Viscous compressible barotropic symmetric flows with free boundary under general mass force Part I: Uniform‐in‐time bounds and stabilization

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Abstract
We consider symmetric flows of a viscous compressible barotropic fluid with a free boundary, under a general mass force depending both on the Eulerian and Lagrangian co‐ordinates, with arbitrarily large initial data. For a general non‐monotone state function p, we prove uniform‐in‐time energy bound and the uniform bounds for the density ρ, together with the stabilization as t → ∞ of the kinetic and potential energies. We also obtain H1‐stabilization of the velocity v to zero provided that the second viscosity is zero. For either increasing or non‐decreasing p, we study the Lλ‐stabilization of ρ and the stabilization of the free boundary together with the corresponding ω‐limit set in the general case of non‐unique stationary solution possibly with zones of vacuum. In the case of increasing p and stationary densities ρS separated from zero, we establish the uniform‐in‐time H1‐bounds and the uniform stabilization for ρ and v. All these results are stated and mainly proved in the Eulerian co‐ordinates. They are supplemented with the corresponding stabilization results in the Lagrangian co‐ordinates in the case of ρS separated from zero. Copyright © 2005 John Wiley & Sons, Ltd.

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