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Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories. (English) Zbl 0943.76097
The authors consider dissipative relativistic fluid theories on fixed flat, compact, globally hyperbolic, Lorentzian manifold. It is shown that for all initial data in a small enough neighborhood of the equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires that the system of equations is symmetric and hyperbolic, a fundamental requisite to have a well-posed and physically consistent initial value formulation. The second condition is a generic consequence of the entropy law, and is imposed on the non-principal part of the equations. The third condition is imposed on the principal part of the equations and implies that the dissipation affects all the fields of the theory. With these requirements the authors prove that all the eigenvalues of the symbol associated to the system of equations of the fluid theory have strictly negative real parts, which in fact, is an alternative characterization for the theory to be totally dissipative. Once this result has been obtained, a straightforward application of a general stability theorem due to Kreiss, Ortiz and Reula, implies the above-mentioned results.

MSC:
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
35Q75 PDEs in connection with relativity and gravitational theory
58J45 Hyperbolic equations on manifolds
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