ON THE PRODUCT IN BESOV-LORENTZ-MORREY SPACES AND EXISTENCE OF SOLUTIONS FOR THE STATIONARY BOUSSINESQ EQUATIONS

. This paper is devoted to the Boussinesq equations that models natural convection in a viscous ﬂuid by coupling Navier-Stokes and heat equa- tions via a zero order approximation. We consider the problem in R n and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.


(Communicated by Alain Miranville)
Abstract. This paper is devoted to the Boussinesq equations that models natural convection in a viscous fluid by coupling Navier-Stokes and heat equations via a zero order approximation. We consider the problem in R n and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.
1. Introduction. The Boussinesq equations of hydrodynamics arises from zero order approximation to the coupling between the Navier-Stokes equations and the heat equation, modeling the fluid movement by the natural convection. The stationary Boussinesq system is given by the following system of PDEs:    −ν∆u + u · ∇u + ∇π = θf + F in R n , ∇ · u = 0 in R n , −κ∆θ + u · ∇θ = G in R n , (1.1) where u = (u j ) n j=1 , π and θ are respectively the velocity of the fluid, scalar pressure and temperature. The fields F and f denote given external forces and G represents a given reference temperature. The coefficient ν denotes the dynamic viscosity and κ is the molecular diffusivity; throughout this paper, they are assumed to be positive and will be taken to be 1. The dynamic model consists in the heat advection-diffusion equation of the temperature coupled with the incompressible Navier-Stokes equations through the gravitational term θf . The coupling term θf arises from the Boussinesq approximation which establishes that the density variation can be neglected in the system except for a buoyancy force proportional to the local temperature in the momentum balance [9]. As usual, system (1.1) can be rewritten as −∆u + P (u · ∇u) = P (θf ) + PF in R n , −∆θ + u · ∇θ = G in R n , (1.2) where the so-called Leray-Hopf projector P is defined as (P i,j ) n×n , where P i,j := δ i,j +R i R j and R i = (−∆) −1/2 ∂ i is the i-th Riesz transform. The issues of existence and long-time behavior of solutions for the corresponding non-stationary Boussinesq equations have attracted the attention of many authors (see for instance, [1,6,7,11] and references therein). Most of the existence results for the stationary Boussinesq equations have been obtained in classical Sobolev spaces and considering bounded domains [20,22]. The existence of a class of stable stationary solutions for the Boussinesq equations in the scaling invariant class L (n,∞) was obtained in [12] in the whole space R n and in [14] in exterior domains. For f and G satisfying the scale relation f (λx) = λ 2 f (λx) and Typical examples of critical spaces for (1.2) are the Lebesgue space L n (R n ), the weak-L p space L (n,∞) (R n ), the homogeneous Besov spaceḂ n p −1 p,∞ , p > n, among others. A natural problem is to find large critical spaces in which PDEs presents a good existence theory.
In this paper, we obtain existence of stationary solutions in the class of Besov-Lorentz-Morrey spacesḂM l,s (p,d),r ×ḂM l1,s1 (p1,d1),r1 , whereḂM l,s (p,d),r is defined bẏ for some parameters s, l, p, d, r, where M s (p,d) stands for the Lorentz-Morrey spaces (see definitions in Section 2). Considering in particular d = ∞ and r = ∞, we desire to analyze the existence of solutions in the larger critical spacesḂM l,n/l−1 (p,∞),∞ in (1.3). The case d = ∞ corresponds to the Besov-weak-Morrey spaceḂM l,s (p,∞),r and is denoted byḂW M l,s p,r . Using the inverse of ∆ and the identities u·∇u = div (u ⊗ u) and u·∇θ = div (θu) , we rewrite system (1.2) as follows In order to analyze the existence of solution in Besov-Lorentz-Morrey spaces for (1.2), observing the system (1.5), one needs to establish product estimates that permit us to deal with u ⊗ u and the coupling terms θu and θf . Naturally, a multiplier result of Mihlin type is also needed to handle P, ∆ −1 and derivatives terms in (1.5) (see Lemmas 2.7 and 2.8).
Then, the conditions in Proposition 1.3 can be verified ifl is close to n 3 (so l 2 < n is close to n).

Let n ≥ 3 and 1 < p
contain homogeneous functions of degree −2. In fact, using the Hölder and Bernstein inequalities in Morrey spaces, it follows that In what follows, we state our existence result and afterwards, in connection with the above product estimates, we present the main examples in two corollaries.
Some further comments about our results are in order.
where G is the gravitational constant. This case can be regarded as a mathematical version in the whole space of the Bénard problem (see [9]). 2. (Large f ) In Theorem 1.4 and Corollaries 1.5 and 1.6, no smallness condition is needed on the factor f of the coupling term θf . Here, after obtaining the needed estimates, we use an iterative scheme (see (4.1)-(4.2)) that is equivalent to the contraction mapping argument for continuous bilinear mappings, when the problem is formulated in the variables u and f X θ and with the data F and f X G.
3. (Navier-Stokes equations) Considering θ 0 = 0 and θ = 0, the system (1.2) becomes the stationary Navier-Stokes equations (S-NS). Existence results for (S-NS) in R n are known in the framework of Sobolev [13,24], pseudomeasure [8], weak-L p [18] and Morrey spaces [17] (see also [16] for solutions in Besov-Morrey spaces in the non-stationary case). In comparison with previous results, our existence results provided a new class of existence-uniqueness of solutions for (S-NS) in R n . For results in other types of domains Ω (e.g. bounded and exterior ones), the reader is referred to [5,13,18,24] This paper is organized as follows. In Section 2, we give some preliminaries about Lorentz, Lorentz-Morrey and Besov-Lorentz-Morrey spaces. Section 3 is devoted to the proof of the product estimates stated in Propositions 1.1, 1.2 and 1.3. In Section 4, we prove our results about existence of stationary solutions for the Boussinesq system.

2.1.
Lorentz and Lorentz-Morrey spaces. Briefly, we introduce some preliminaries about Lorentz spaces L (p,q) (Ω), Ω ⊆ R n . The reader interested in more details about these spaces is refereed to [2]. A measurable function f defined on Ω belongs to the Lorentz space L (p,q) (Ω) if the quantity with m denoting the Lebesgue measure in R n . The space L (p,q) with the norm f (p,q) is a Banach space. In particular, L p (Ω) = L (p,p) (Ω) and L (p,∞) (Ω) is called the Marcinkiewicz spaces or weak-L p space (q = ∞). Moreover, The following two propositions are Hölder and convolution inequalities in Lorentz spaces, respectively. d1) (Ω) and g ∈ L (p2,d2) (Ω) , then h = f g ∈ L (p,d) (Ω) and
The following lemma is a convolution estimate in Lorentz-Morrey spaces (see [10]).

Lemma 2.4 (Convolution in Lorentz-Morrey spaces)
. Let 1 < p ≤ l ≤ ∞, 1 ≤ d ≤ ∞ and θ ∈ L 1 (R n ). Then, there exists C > 0 (independent of θ) such that 3) for all f ∈ M l (p,d) . For the remainder of this paper ϕ denotes a radially symmetric function such that The localization operators ∆ j and S j are defined by One can check easily the identities Moreover, we have the Bony's decomposition (see [4]) For simplicity, in some calculations we also denoteφ j = ϕ j−1 + ϕ j + ϕ j+1 and D j = D j−1 ∪ D j ∪ D j+1 where j ∈ Z and D j = x ; 3 4 2 j ≤ |x| ≤ 8 3 2 j . Notice that ϕ j = 1 in D j , so we always have ϕ j = ϕ jφj .
The following lemma corresponds to a Bernstein type inequality in Lorentz-Morrey spaces.

(2.6)
Note that the estimate (2.6) is also true in the case p 0 = l 0 = d 0 = 1. In any case, we obtain that for any x ∈ R n and R > 0 Now consider the case 1 ≤ p 0 < p < ∞ and p l = p0 l0 (d 0 = 1 if p 0 = 1). For x ∈ R n and R > 0, we interpolate the estimate f L p 0 ,d 0 (D(x,R)) ≤ CR and (2.7), in order to get , which concludes the proof of the lemma.

2.2.
Besov-Lorentz-Morrey spaces. Below we give a general definition of Besovtype spaces based on a Banach space E. This definition has been used by some authors, see e.g. [21,19].
Definition 2.6. Let E ⊂ S be a Banach space, 1 ≤ r ≤ ∞ and s ∈ R. The homogeneous Besov-E spaceḂE s r is defined aṡ There exist other notations for the Besov-E spaceḂE s r , for exampleḂ s,r E oṙ B s E,r (see [19]). If we take E = M p (p,p) = L p then we obtain the usual homogeneous Besov spaceḂ s p,r . Taking E = M l (p,p) = M l p we obtain the homogeneous Besov-Morrey spaceḂM l,s (p,p),r = N s l,p,r introduced in [16]. Here we use E = M l (p,d) that corresponds to the homogeneous Besov-Lorentz-Morrey spaceḂE s r =ḂM l,s (p,d),r . As already pointed in the Introduction, in the case d = ∞ we obtain the Besovweak-Morrey space and use the notationḂM l,s (p,∞),r =ḂW M l,s p,r .
In order to finish the proof, it is enough to show that K L 1 ≤ C2 mj . Taking N ∈ N such that n 2 < N ≤ n, we can estimate The following lemma is a multiplier result of Mihlin type. This can be regarded as an extension of [16,Theorem 2.9] to our setting. As a direct consequence (taking m = 0), we obtain the boundedness of the Leray-Hopf projector P in Besov-Lorentz-Morrey spaces.
Proof of Proposition 1.1. Using Bony's decomposition (2.4), we have Now we estimate I j 1 , I j 2 and I j 3 separately. For I j 1 we have .

ON THE STATIONARY BOUSSINESQ EQUATIONS 2433
Since j ∼ k, we can take the l r -norm in order to obtain In what follows, we prove Proposition 1.2.
As before, we have that (3.5) The term I j 1 can be estimated as For I j 2 , we proceed similarly in order to estimate