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From the highly compressible Navier-Stokes equations to the Porous Medium equation - rate of convergence
We consider the one-dimensional Cauchy problem for the Navier-Stokes
equations with degenerate viscosity coefficient in highly compressible regime.
It corresponds to the compressible Navier-Stokes system with large Mach number
equal to for going to . When
the initial velocity is related to the gradient of the initial density, a
solution to the continuity equation- converges to the unique
solution to the porous medium equation [13,14]. For viscosity coefficient
with , we obtain a
rate of convergence of in ; for the solution
converges in . For compactly
supported initial data, we prove that most of the mass corresponding to
solution is located in the support of the solution to the
porous medium equation. The mass outside this support is small in terms of
.Comment: 19 page
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