## Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass.(English)Zbl 1481.35330

Summary: Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., $$\alpha = 1$$, $$\gamma = 2$$).

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35Q31 Euler equations 35B25 Singular perturbations in context of PDEs 35B44 Blow-up in context of PDEs 35L65 Hyperbolic conservation laws 35L67 Shocks and singularities for hyperbolic equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35R09 Integro-partial differential equations 35R35 Free boundary problems for PDEs 35D30 Weak solutions to PDEs 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76N17 Viscous-inviscid interaction for compressible fluids and gas dynamics
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### References:

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