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

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 $L^1(0,T; L^\infty({\bf T}^2))$.