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Global mild solutions in a critical setting for a forced fractional Boussinesq system

Diego Chamorro, Maxence Mansais

TL;DR

We address global mild solutions for a forced fractional Boussinesq system with $1<\alpha<2$ in $\mathbb{R}^d$. By formulating the problem in a mild integral form and exploiting a scaling-invariant, critical Banach space built from parabolic Morrey and Triebel-Lizorkin-Morrey spaces, we prove a small-data global existence result via a fixed-point argument. The authors also establish equivalences between Triebel-Lizorkin-Morrey initial-data spaces and the corresponding parabolic Morrey resolution norms, providing a coherent critical framework for the fractional Boussinesq dynamics. These results extend the understanding of nonlocal convection-diffusion systems in large critical spaces and furnish robust tools for analyzing fractional PDEs with forcing terms.

Abstract

We study here mild solutions for the forced, incompressible fractional Boussinesq system. Under suitable estimates for the terms involved (in an adapted functional framework) we can invoque a fixed point argument in order to obtain mild solutions. Although many functional spaces can be considered, we are interested here in a critical setting which ensures the existence of global solutions and we will work in particular with parabolic Morrey spaces which provide one of the largest critical functional frameworks available for constructing mild solutions for the fractional Boussinesq equations.

Global mild solutions in a critical setting for a forced fractional Boussinesq system

TL;DR

We address global mild solutions for a forced fractional Boussinesq system with in . By formulating the problem in a mild integral form and exploiting a scaling-invariant, critical Banach space built from parabolic Morrey and Triebel-Lizorkin-Morrey spaces, we prove a small-data global existence result via a fixed-point argument. The authors also establish equivalences between Triebel-Lizorkin-Morrey initial-data spaces and the corresponding parabolic Morrey resolution norms, providing a coherent critical framework for the fractional Boussinesq dynamics. These results extend the understanding of nonlocal convection-diffusion systems in large critical spaces and furnish robust tools for analyzing fractional PDEs with forcing terms.

Abstract

We study here mild solutions for the forced, incompressible fractional Boussinesq system. Under suitable estimates for the terms involved (in an adapted functional framework) we can invoque a fixed point argument in order to obtain mild solutions. Although many functional spaces can be considered, we are interested here in a critical setting which ensures the existence of global solutions and we will work in particular with parabolic Morrey spaces which provide one of the largest critical functional frameworks available for constructing mild solutions for the fractional Boussinesq equations.

Paper Structure

This paper contains 6 sections, 5 theorems, 72 equations.

Table of Contents

  1. Introduction and presentation of the results
  2. Kernel estimates, parabolic Riesz potentials and some inequalities
  3. Proof of the Theorem \ref{['Theorem_Triebel_Morrey']}
  4. Proof of the Theorem \ref{['Theorem_Equivalences']}
  5. Proof of the Proposition \ref{['Propo_Inclusion_MorreyParabolique_Besov']}
  6. A functional inequality

Key Result

Theorem 1

Let $d\geq2$ denote the dimension and consider a fractional power such that $1<\alpha < 2$. Fix now a real parameter $1\leq p<+\infty$ such that $4<\frac{3\alpha-2}{\alpha-1}<p\leq \frac{d+\alpha}{\alpha-1}$. If the quantity is small enough, then there exists a global mild solution $(\vec{u},\theta)$ of the fractional integral problem (Formulation_Integrale) such that

Theorems & Definitions (5)

  • Theorem 1: Initial data in Triebel-Lizorkin-Morrey spaces
  • Proposition 1.1
  • Theorem 2
  • Lemma 3.1
  • Proposition B.1