e^eC0t.

It remains to prove the third point of the proposition. The first smoothing effect onθis a direct consequence of (1) and the above inequality,

kθk

eL¹_tBr,∞² ≤C₀e^eC0t.

For the second one we apply Proposition 2.2 to the temperature equation . To establish the velocity estimate we write

kvk

eL^∞_t B¹_p,1^+2p .kvk_L∞

t L^p+kωk

eL^∞_t B_p,1^2p . Using the velocity equation, we obatin

kv(t)k_L^p .kv⁰k_L^p+tkθ⁰k_L^p+

Z _t

0

kP(v· ∇v)(τ)k_L^pdτ.

wherePdenotes Leray projector. It is well-known that Riesz transforms act continuously onL^p, with 1< p<∞, which yields

kP(v· ∇v)k_L^p .kv· ∇vk_L^p . kvk_L^pk∇vk_L∞. Thus we get in view of Gronwall’s inequality and (21)

(22) kvk_L∞

t L^p ≤C₀e^eeC0

t

. It remains to estimatekω(t)k

Bp,1^2p

. We apply Proposition 2.2 to the vorticity equation com- bined with the temperature smoothing, (18) and (21),

kωk

eL^∞_t B_p,1^2p . ^eCV(t)(kω⁰k

B_p,1^2p +kθk

L¹_tB¹_p,1^+2p).

5. PROOF OFTHEOREM1.2

The case p = +_∞ is more subtle and the difficulty comes from the term k∇_∆−1vk_L∞, since Riesz transforms do not mapL^∞to itself. To avoid this problem we use a frequency interpolation method. The main result is the following:

Proposition 5.1. There exists a constant C0 depending onkv⁰k_B0

∞,1 andkθ⁰k_B∞ such that for t ∈[0,∞[

kθ(t)k_L∞ ≤ kθ⁰k_L∞; kθk_L1

tB_∞,1¹ ≤ C0e^C0t3 and

kvk

eL^∞_t B¹_∞,1+kθk

eL¹_tB_∞,∞² +kθ(t)k_B∞ ≤ C₀e^eC0t3.

Proof. TheL^∞-bound of the temperature can be easily obtained from the maximum prin- ciple. To give the other bounds we start with the following estimate for the vorticity, which is again a direct consequence of the maximum principle,

(23) kω(t)k_L∞ ≤ kω⁰k_L∞+k∇θk_L1

tL^∞ ≤ kω⁰k_L∞+kθk_L1 tB_∞,1¹ .

LetN∈ _N^∗, then we get by definition of Besov spaces and the maximum principle kθk_L1

tB¹_∞,1 =

∑

q≤N−1

2^qk_∆qθk_L1

tL^∞+

∑

q≥N

2^qk_∆qθk_L1 tL^∞

. ^2Ntkθ⁰k_L∞+

∑

q≥N

2^qk_∆qθk_L1 tL^∞. By virtue of Theorem 3.1 one has

kθk_L1

tB¹_∞,1 . ^2Ntkθ⁰k_L∞+2^−Nkθ⁰k_L∞

1+t+k∇_∆−1vk_L1

tL^∞+Nkωk_L1 tL^∞

. ^2Ntkθ⁰k_L∞+2^−Nkθ⁰k_L∞

1+t+k∇_∆−1vk_L1 tL^∞

+kωk_L1 tL^∞. Choosing judiciously Nwe get

(24) kθk_L1

tB¹_∞,1.kωk_L1

tL^∞+t¹²kθ⁰k_L∞

1+t+k∇_∆−1vk_L1 tL^∞

¹₂

The following lemma gives an estimate of the low frequency of the velocity.

Lemma 5.2. For all t≥0,wed have

k∇_∆−1v(t)k_L∞ .¹+C₀ 1+t kωk_L∞

t L^∞+tkωk²_L∞ t L^∞, with C₀a constant depending on the norms of the initial data.

Proof. FixN∈_N^∗. Since∆−1 =_∆−1(S^˙−N+_∑⁰_q=−N∆^˙_q)then we have k∇_∆−1vk_L∞ . k∇S^˙−Nvk_L∞+

∑

k∇_∆^˙_qvk_L∞

. ^2−Nkvk_L∞+

∑

−N

k_∆^˙_qωk_L∞

. 2^−Nkvk_L∞+Nkωk_L∞. TakingN≈_log(e+kvk_L∞)_{we get}

(25) k∇_∆−1vk_L∞ .1+kωk_L∞log(e+kvk_L∞)_.

It remains to estimatekvk_L∞. Let M∈_Nthen we have kvk_L∞ .kS^˙−Mvk_L∞+2^Mkωk_L∞. Now using the equation of the velocity we get

kS^˙−Mv(t)k_L∞ ≤ kS^˙−Mv⁰k_L∞+kPS^˙−Mθk_L1 tL^∞

+

Z _t

0

kS^˙−MdivP(v⊗v)(τ)k_L∞dτ . kv⁰k_L∞+k_∆−1θk_L1

tB˙⁰_∞,1+2^−M Z _t

0

kv(τ)k²_L∞dτ.

We have used the following inequality based on the uniform continuity with respect toq of the operator ˙∆qP : L^∞ →L^∞ :

kS^˙−MdivP(v⊗v)k_L∞ ≤

∑

q≤−M−1

k_∆^˙_qdivP(v⊗v)k_L∞

.

∑

q≤−M−1

2^qkv⊗vk_L∞. Thus we obtain

kvk_L∞.kv⁰k_L∞+k_∆−1θk_L1

tB˙⁰_∞,1+₂^−M

Z _t

0

kv(τ)k²_L∞dτ+₂^Mkω(t)k_L∞. To estimatek_∆−1θk_L1

tB˙_∞,1⁰ we use the temperature equation, k_∆^˙_qθ(t)k_L∞ ≤ k_∆^˙_qθ⁰k_L∞+k_∆^˙_q(v· ∇θ)k_L1

tL^∞+k_∆^˙_q∆θk_L1 tL^∞

. k_∆^˙_qθ⁰k_L∞+2^qkvθk_L1

tL^∞+2^2qkθk_L1 tL^∞

. k_∆^˙_qθ⁰k_L∞+2^qkθ⁰k_L∞kvk_L1

tL^∞+2^2qtkθ⁰k_L∞. Therefore we get

q

∑

k_∆^˙_qθ(t)k_L∞ .

∑

q≤0

k_∆^˙_qθ⁰k_L∞+tkθ⁰k_L∞+kθ⁰k_L∞

Z _t

0

kv(τ)k_L∞dτ.

TakingMsuch that

2^2M ≈ Rt

0kvk²_L∞dτ kωk_L∞ , we find

kvk_L∞ .^C0(1+t) +kθ⁰k_L∞

Z _t

0

kv(τ)k_L∞dτ+kω(t)k_L¹²_∞

Z _t

0

kv(τ)k²_L∞dτ¹₂ . According to Gronwall’s inequality we get

(26) kvk_L∞ ≤C₀e^C0te^{CtkωkL∞t L∞}.

Inserting this estimate into (25) we find the desired inequality.