Stable Viscosities and Shock Profiles for Systems of Conservation Laws.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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Many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws. With small dissipative effects neglected, typically smooth solutions must develop discontinuities shocks in finite time. Re-incorporating dissipation helps select those discontinuities which are physically relevant. For this purpose, many different sorts of dissipation will do in particular, the physical viscosity is typically degenerate and not convenient. In this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities. A natural class of admissible viscosity terms is determined based on a simple linearized stability criterion. In addition, they determine a class of degenerate second order viscosity terms of physical type which are admissible. These results are applied to the equations of compressible fluid dynamics, to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction. This should be of interest to others interested in general equations of state for compressible fluids, such as those investigating phase transitions.
- Numerical Mathematics
- Fluid Mechanics