On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations

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.

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

“…For instance, Leray [15] and Ladyzhenskaya [14] showed the existence of strong solutions to (SNS), and Heywood [10] constructed solutions of (SNS) as a limit of solutions of the nonstationary Navier-Stokes equations. Later on, Chen [4] showed that for every small external force having a divergence-form 𝑓 = ∇ ⋅ 𝐹, 𝐹 ∈ 𝐿 TSURUMI [13] and weak 𝐿 𝑛 (ℝ 𝑛 ) space [12], which is later generalized by Ferreira, Pérez-López, and Villamizar-Roa [5] who studied solutions to the stationary Boussinesq equations in critical Besov-Lorentz-Morrey spaces. Recently, the existence and uniqueness of solutions to (SNS) in the homogeneous Besov spaces on ℝ 𝑛 were well studied by Kaneko-Kozono-Shimizu [11].…”

“…Later on, Chen [4] showed that for every small external force having a divergence‐form $$f=\nabla \cdot F$$, $$F\in L^{\frac{n}{2}}(\mathbb {R}^n)$$, there exists a unique strong solution u of (SNS) in $$L^n (\mathbb {R}^n)$$. After that, Kozono–Yamazaki showed the existence of solutions in critical Morrey spaces [13] and weak $$L^n(\mathbb {R}^n)$$ space [12], which is later generalized by Ferreira, Pérez‐López, and Villamizar‐Roa [5] who studied solutions to the stationary Boussinesq equations in critical Besov–Lorentz–Morrey spaces. Recently, the existence and uniqueness of solutions to (SNS) in the homogeneous Besov spaces on $$\mathbb {R}^n$$ were well studied by Kaneko–Kozono–Shimizu [11].…”

“…However, it seems that there is no satisfactory theory on the existence, uniqueness, and regularity of solutions to general external forces in the whole plane $$\mathbb {R}^2$$. For example, the well‐posedness of (SNS) in $$\mathbb {R}^n$$ for $$n\ge 3$$ established by [5, 12, 13], and [11] are still open problems for $$n=2$$. Actually, these difficulties seem to occur from the bilinear estimates in critical spaces of solutions as in (11) in the third section below.…”

A rescaled approach for the 3D-Boussinesq system in critical Fourier -Besov spaces

“…Finally, they established a decay estimative for this global solution. In the work ( [19], 2018) L.C.F. Ferreira, J.E.…”

Adams’ trace principle in Morrey–Lorentz spaces on β-Hausdorff dimensional surfaces

“…without invoke Kato's approach, see [13] for weak-Morrey spaces, see [14] for Besovweak-Morrey spaces and see [36] for weak-L p spaces. For stationary Boussinesq equations, see [15] for Besov-weak-Morrey spaces and see [16] for weak-L p spaces. Choosing a specific h λ p -atom and using discrete Calderón reproducing formula in Hardy-Morrey spaces, from atomic decomposition theorem the authors [23] characterized the continuity of I α : h λ p (dν) → h λ * q (dµ) by using the growth condition µ β < ∞, provided that 0 < p < q < 1 satisfies (1.4).…”