The Oberbeck-Boussinesq approximation in critical spaces

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Submitted on 16 Apr 2013

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The Oberbeck-Boussinesq approximation in critical spaces

Raphaël Danchin, Lingbing He

To cite this version:

Raphaël Danchin, Lingbing He. The Oberbeck-Boussinesq approximation in critical spaces. Asymp-

totic Analysis, IOS Press, 2013, 84, pp.61-102. �hal-00795419�

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SPACES

RAPHA¨EL DANCHIN AND LINGBING HE

Abstract. In this paper we study the validity of the so-called Oberbeck-Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space.

Both the cases of fluids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.

1. Introduction

This work aims at giving a mathematical justification of the Oberbeck-Boussinesq approximation that is commonly used to model stratified fluids such as e.g. atmosphere or oceans.

One of the characteristics of the this approximation is that, although the primitive system is the full compressible Navier-Stokes system, the limit equations are incompressible, and the density is a constant. In fact, the velocity field just convects an active scalar creating buoyancy force, proportional to the discrepancy between the temperature and its equilibrium.

1.1. Formal derivation. The starting point of our analysis is the full Navier-Stokes system for compressible viscous fluids, namely

 

 

 



∂

_t

ρ + div (ρu) = 0,

∂

_t

(ρu) + div (ρu ⊗ u) − div τ + 1

Ma

²

∇ P = 1 Fr

²

ρ ∇ V,

∂

_t

(ρs) + div (ρs) + div (q/ T ) = σ.

Above ρ = ρ(t, x) ∈ R

⁺

, u = u(t, x) ∈ R

³

and T = T (t, x) ∈ R

⁺

stand for the density, velocity field and temperature, respectively. The scalar function V stands for some (given) external potential (e.g. the gravity potential). We concentrate on the study of the evolution toward the future in the whole space R

³

(hence the time variable t belongs to R

⁺

and the space variable x, to R

³

).

In the Newtonian case that we shall consider, the stress tensor τ is given by τ = µ( ∇ u + Du) + λdiv u Id.

For simplicity, the viscosity coefficients λ and µ are assumed to be constant. As we only consider viscous fluids, those two coefficients satisfy

µ > 0 and ν := λ + 2µ > 0.

This ensures ellipticity for the second order operator A := µ∆ + (λ + µ) ∇ div .

1

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The heat flux q is equal to − κ ∇T for some constant conductivity coefficient κ ≥ 0. The pressure P, the internal energy e and the specific entropy s are related to ρ and T through the Gibbs relation

T ds = de + P d(1/ρ).

We focus on perfect gases, namely we assume that for some a > 0 and b > 0, P = aρ T and e = b T . After rescaling, it is non restrictive to take a = b = 1.

Finally, in the velocity equation, the Mach number Ma and the Froude number Fr are two dimensionless small parameters accounting for the compressibility and the stratification of the fluid. Formally, Oberbeck-Boussinesq approximation is obtained in the asymptotics ε → 0 if

Ma = ε and Fr = √ ε, an assumption that we shall make from now on.

Gathering all the above assumptions over the coefficients and state laws, we end up with the following system (with exponents ε emphasizing the dependency with respect to ε):

(1.1)

 

 



 

 

∂

_t

ρ

^ε

+ div (ρ

^ε

u

^ε

) = 0,

∂

_t

(ρ

^ε

u

^ε

) + div (ρ

^ε

u

^ε

⊗ u

^ε

) − µ∆u

^ε

− (λ + µ) ∇ div u

^ε

+ ∇ P

^ε

ε

²

= 1

ε ρ

^ε

∇ V

^ε

,

∂

_t

(ρ

^ε

T

^ε

) + div (u

^ε

ρ

^ε

T

^ε

) − κ∆ T

^ε

= ε

²

[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

].

Let us first provide a formal derivation of the Oberbeck-Boussinesq approximation in the case where the heat conductivity κ is positive. We want to consider so-called ill-prepared data of the form ρ

^ε₀

= 1 + εa

^ε₀

, u

^ε₀

and T

0^ε

= 1 + εθ

₀^ε

where (a

^ε₀

, u

^ε₀

, θ

^ε₀

) are bounded in a sense that will be specified later on. Setting ρ

^ε

= 1 + εa

^ε

and T

^ε

= 1 + εθ

^ε

, we get the following governing equations for (a

^ε

, u

^ε

, θ

^ε

):

(1.2)

 

 

 

 



∂

_t

a

^ε

+ div u

^ε

ε = − div (a

^ε

u

^ε

),

∂

_t

u

^ε

+ u

^ε

· ∇ u

^ε

− A u

^ε

1 + εa

^ε

+ ∇ (a

^ε

+ θ

^ε

+ εa

^ε

θ

^ε

) ε(1 + εa

^ε

) = 1

ε ∇ V

^ε

,

∂

_t

θ

^ε

+ div u

^ε

ε + div (θ

^ε

u

^ε

) − κ∆θ

^ε

1 + εa

^ε

= ε

1 + εa

^ε

[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

].

In order to handle the singular potential term in the r.h.s. of the velocity equation, it is usual to work with the modified deviation of density b

^ε

:= a

^ε

− V

^ε

. We get

(1.3)

 

 



 

 

∂

_t

b

^ε

+ u

^ε

· ∇ b

^ε

+ div u

^ε

ε = − ∂

_t

V

^ε

− div (V

^ε

u

^ε

) − b

^ε

div u

^ε

,

∂

t

u

^ε

+ u

^ε

· ∇ u

^ε

− A u

^ε

+ ∇ (b

^ε

+ θ

^ε

)

ε =

a

^ε

− θ

^ε

1 + εa

^ε

∇ a

^ε

− εa

^ε

1 + εa

^ε

A u

^ε

,

∂

_t

θ

^ε

+ u

^ε

· ∇ θ

^ε

+ div u

^ε

ε − κ∆θ

^ε

= ε

1 + εa

^ε

[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

]

− κ εa

^ε

1 + εa

^ε

∆θ

^ε

− θ

^ε

div u

^ε

, which may formally written as follows:

∂

∂t



 b

^ε

u

^ε

θ

^ε



 + 1 ε





0 div 0

∇ 0 ∇

0 div 0







 b

^ε

u

^ε

θ

^ε



 = O (1).

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The notation O (1) designates terms that are expected to be bounded uniformly with respect to ε.

As a consequence of our considering ill-prepared data, the first order time derivatives are likely to blow-up like 1/ε for ε going to 0. At the ‘physical’ level, this means that highly oscillating acoustic waves may propagate in the fluid.

In order to better understand the action of those singular terms, we may first look at the kernel Ker L of the 5 × 5 first order antisymmetric differential matrix operator L above.

The basic idea is that modes that are in Ker L will not be affected, while modes that are in (Ker L)

^⊥

may experience wild oscillations. A straightforward computation shows that

Ker L = n

(b, u, θ) : div u = 0 and ∇ (b + θ) = 0 o , (Ker L)

^⊥

= n

(b, u, θ) : curl u = 0 and ∇ (b − θ) = 0 o

· Hence it is natural to look more closely at the equations satisfied by (q

^ε

, Q u

^ε

) and (Θ

^ε

, P u

^ε

) where P and Q stand for the orthogonal projectors over divergence-free and curl-free vector fields, respectively, and

q

^ε

:= θ

^ε

+ b

^ε

√ 2 , Θ

^ε

:= θ

^ε

− b

^ε

√ 2 ·

As L is antisymmetric, we expect the oscillating components of the solution, namely Q u

^ε

and q

^ε

to be dispersed whereas L will have no effect on P u

^ε

and Θ

^ε

. Let us be more accurate: we see that (q

^ε

, Q u

^ε

) satisfies

(1.4)

 

 



 

 

∂

_t

q

^ε

+

√ 2

ε div Q u

^ε

= − div (q

^ε

u

^ε

) −

√ 2 2

∂

_t

V

^ε

+ div (V

^ε

u

^ε

) + κ ∆θ

^ε

1 + εa

^ε

+

√ 2 2

ε

1 + εa

^ε

[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

],

∂

_t

Q u

^ε

+

√ 2

ε ∇ q

^ε

= Q

a

^ε

− θ

^ε

1 + εa

^ε

∇ a

^ε

− A u

^ε

1 + εa

^ε

− u

^ε

· ∇ u

^ε

whereas (Θ

^ε

, P u

^ε

) fulfills

(1.5)

 

 



 

 

∂

_t

Θ

^ε

+ P u

^ε

·∇ Θ

^ε

− κ

2 ∆Θ

^ε

= − div (Θ

^ε

Q u

^ε

)+

√ 2

2 ∂

_t

V

^ε

+ P u

^ε

·∇ V

^ε

+div (V

^ε

Q u

^ε

) + κ

2 ∆q

^ε

−

√ 2κ 2

εa

^ε

1 + εa

^ε

∆θ

^ε

+

√ 2 2

ε

1 + εa

^ε

[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

],

∂

_t

P u

^ε

− µ∆ P u

^ε

+ P ( P u

^ε

· ∇P u

^ε

) + P (θ

^ε

∇ a

^ε

) = −P u

^ε

· ∇Q u

^ε

+ Q u

^ε

· ∇P u

^ε

− P

εa

^ε

1 + εa

^ε

A u

^ε

+ P

εa

^ε

(θ

^ε

− a

^ε

) 1 + εa

^ε

∇ a

^ε

· If we assume the solution (b

^ε

, u

^ε

, θ

^ε

) and the data to be bounded independently of ε then

the right-hand side of (1.4) is bounded, too. Hence, owing to the antisymmetric (and nonde-

generate) structure of the left-hand side of (1.4), one may expect (q

^ε

, Q u

^ε

) to tend weakly to

0. We shall see later on that in the whole space setting that is here considered, it is possible

to get strong convergence (for suitable negative Besov norms), with an explicit rate.

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In order to find out what the limit system for (1.5) is, let us observe that

√ 2 P (θ

^ε

∇ a

^ε

) = P (Θ

^ε

∇ V

^ε

) + P (q

^ε

∇ V

^ε

) + √

2 P (θ

^ε

∇ b

^ε

)

= P (Θ

^ε

∇ V

^ε

) + P (q

^ε

∇ V

^ε

) + √

2 P ((θ

^ε

+ b

^ε

) ∇ b

^ε

)

= P (Θ

^ε

∇ V

^ε

) + P (q

^ε

∇ V

^ε

) + 2 P (q

^ε

∇ b

^ε

).

(1.6)

Because q

^ε

tends to 0, we expect that

√ 2 P (θ

^ε

∇ a

^ε

) − P (Θ

^ε

∇ V

^ε

) → 0 for ε going to 0.

Hence, if we assume in addition that V

^ε

→ V, P u

^ε₀

→ v

₀

and Θ

^ε₀

→ Θ

₀

, then (Θ

^ε

, P u

^ε

) should tend to the solution (Θ, v) to the following Boussinesq system:

(1.7)

 

 



 

 

∂

_t

Θ + v · ∇ Θ − κ 2 ∆Θ =

√ 2

2 (∂

_t

+ v · ∇ )V,

∂

_t

v + v · ∇ v − µ∆v + ∇ Π = −

√ 2

2 Θ ∇ V, div v = 0, (Θ, v) |

t=0

= (Θ

0

, v

0

).

Setting ˜ Θ = Θ − √

2/2 V, and changing ∇ Π accordingly, we see that this system is equivalent to the following one, which is commonly used:

(1.8)

 

 

 



∂

_t

Θ + ˜ v · ∇ Θ ˜ − κ 2 ∆ ˜ Θ =

√ 2 4 κ∆V,

∂

_t

v + v · ∇ v − µ∆v + ∇ Π = ˜ −

√ 2

2 Θ ˜ ∇ V, div v = 0.

Note that although the density is constant in the limit system, it comes into play in the buoyancy force where it is related to the temperature and the potential.

We end this paragraph with a formal derivation in the case κ = 0. It turns out to be easier to work with the pressure rather than with the temperature. We thus set ρ

^ε

= 1 + εa

^ε

and P

^ε

= ρ

^ε

T

^ε

= 1 + ε( R

^ε

+ V

^ε

), and obtain that

(1.9)

 

 

 

 



∂

_t

a

^ε

+ div u

^ε

ε = − div (a

^ε

u

^ε

),

∂

_t

u

^ε

+ u

^ε

· ∇ u

^ε

− A u

^ε

1 + εa

^ε

+ ∇R

^ε

ε(1 + εa

^ε

) = a

^ε

1 + εa

^ε

∇ V

^ε

,

∂

_t

R

^ε

+ div u

^ε

ε + div ( R

^ε

u

^ε

) = ε[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

] − ∂

_t

V

^ε

− div (V

^ε

u

^ε

).

Setting Θ

^ε

:= a

^ε

− R

^ε

− V

^ε

, we thus get

(1.10)

 

 



 

 

∂

_t

Θ

^ε

+ div (Θ

^ε

u

^ε

) = − ε[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

],

∂

_t

P u

^ε

+ P (u

^ε

· ∇ u

^ε

) − µ∆ P u

^ε

= −P

εa

^ε

1 + εa

^ε

A u

^ε

+ P

a

^ε

1 + εa

^ε

∇ (V

^ε

+ R

^ε

)

,

∂

t

Q u

^ε

+ Q (u

^ε

·∇ u

^ε

) − ν∆ Q u

^ε

+ ∇R

^ε

ε = −Q

εa

^ε

1+εa

^ε

A u

^ε

+ Q

a

^ε

1+εa

^ε

∇ (V

^ε

+ R

^ε

)

,

∂

_t

R

^ε

+ div u

^ε

ε + div ( R

^ε

u

^ε

) = ε[2µ | Du

^ε

|

²

+ λ(div u

^ε

)

²

] − ∂

_t

V

^ε

− div (V

^ε

u

^ε

).

As before, owing to the first order antisymmetric terms, we expect ( Q u

^ε

, R

^ε

) to go to 0.

Concerning (Θ

^ε

, P u

^ε

), we notice that

P (a

^ε

∇ (V

^ε

+ R

^ε

)) = P (Θ

^ε

∇ V

^ε

) + P (Θ

^ε

∇R

^ε

).

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Therefore the limit system for (Θ

^ε

, P u

^ε

) reads (1.11)

 



 

∂

_t

Θ + v · ∇ Θ = 0,

∂

_t

v + v · ∇ v − µ∆v + ∇ Π = Θ ∇ V, div v = 0.

Note that in contrast with (1.7), this system is not fully parabolic.

1.2. Some related works. There is an important literature dedicated to the limit system, that is the Oberbeck-Boussinesq equations (1.7), (1.8) and (1.11), under various hypotheses over the coefficients κ and µ, and the potential V (although the most common assumption is that V = x

₃

). Loosely speaking the classical results concerning the existence issue are (see e.g. [8, 9, 16] and the references therein):

• Dimension 2: Global existence of strong solutions if (µ, κ) 6 = (0, 0).

• Dimension 3 with µ 6 = 0: Global weak solutions and local strong solutions (which become global if the data are small).

• Dimension 3 with µ = 0 : only local-in-time strong solutions are available.

In contrast, although the Oberbeck-Boussinesq approximation is commonly used in geo- physics (see e.g. the books by J. Pedlosky [20] or R. K. Zeytounian [21]) there are few results concerning the rigorous justification of the derivation that we presented in the previous subsection. To our knowledge, the first mathematical justification of Oberbeck-Boussinesq approximation in this context has been given only rather recently in the framework of the so-called variational weak solutions (see [11] for a complete presentation of such solutions for the full Navier-Stokes equations). The case of bounded domains with potential V = x

₃

(or more generally, in W

^1,∞

(Ω)) has been treated by E. Feireisl and A. Novotny in [12, 13], while the exterior domain case has been studied by E. Feireisl and M. Schonbek in [14] (still under the assumption V ∈ W

^1,∞

(Ω), thus ruling out the common but not so physical assumption that V = x

3

). For passing to the limit, all those works borrow some seminal ideas that have been introduced by P.-L. Lions in his book [19] and B. Desjardins et al in [10] in the related context of low Mach number limit for the isentropic Navier-Stokes equations

¹

.

On the one hand, those results are very general for one may consider any finite energy data. On the other hand, the convergence results are not very accurate for they strongly rely on compactness methods : in particular convergence holds up to extraction only, and no rate may be given.

1.3. Aim of the paper. Getting stronger results of convergence that is in particular convergence of the whole sequence with an explicit rate, is the main purpose of the present work. Considering general variational solutions is hopeless. We shall focus on strong solutions with the so-called critical regularity, a framework which is nowadays classical for the study of viscous compressible fluids (see e.g. [2, 4, 5]). Of course, this will enforce us to restrict considerably the set of admissible data, but we will get much more accurate results of convergence.

Working in a functional framework that has the same scaling invariance as (1.2), if any, is the basic idea. Here we see that (if V

^ε

≡ 0 to simplify the presentation), the system is

“almost” invariant for all ℓ > 0 by the rescaling

a

^ε

(t, x) → a

^ε

(λ

²

t, λx), u

^ε

(t, x) → λu

^ε

(λ

²

t, λx), θ

^ε

(t, x) → λ

²

θ

^ε

(λ

²

t, λx).

1For other recent results concerning the low Mach number asymptotics for thefullNavier-Stokes equations, the reader may refer to [1, 15, 17, 18].

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If we believe in an energy type method then a good candidate for initial data is thus the homogeneous Sobolev space

H ˙

³²

( R

³

) × H ˙

¹²

( R

³

)

3

× H ˙

⁻¹²

( R

³

), or rather the slightly smaller following homogeneous Besov space:

B ˙

3 2

2,1

( R

³

) × B ˙

1 2

2,1

( R

³

)

3

× B ˙

⁻

1 2

2,1

( R

³

) which has nicer embedding properties ( ˙ B

3 2

2,1

is embedded in bounded functions for instance) and better behaves with respect to maximal parabolic estimates.

However, owing to the lower order pressure term, the above scaling invariance is not quite respected. Consequently, we have to work at a different level of regularity for the low frequencies of a

^ε

and θ

^ε

, to compensate this scaling defect. All this is now well understood and already occurs in the isentropic case [4].

Finally, in the case κ = 0 that we shall also consider (and that cannot be studied in the framework of variational solutions), only the velocity is smoothed out during the evolution, and it is no longer possible to use a critical regularity framework: we will have to assume much more regularity.

We end this introductory part with a short description of the rest of the paper. After an unavoidable introduction of some notations and functional spaces, the next section is devoted to the presentation of the main results of the paper. The analysis of the heat conducting case is carried out in Section 3 while κ = 0 is considered in Section 4. Some technical estimates are postponed in the Appendix.

2. Results

Before presenting the main statements of the paper, we briefly introduce some notations and function spaces. We are given an homogeneous Littlewood-Paley decomposition ( ˙ ∆

j

)

j∈Z

that is a dyadic decomposition in the Fourier space for R

³

. One may for instance set ˙ ∆

_j

:=

ϕ(2

^−j

D) with ϕ(ξ) := χ(ξ/2) − χ(ξ), and χ a non-increasing nonnegative smooth function supported in B(0, 4/3), and value 1 on B (0, 3/4) (see [2], Chap. 2 for more details).

We then define, for p ∈ [1, + ∞ ] and s ∈ R , the semi-norms k z k

B˙^s_p,1

:= X

j∈Z

2

^js

k ∆ ˙

j

z k

L^p

.

In order to avoid complications due to polynomials, we adopt the following definition of homogeneous Besov spaces:

B ˙

_p,1^s

= n

z ∈ S

^′

( R

³

) : k z k

B˙_p,1^s

< ∞ and lim

j→−∞

S ˙

j

z = 0 o

with ˙ S

j

:= χ(2

^−j

D).

To compensate the lack of strict scaling invariance of the system under consideration (as pointed out in the previous section), we also need to introduce the following hybrid Besov spaces with different regularity exponent in low and high frequencies:

Definition 2.1. For s ∈ R , p ∈ [1, ∞ ] and α > 0, we set k z k

B˜p,α^s,^±

:= X

j∈Z

2

^js

min(α

⁻¹

, 2

^j

)

±1

k ∆ ˙

_j

z k

L^p

(8)

and define

B ˜

_p,α^s,±

:= n

z ∈ S

^′

( R

³

) : k z k

B˜p,α^s,^±

< ∞} and lim

j→−∞

S ˙

_j

z = 0 o

· We shall mainly use the above definition with p = 2, in which case, the corresponding hybrid Besov space will be simply denoted by ˜ B

α^s,±

, if the fact that p = 2 is clear from the context.

We agree that

²

:

(2.12) z

^ℓ

:= X

2^jα≤1

∆ ˙

_j

z and z

^h

:= X

2^jα>1

∆ ˙

_j

z.

With this notation, we have

k z k

B˜^s,p,α^±

= k z

^ℓ

k

B˙_p,^s^±11

+ α

^∓1

k z

^h

k

B˙^s_p,1

.

Therefore ˙ B

_p,1^s

is the bulk regularity of a function in ˜ B

_p,α^s,±

while the behavior at infinity is given by the low frequency part which is in ˙ B

_p,1^s±1

. Of course, changing the value of α does not affect the space, and the corresponding norms are equivalent. However a suitable choice of α will enable us to get uniform estimates with respect to ε.

As we shall work with time-dependent functions with values in Besov spaces, we introduce the norms:

k u k

_L^q_T_{( ˙}_B^s_p,₁₎

:= k u(t, · ) k

B˙^s_p,1

L^q(0,T)

and k u k

_L^q

T( ˜B^s,p,α^±1)

:= k u(t, · ) k

B˜p,α^s,^±1

L^q(0,T)

. As in many works using parabolic estimates in Besov spaces, it is somehow natural to take the time-Lebesgue norm before performing the summation for computing the Besov norm.

This motivates us to introduce the following quantities:

k u k

L˜^q_T( ˙B^s_p,1)

:= X

j∈Z

2

^js

k ∆ ˙

_j

u k

L^q_T(L^p)

and k u k

L˜^q_T( ˜B^s,_p,α^±)

:= X

j∈Z

2

^js

min(α

⁻¹

, 2

^j

)

±1

k ∆ ˙

_j

u k

L^q_T(L^p)

. The index T will be omitted if T = + ∞ and we shall denote by ˜ C( ˙ B

_p,1^s

) (resp. ˜ C( ˜ B

^s,±_p,α

)) the subset of ˜ L

^∞

( ˙ B

_p,1^s

) (resp. ˜ L

^∞

( ˜ B

p,α^s,±

)) constituted by continuous functions over R

⁺

with values in ˙ B

_p,1^s

(resp. ˜ B

^s,±_p,α

).

Let us emphasize that, owing to Minkowski inequality, we have k u k

_L^q

T( ˙B_p,1^s )

≤ k u k

L˜^q_T( ˙B_p,1^s )

with equality if and only if q = 1. Similar properties hold for hybrid Besov spaces.

Throughout, we shall denote

(2.13) ˜ κ := κ/ν, λ ˜ := λ/ν, µ ˜ := µ/ν with ν := λ + 2µ.

One can state our first main result : the global existence of solutions corresponding to small (critical) data with estimates independent of ε in the case κ > 0.

Theorem 2.1. Assume that the initial data (b

^ε₀

, u

^ε₀

, θ

^ε₀

) and that the potential term V

^ε

satisfy, for a small enough constant η depending only on ˜ κ and ˜ µ:

k b

^ε₀

k

_˜

B

3 2,−

εν

+ k u

^ε₀

k

_˙

B

1 2 2,1

+ k θ

^ε₀

k

_˜

B⁻

1 2,+

εν

≤ ην, (2.14)

ν

¹²

k∇ V

^ε

k

L²( ˜B

3 2,−

εν )

+ k V

^ε

k

_˜

L^∞( ˜B

3 2,−

εν )

+ k ∂

_t

V

^ε

k

L¹( ˜B

3 2,− εν )

≤ ην.

(2.15)

2We omit the dependency with respect to the thresholdαin the above notation because the value ofαwill be always clear from the context.

(9)

Let a

^ε₀

:= b

^ε₀

+ V

^ε

(0). Then System (1.2) with initial data (1 + εa

^ε₀

, u

^ε₀

, 1 + εθ

₀^ε

) has a unique global solution (a

^ε

, u

^ε

, θ

^ε

) (with a

^ε

= b

^ε

+ V

^ε

) which satisfies

b

^ε

∈ C( ˜ ˜ B

3 2,−

εν

) ∩ L

¹

( ˜ B

3 2,+

εν

), u

^ε

∈ C( ˙ ˜ B

1 2

2,1

) ∩ L

¹

( ˙ B

5 2

2,1

), θ

^ε

∈ C( ˜ ˜ B

⁻

1 2,+

εν

) ∩ L

¹

( ˜ B

3 2,+

εν

) and, for a constant K depending only on κ ˜ and µ, ˜

k b

^ε

k

_˜

L^∞( ˜B

3 2,−

εν )

+ ν k b

^ε

k

L¹( ˜B

3 2,+

εν )

+ k u

^ε

k

_˜

L^∞( ˙B

1 2

2,1)

+ ν k u

^ε

k

L¹( ˙B

5 2

2,1)

+ k θ

^ε

k

_˜

L^∞( ˜B⁻

1 2,+

εν )

+ν k θ

^ε

k

L¹( ˜B

3 2,+

εν )

≤ K k b

^ε₀

k

_˜

B

3 2,− εν

+ k u

^ε₀

k

_˙

B

1 2 2,1

+ k θ

^ε₀

k

_˜

B⁻

1 2,+ εν

+ k ∂

_t

V

^ε

k

L¹( ˜B

3 2,− εν )

. Remark 2.1. Smoother data give rise to smoother solutions. For example if in addition to the above hypotheses, we have

ε k b

^ε₀

k

_˜

B

5 2,− εν

+ ν

⁻¹

k (θ

₀^ε

, u

^ε₀

) k

_˜

B

3 2,− εν

+ ε k ∂

_t

V

^ε

k

L¹_t( ˜B

5 2,−

εν )

+ ε k∇ V

^ε

k

L²( ˜B

5 2,− εν )

≤ η, then the above solution also satisfies

ε k b

^ε

k

_˜

L^∞( ˜B

5 2,−

εν )

+ ν

⁻¹

k (θ

^ε

, u

^ε

) k

_˜

L^∞( ˜B

3 2,−

εν )

+ εν k b

^ε

k

L¹( ˜B

5 2,+

εν )

+ εν k (u

^ε

, θ

^ε

) k

L¹( ˜B

7 2,−

εν )

≤ Kη.

Next, combining this result with Strichartz estimates, we shall prove the following result of convergence to the Boussinesq system.

Theorem 2.2. Consider a family of data (b

^ε₀

, u

^ε₀

, θ

₀^ε

, V

^ε

)

ε>0

satisfying the conditions of The- orem 2.1 with in addition

(2.16)

M

₀

:= sup

ε>0

k b

^ε₀

k

_˜

B

3 2,−

εν

+ k u

^ε₀

k

_˙

B

1 2 2,1

+ k θ

₀^ε

k

_˜

B⁻

1 2,+

εν

+ ν

¹²

k∇ V

^ε

k

L²( ˜B

3 2,−

εν )

+ k V

^ε

k

_˜

L^∞( ˜B

3 2,−

εν )

+ k ∂

t

V

^ε

k

L¹( ˜B

3 2,− εν )

≤ ην.

Let q

^ε

:= (θ

^ε

+ b

^ε

)/ √

2 and Θ

^ε

:= (θ

^ε

− b

^ε

)/ √

2. Assume that ( P u

^ε₀

, Θ

^ε₀

, V

^ε

) converges (in the sense of distributions) to some triplet (v

₀

, Θ

₀

, V ) such that

v

₀

∈ B ˙

1 2

2,1

, Θ

₀

∈ B ˙

1 2

2,1

, ∇ V ∈ L

²

( ˙ B

1 2

2,1

), ∂

_t

V ∈ L

¹

( ˙ B

1 2

2,1

).

Then the following properties hold true :

(1) System (1.2) with initial data (1 + εa

^ε₀

, u

^ε₀

, 1 + εθ

^ε₀

) has a unique global solution with the properties described in Theorem 2.1;

(2) Boussinesq system (1.7) admits a unique global solution (v, Θ) in C ˜ ( ˙ B

1 2

2,1

) ∩ L

¹

( ˙ B

5 2

2,1

) satisfying for some constant K = K(˜ κ, µ): ˜

k (v, Θ) k

_˜

L^∞( ˙B

1 2

2,1)

+ ν k (v, Θ) k

L¹( ˙B

5 2

2,1)

≤ K k (v

₀

, Θ

₀

) k

_˙

B

1 2 2,1

+ k ∂

_t

V k

L¹( ˙B

1 2 2,1)

.

(3) The functions q

^ε

and Q u

^ε

go to 0 in the following meaning for all p ∈ [2, ∞ ] and s ∈ [ − 1/2 + 4/p, 3/p]:

ν

¹²

k q

^ε

k

L˜²( ˜Bp,εν^s−1,+)

+ ν

¹²

kQ u

^ε

k

L˜²( ˙B^s_p,1)

≤ K(εν)

³^p^−s

M

₀

.

(4) The couple ( P u

^ε

, Θ

^ε

) tends to (v, Θ) in the following meaning for all p ∈ [2, ∞ ] and s ∈ [ − 1/2 + 4/p, 3/p] with s > 1/2 :

ν

^1/2

k δ Θ

^ε

k

L˜²( ˜Bp,εν^s⁻¹^,⁺)

+ k δ Θ

^ε

k

L˜^∞( ˜B^sp,εν⁻²^,⁺)

+ ν k δv

^ε

k

_L¹_{( ˜}_B_p,εν^s,⁺₎

+ k δv

^ε

k

L˜^∞( ˜B^sp,εν⁻²^,⁺)

≤ C k (δ Θ

^ε₀

, δv

^ε₀

) k

B˜^s−2,+p,εν

+ k ∂

_t

δV

^ε

k

_L¹_{( ˜}_B^s−2,+_p,εν _)+L²_{( ˜}_B_p,εν^s−3,+₎

+ M

₀²

ε

³^p^−s

+ M

₀

k∇ δV

^ε

k

_L²_{( ˙}_B^s−1

p,1 )

with δ Θ

^ε

:= Θ

^ε

− Θ, δv

^ε

:= P u

^ε

− v, δV

^ε

:= V

^ε

− V and C = C(˜ µ, ˜ κ, s, p).

(10)

Remark 2.2. If the data are smoother, e.g. as in Remark 2.1 then the results of convergence hold for stronger norms. For instance, it may be shown that ( Q u

^ε

, q

^ε

) → 0 in L ˜

²

( ˙ B

4 p−¹₂ p,1

), that P u

^ε

→ v in L

¹

( ˙ B

4 p+¹₂

p,1

) ∩ L ˜

^∞

( ˙ B

4 p−³₂

p,1

), and that Θ

^ε

→ Θ in L ˜

²

( ˙ B

4 p−¹₂

p,1

) ∩ L ˜

^∞

( ˙ B

4 p−³₂ p,1

), with the decay rate ε

¹²⁻¹^p

.

Let us finally state our main global existence and convergence result for nonconducting fluids.

Theorem 2.3. Assume that the initial data (a

^ε₀

, u

^ε₀

, R

^ε0

) and the force term V

^ε

verify that C

₀^ε

:= k (a

^ε₀

, u

^ε₀

, R

^ε0

) k

_˙

B

1 2 2,1

+(εν)

³

k (a

^ε₀

, R

^ε0

) k

_˙

B

7 2 2,1

+ (εν)

²

k u

^ε₀

k

_˙

B

5 2 2,1

+ k ∂

t

V

^ε

k

L¹( ˙B

1 2

2,1)

+(εν)

³

k ∂

t

V

^ε

k

L¹( ˙B

7 2

2,1)

≤ ην, k V

^ε

k

_˜

L^∞( ˙B

1 2

2,1)

+ (εν)

³

k V

^ε

k

_˜

L^∞( ˙B

7 2

2,1)

+ ν k V

^ε

k

L¹( ˙B

5 2

2,1)

+ ν(εν)

²

k V

^ε

k

L¹( ˙B

9 2

2,1)

≤ ην where the constant η is sufficiently small and depends only on µ. ˜

Then System (1.9) admits a unique global solution (a

^ε

, u

^ε

, R

^ε

) which satisfies a

^ε

∈ C( ˙ ˜ B

1 2

2,1

∩ B ˙

7 2

2,1

), u

^ε

∈ C( ˙ ˜ B

1 2

2,1

∩ B ˙

5 2

2,1

) ∩ L

¹

( ˙ B

5 2

2,1

∩ B ˙

9 2

2,1

), R

^ε

∈ C( ˙ ˜ B

1 2

2,1

∩ B ˙

7 2

2,1

) ∩ L

¹

( ˜ B

3 2,+

εν

∩ B ˜

7 2,+

εν

) and, for some constant K depending only on µ, ˜

k (a

^ε

, R

^ε

) k

_˜

L^∞( ˙B

1 2

2,1)

+ (εν)

³

k (a

^ε

, R

^ε

) k

_˜

L^∞( ˙B

7 2

2,1)

+ k u

^ε

k

_˜

L^∞( ˙B

1 2

2,1)

+ (εν)

²

k u

^ε

k

_˜

L^∞( ˙B

5 2 2,1)

+ν kR

^ε

k

L¹( ˜B

3 2,+

εν )

+ ν (εν)

²

kR

^ε

k

L¹( ˜B

7 2,+

εν )

+ ν k u

^ε

k

L¹( ˙B

5 2

2,1)

+ ν(εν)

²

k u

^ε

k

L¹( ˙B

9 2

2,1)

≤ KC

₀^ε

. Suppose in addition that Θ

^ε₀

→ Θ

₀

, that P u

^ε₀

→ v

₀

and that V

^ε

→ V with

(2.17) k v

₀

k

_˙

B

1 2 2,1

+ k∇ V k

L¹( ˙B

3 2

2,1)

+ k Θ

₀

k

_˙

B

1 2 2,1

≤ ηµ.

Then the corresponding limit Boussinesq system (1.11) admits a unique global solution (Θ, v) in C( ˙ ˜ B

1 2

2,1

) × C( ˙ ˜ B

1 2

2,1

) ∩ L

¹

( ˙ B

5 2

2,1

)

. Furthermore we have (2.18) k (Θ, v) k

_˜

L^∞( ˙B

1 2

2,1)

+ µ k v k

L¹( ˙B

5 2

2,1)

≤ K k (Θ

₀

, v

₀

) k

_˙

B

1 2 2,1

.

In addition, if C

₀^ε

is bounded by some constant C

₀

when ε goes to 0 then ( Q u

^ε

, R

^ε

) goes to zero with the following rates of convergence for all p ∈ [2, ∞ ) :

k ( Q u

^ε

, R

^ε

) k

L˜

2p p−2( ˙B

2 p−1

2

p,1 )

≤ KC

₀

ε

¹²⁻¹^p

(2.19)

ν

¹²

k ( Q u

^ε

, R

^ε

) k

L˜²( ˙B^s_p,1)

≤ KC

₀

(εν)

³^p^−s

if s ∈ [ − 1/2 + 4/p, 3/p].

(2.20)

Finally, if Θ

^ε₀

and P u

^ε₀

are independent

³

of ε then for all p and s as above (with in addition s > 1/2), and T > 0,

Θ

^ε

− Θ → 0 in C ˜

_T

( ˙ B

_p,1^s−2

) and P u

^ε

− v → 0 in C ˜

_T

B ˙

_p,1^s−1

+ ˙ B

_p,1^s−2

∩ L ˜

²_T

( ˙ B

_p,1^s

)+L

¹_T

( ˙ B

^s_p,1

) , and the rate of convergence is ε

³^p^−s

.

The above statements deserve some comments:

3The reader may refer to Inequalities (4.116) and (4.117) for the general case.