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On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness

Nestor Acevedo, Manuel Fernando Cortez, Oscar Jarrín

Abstract

The stationary version of the Boussinesq system with a general gravitational acceleration term is considered. Under suitable assumptions on this term, as well as on the external forces acting on each equation of this coupled system, we first establish the existence of weak solutions in the natural energy space $\dot{H}^1(\mathbb{R}^3)$. The uniqueness of these solutions is a challenging open problem. Within this framework, our first main contribution is to show that \emph{any} weak $\dot{H}^1$-solution exhibits an analytic smoothing effect in the Gevrey class. Our second main contribution is to show that the Gevrey class regularity can also be used to study the uniqueness problem, provided that these solutions satisfy a suitable low-frequency control. As a by-product, we also obtain new regularity results and a \emph{new Liouville-type result} for weak $\dot{H}^1$-solutions of the classical Navier--Stokes equations.

On Stationary Gevrey Solutions to the Gravitational Boussinesq System and Applications to Uniqueness

Abstract

The stationary version of the Boussinesq system with a general gravitational acceleration term is considered. Under suitable assumptions on this term, as well as on the external forces acting on each equation of this coupled system, we first establish the existence of weak solutions in the natural energy space . The uniqueness of these solutions is a challenging open problem. Within this framework, our first main contribution is to show that \emph{any} weak -solution exhibits an analytic smoothing effect in the Gevrey class. Our second main contribution is to show that the Gevrey class regularity can also be used to study the uniqueness problem, provided that these solutions satisfy a suitable low-frequency control. As a by-product, we also obtain new regularity results and a \emph{new Liouville-type result} for weak -solutions of the classical Navier--Stokes equations.
Paper Structure (11 sections, 28 theorems, 112 equations)

This paper contains 11 sections, 28 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of finite-energy weak solutions: proof of Proposition \ref{['Prop:Existence-Solutions']}
  4. Analyticity of weak $\dot{H}^1$-solutions in the non-homogeneous case
  5. Proof of Theorem \ref{['Th:Gevrey']}
  6. Proof of Corollary \ref{['Corollary:Holder']}
  7. First part
  8. Second part
  9. Analyticity of weak $\dot{H}^1$-solutions in the homogeneous case: proof of Theorem \ref{['Th:Gevrey-Homogeneous']}
  10. Liouville-type theorem for weak $\dot{H}^1$-solutions: proof of Theorem \ref{['Th:Liouville']}
  11. Statements and Declaration

Key Result

Proposition 1.1

Let $\vec{f}\in \dot H^{-1}(\mathbb R^3)$ be such that $\text{div} (\vec{f}) = 0$, $g\in \dot H^{-1}(\mathbb{R}^3)$, and $\vec{{\bf g}}\in L^{\frac{3}{2}}(\mathbb R^3)\cap \dot{H}^{\frac{1}{2}}(\mathbb{R}^3).$ Then the system (Boussinesq) has at least one finite-energy weak solution Moreover, for a numerical constant $C>0$, this solution satisfies the following energy estimate: and

Theorems & Definitions (44)

  • Proposition 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: Heat kernel estimates
  • Lemma 2.2: Fractional Leibniz rule
  • Lemma 2.3: Hölder regularity
  • Theorem 3.1: Schaefer's fixed point
  • Lemma 3.1
  • ...and 34 more