The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion

TL;DR

This paper investigates the global well-posedness and stability of the 3D regularized Boussinesq equations with fractional Laplacian, establishing conditions for existence, uniqueness, and continuous dependence on initial data.

Contribution

It introduces a new analysis for the 3D regularized Boussinesq equations with fractional Laplacian, proving global well-posedness and stability under specific regularity assumptions.

Findings

Global well-posedness for initial data in specified Sobolev spaces.

Uniqueness of solutions under certain regularity conditions.

Continuous dependence of solutions on initial conditions.

Abstract

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized \`a la Leray through a smoothing kernel of order $\alpha$ in the nonlinear term and a $\beta$-fractional Laplacian; we consider the critical case $\alpha+\beta=\frac{5}{4}$ and we assume $\frac 12 <\beta<\frac 54$. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order $\alpha$. We prove global well posedness when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-\beta}$ for $r>\max(2\beta,\beta+1)$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of the solutions on the initial conditions.