Uniform regularity in the low Mach number and inviscid limits for the full Navier-Stokes system in domains with boundaries

Abstract

Motivated by the studies on the low Mach number limit problem, this manuscript establishes uniform regularity estimates with respect to the Mach number for the non-isentropic compressible Navier-Stokes system in smooth domains with Navier-slip boundary conditions, in the general case of ill-prepared initial data. The thermal conduction is taken into account and the large variation of temperature is allowed. Moreover, the obtained regularity estimates are also uniform in the Reynolds number $\text{Re}\in[1,+\infty),$ P\'eclet number $\text{Pe}\in [1,+\infty),$ provided $$\big|\frac{1}{\text{Re}}-\frac{\iota_0}{\text{Pe}}\big|\lesssim \frac{1}{\text{Pe}^{\frac{1}{2}}}\frac{1}{\text{Re}},$$ where $\iota_0$ is a fixed constant independent of Mach number, Reynolds number and P\'eclet number. The convergence to the limit system when the Mach number tends to zero is then justified for an exterior domain outside a smooth compact set in $\mathbb{R}^3$ in the spirit of \cite{MR2106119}.Comment: Comments are welcome