Long-time behavior of weak solutions for compressible Navier-Stokes equations with degenerate viscosity

TL;DR

This paper investigates the long-time behavior of weak solutions to the compressible Navier-Stokes equations with degenerate viscosity, demonstrating positivity of density, exponential decay to equilibrium, and conditions under which solutions become strong.

Contribution

It establishes the positivity of density after finite time, derives exponential decay rates, and shows conditions for weak solutions to become strong in two dimensions.

Findings

Density remains strictly positive after finite time.

Weak solutions decay exponentially to equilibrium in L^2 norm.

Under zero initial momentum, solutions become strong in 2D.

Abstract

The long-time regularity and asymptotic of weak solutions are studied for compressible Navier-Stokes equations with degenerate viscosity in a bounded periodic domain in two and three dimensions. It is shown that the density keeps strictly positive from below and above after a finite period of time. Moreover, higher velocity regularity is obtained via a parabolic type iteration technique. Since then the weak solution conserves its energy equality, and decays exponentially to the equilibrium in $L^{2}$-norm as time goes to infinity. In addition, assume that the initial momentum is zero, the exponential decay rate is derived for the derivative functions, and the weak solution becomes a strong one in two dimensional space.