Global strong solutions and asymptotic behavior for arbitrarily large initial data of the 2D compressible Navier-Stokes equations with transport entropy

TL;DR

This paper proves the global existence of strong solutions for large initial data in 2D compressible Navier-Stokes equations with variable entropy, even with vacuum, and analyzes their long-term behavior.

Contribution

It extends the existence theory to non-isentropic fluids with variable entropy and slip boundary conditions, allowing large initial data and vacuum states.

Findings

Global strong solutions exist for large initial data.

Density remains uniformly bounded over time.

Solutions converge to equilibrium as time approaches infinity.

Abstract

In 1995, Kazhikhov and Vaigant introduced a particular class of isentropic compressible Navier-Stokes equations with variable viscosity coefficients and, for the first time, established the existence of global smooth solutions for arbitrarily large initial data in bounded two-dimensional domains. This result was subsequently extended and refined to accommodate more general constraints on the viscosity coefficients. However, because the proofs in this line of work [17,14,15,8] rely heavily on the structure of the isentropic equations, they could not be generalized to the broader setting of multidimensional compressible heat-conductive Navier-Stokes-Fourier systems. In this paper, we consider a special class of non-isentropic compressible fluids governed by the two-dimensional compressible Navier-Stokes equations with variable entropy. In this system, the pressure depends nonlinearly on both density and entropy, and the entropy evolves solely through a transport equation-a feature that distinguishes it from the standard Navier-Stokes-Fourier model. We establish, for the first time, the global existence of strong solutions for arbitrarily large initial data on both two-dimensional periodic domains and bounded domains endowed with Navier-slip boundary conditions. For the bounded-domain case, a key step in our analysis is the derivation of new commutator estimates compatible with the slip condition. Our results hold even when the initial density may contain vacuum and require no smallness assumption on the initial data, provided the shear viscosity is constant and the bulk viscosity follows a power-law form $\lambda(\rho)=\rho^\beta$ with $\beta > 4/3$. Moreover, we demonstrate that the density remains uniformly bounded for all time. Consequently, the solution converges to an equilibrium state as time tends to infinity.