Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations

Details Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations

2010-11-09

arxiv.org/abs/1011.1991v2

Mathematics

Analysis of PDEs

23 pages

Scientific paper

It is well-known that one-dimensional isentropic gas dynamics has two elementary waves, i.e., shock wave and rarefaction wave. Among the two waves, only the rarefaction wave can be connected with vacuum. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we can construct a sequence of solutions to one-dimensional compressible isentropic Navier-Stokes equations which converge to the above rarefaction wave with vacuum as the viscosity tends to zero. Moreover, the uniform convergence rate is obtained. The proof consists of a scaling argument and elementary energy analysis based on the underlying rarefaction wave structures.

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