Stability of Shock Waves under Hyperbolic Dissipation

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This project studies PDE systems of compressible fluid dynamics that model the dissipative mechanisms of viscosity and heat conduction not parabolically as the classical Navier-Stokes-Fourier (NSF) description, but, like the non-dissipative part of the evolution, through hyperbolic operators. The issues of possible non-uniqueness, conceivable wild solutions, likely turbulence, and hoped for vanishing-viscosity limit arise for these models just as for the classical one. However, they appear on different, hyperbolic scales. To preexamine these scales, the project exemplarily investigates the stability of prototypical shock waves in hyperbolic-hyperbolic models, aiming at (a) an understanding as good as the presently existing theory for shock waves in the hyperbolic-parabolic setting and (b) a description of limits from the hyperbolic-hyperbolic scales to the classical hyperbolic-parabolic scale. The main emphasis is on second-order systems that consist exclusively of the five conservation laws for mass, momentum and energy, and incorporate viscosity and heat conduction through operators using second-order space and time derivatives of the state variables. In recent years, the literature has discussed such models notably in the relativistic setting. A smaller part of the project deals with (non-relativistic) hyperbolic models of first order in which the dissipation is described by relaxing balance laws for additional fields, in the spirit of Extended Thermodynamics. The scaling limits we plan to study from the point of view of shock wave stability are that of infinite light speed, i.e., from second-order hyperbolic-hyperbolic to classical compressible NSF, and that of vanishing relaxation times, i.e., from first-order hyperbolic-hyperbolic to classical compressible NSF.