Stability with respect to domain of the low Mach number limit of compressible heat-conducting viscous fluid

TL;DR

This paper analyzes the low Mach number limit of compressible heat-conducting viscous fluids in large domains with rough surfaces, showing convergence to the incompressible Oberbeck-Boussinesq system using spectral analysis of acoustic waves.

Contribution

It establishes the asymptotic behavior of solutions to the Navier-Stokes-Fourier system in complex domains, linking compressible and incompressible models through spectral methods.

Findings

Limit velocity field is divergence-free and satisfies Oberbeck-Boussinesq equations.

Convergence is achieved using spectral analysis of the wave propagator.

Handles large domains with rough surfaces as the Mach number tends to zero.

Abstract

We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $\varepsilon \to 0$, the Froude number proportional to $\sqrt{\varepsilon}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $\varepsilon \to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck-Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.