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Title of Journal: Commun Math Phys

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Abbravation: Communications in Mathematical Physics

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Springer Berlin Heidelberg

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DOI

10.1007/s12288-012-0214-3

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1432-0916

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Vanishing Viscosity Solutions of the Compressible

Authors: GuiQiang G Chen Mikhail Perepelitsa
Publish Date: 2015/05/15
Volume: 338, Issue: 2, Pages: 771-800
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Abstract

We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids which are motivated by many important physical situations Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at a certain time under some circumstance The central feature is the strengthening of waves as they move radially inward A longstanding open fundamental problem is whether concentration could be formed at the origin In this paper we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions and establish the convergence of the approximate solutions to a global finiteenergy entropy solution of the isentropic Euler equations with spherical symmetry and large initial data This indicates that concentration is not formed in the vanishing viscosity limit even though the density may blow up at a certain time To achieve this we first construct global smooth solutions of appropriate initialboundary value problems for the Euler equations with designed viscosity terms approximate pressure function and boundary conditions and then we establish the strong convergence of the viscosity approximate solutions to a finiteenergy entropy solution of the Euler equations

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