Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain

TL;DR

This paper proves that solutions to the heat conductive Boussinesq equation on a 2D periodic domain converge to Euler--Boussinesq solutions as viscosity approaches zero, under specific initial conditions and regularity assumptions.

Contribution

It extends the inviscid limit analysis for the Boussinesq system with heat conduction, including forcing terms, in a periodic setting.

Findings

Convergence of solutions in $L^ abla(0,T; W^{1,p})$ as viscosity tends to zero.

Extension of previous inviscid limit results to heat conductive Boussinesq equations.

Applicability to initial data with vorticity in $L^\infty$ and velocity, temperature in $L^2$.

Abstract

We establish the inviscid limit of the Yudovich solution to the heat conductive Boussinesq equation with initial velocity and temperature/buoyancy in $L^2$ and initial vorticity in $L^\infty$ on the two-dimensional periodic domain ${\bf T}^2$. Given any finite time $T>0$ and $p \in [1,\infty[$, we show that the solution to the diffusive Boussinesq equation converges in $L^\infty(0,T; W^{1,p}({\bf T}^2))$ to the solution to the Euler--Boussinesq equation as the viscosity tends to zero, provided that the initial vorticity, velocity, and temperature/buoyancy converge strongly in $L^2$. Our proof adapts and extends the arguments in [P. Constantin, T. D. Drivas, and T. M. Elgindi, Comm. Pure Appl. Math. 75 (2022), 60--82] to forcing terms in $L^1(0,T; L^\infty({\bf T}^2))$.