Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

By Hai-Liang Li, Jing Li, Zhouping Xin. 2008

254 citations recorded by the source corpus.

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 the 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/272e6ec7740aa641a80942e99e9df2b7987ec9a0

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Well‐posedness of Compressible Euler Equations in a Physical Vacuum (2010, background): “Since then, there has been a lot of studies on global weak solutions for various models and stabilization results under gravitation and external forces: see [39, 68] and the references therein.”
Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities (2015, background): “In particular, for one-dimensional case, Li-Li-Xin [25] obtained the global existence of weak solutions to (1.1) (1.3) with h(ρ) = g(ρ) = ρα(α > 1/2) and proved that for any global entropy weak solution, any vacuum state must vanish within the finite time.”