Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations

TL;DR

This paper investigates the behavior of solutions to compressible Navier-Stokes equations with vacuum states, proving vacuum vanishing in finite time, velocity blow-up, and convergence to equilibrium, with implications for shallow water models.

Contribution

It establishes the global existence, uniqueness, and regularity of solutions, and rigorously analyzes vacuum dynamics, including finite-time vacuum vanishing and solution blow-up.

Findings

Vacuum states vanish within finite time.

Velocity blows up as vacuum states vanish.

Solutions tend to equilibrium exponentially after vacuum disappearance.

Abstract

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.