Inviscid limit for the viscous 2D Boussinesq system with temperature-dependent diffusivity

TL;DR

This paper proves the global existence of solutions for the 2D viscous Boussinesq system with temperature-dependent diffusivity and analyzes the inviscid limit, showing how solutions converge to the inviscid system as viscosity vanishes.

Contribution

It establishes global well-posedness for the 2D viscous Boussinesq equations with temperature-dependent diffusivity and quantifies the inviscid limit convergence rate.

Findings

Global well-posedness in a smooth vortex patch framework

Convergence of viscous solutions to inviscid solutions as viscosity approaches zero

Quantitative rate of convergence in the inviscid limit

Abstract

We establish global-posedness in time for the viscous Boussinesq equations in two dimensions of space with temperature-dependent diffusivity in the framework of a smooth vortex patch. We also provide the inviscid limit for velocity, temperature, and associated flow toward the system studied very recently in \cite{Paicu-Zhu} as soon as the viscosity goes to zero, and quantify the rate of convergence.