The Cauchy problem for a quasilinear system of hyperbolic-parabolic equations is addressed with the method of linearization and fixed point. Coupling between the hyperbolic and parabolic variables is allowed and we do not assume the Friedrich's symmetrizability of the system. This coupling results in weaker linear energy estimates and Banach's fixed point theorem cannot be applied. A metric fixed point theorem is developed in order to conclude the local existence and uniqueness of solutions. We apply our results to the Cattaneo-Christov system for viscous compressible fluid flow, a system of equations whose inviscid part is not hyperbolic.
Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density
On the local existence of solutions to the Navier-Stokes-wave system with a free interface
Local existence of solutions to the Euler-Poisson system, including densities without compact support
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