Global regular solutions for 1-D degenerate compressible Navier-Stokes equations with large data and far field vacuum

TL;DR

This paper proves the global existence of regular solutions for 1-D degenerate compressible Navier-Stokes equations with large data and vacuum at infinity, using novel reformulations and entropy estimates.

Contribution

It establishes the global well-posedness of solutions with density remaining positive and decaying at infinity, addressing degeneracies in the viscosity and singular structures.

Findings

Global-in-time regular solutions with positive density everywhere

Solutions conserve total mass, momentum, and finite energy

Density decays to zero at infinity, consistent with physical models

Abstract

In this paper, the Cauchy problem for the one-dimensional (1-D) isentropic compressible Navier-Stokes equations (\textbf{CNS}) is considered. When the viscosity $\mu(\rho)$ depends on the density $\rho$ in a sublinear power law ($ \rho^\delta$ with $0<\delta\leq 1$), based on an elaborate analysis of the intrinsic singular structure of this degenerate system, we prove the global-in-time well-posedness of regular solutions with conserved total mass, momentum, and finite total energy in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained satisfy that $\rho$ keeps positive for all point $x\in \mathbb{R}$ but decays to zero in the far field, which is consistent with the facts that the total mass of the whole space is conserved, and \textbf{CNS} is a model of non-dilute fluids where $\rho$ is bounded below away from zero. The key to the proof is the introduction of a well-designed reformulated structure by introducing some new variables and initial compatibility conditions, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. Then, combined with the BD entropy estimates and transport properties of the so-called effective velocity $v=u+\varphi(\rho)_x$ ($u$ is the velocity of the fluid, and $\varphi(\rho)$ is a function of $\rho$ defined by $\varphi'(\rho)=\mu(\rho)/\rho^2$), one can obtain the required uniform a priori estimates of corresponding solutions.