Global strong solutions and asymptotic behavior for arbitrarily large initial data of the 2D compressible Navier-Stokes equations with transport entropy
Jie Fan, Xiangdi Huang
TL;DR
This work establishes the global existence of strong solutions for the 2D compressible Navier–Stokes equations with entropy transport and large initial data. By reformulating the system with the Z-transform and proving robust density–entropy coupling bounds, the authors obtain uniform-in-time density bounds under β>4/3 (and time-independent bounds under β>3/2) and derive higher-order regularity through Hoff-type and Beale–Kato–Majda-type techniques. A vacuum-approximation framework and commutator estimates tailored to Navier–slip boundaries enable global existence on both periodic and bounded domains, with the density uniformly bounded for all time and convergence to equilibrium. The results extend previous isentropic/global theories to the non-isentropic, transport-entropy setting and provide a rigorous foundation for long-time behavior in two-dimensional fluid models with variable entropy and density-dependent viscosity. Key technical contributions include new commutator estimates compatible with slip conditions and a detailed density-transportEntropy interplay via the Z-variable formulation.
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 $λ(ρ)=ρ^β$ with $β> 4/3$. 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.