Stability of the compressible Navier–Stokes equations with entropy transport

Abstract

In this paper, we investigate the stability of the compressible Navier–Stokes system coupled with entropy transport in a periodic domain ${\mathbb{T}}^d$ for $d=2,3$. Specifically, we establish the existence of global solutions for this system with small initial data in the Sobolev space $H^3$ and demonstrate that these solutions exhibit exponential decay over time. The primary challenge in this analysis stems from the absence of thermal conduction (i.e., the thermal conduction coefficient is zero) and the transport nature of the entropy equation. Our approach relies on refined energy estimates, leveraging the smoothing effects arising from the intricate coupling and interaction between the entropy and the fluid dynamics.