GLOBAL EXISTENCE AND EXPONENTIAL DECAY FOR HYPERBOLIC DISSIPATIVE RELATIVISTIC FLUID THEORIES

We consider dissipative relativistic fluid theories an a fixed hat, gl obally hyperbolic, Lorentzian manifold (R x T-3,g(ab)). We prove that for all initial data in a small enough neighborhood of the constant eq uilibrium states (in an appropriate Sobolev norm), the solutions evolv e smoothly in time forever and decay exponentially to some, in general undetermined, constant equilibrium state. To prove this, three condit ions are imposed on these theories. The first condition requires the s ystem of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulat ion. For the fiat space-times considered here the equilibrium states a re constant, which is used in the proof. The second condition is a gen eric consequence of the entropy law, and is imposed on the non-princip al part of the equations. The third condition is imposed on the princi pal part of the equations and it implies that the dissipation affects all the fields of the theory. With those requirements we prove that al l 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 dissi pative. Once this result has been obtained, a straightforward applicat ion of a general stability theorem due to Kreiss, Ortiz, and Reula imp lies the results mentioned above. (C) 1997 American Institute of Physi cs.