Quasilinear symmetric hyperbolic-hyperbolic systems of second or mixed order, with applications to relativistic fluid dynamics

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The project characterizes and studies two classes of quasilinear systems of dissipative partial differential equations which are hyperbolic regarding both their leading second order and their first order parts. Applications of both classes occur notably in dissipative relativistic fluid dynamics (DRFD). In the first phase of the project, we successfully identified structural features that make such "symmetric hyperbolic-hyperbolic" systems dissipative in the sense that their linearizations at homogeneous states enjoy decay in L2 based Sobolev spaces; during that step, the scope of the project widened, as these criteria cover broader classes of models than we had foreseen, notably allowing for situations that do not fall under the Hughes-Kato-Marsden framework. The purpose of the work planned for the second phase consists in establishing asymptotic stability in the quasilinear contexts. As we do no longer assume the definiteness encoded in the Hughes-Kato-Marsden condition, this will require combining the dissipativity conditions with the pseudo- and paradifferential calculus of Hörmander, Bony and Meyer used in the spirit of Taylor and Metivier. The resulting general findings should allow us to generalize Sroczinski's theorems on the global existence and long-time behavior of solutions to certain causal descriptions of DRFD to a broad class of such formulations ('relativistic Navier-Stokes').