  We consider dissipative relativistic fluid theories on a fixed flat, compact,
globally hyperbolic, Lorentzian manifold. We prove 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 the system of equations to be symmetric 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 it implies that the
dissipation affects all the fields of the theory. With these requirements we
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 straight forward application of a general
stability theorem due to Kreiss, Ortiz, and Reula, implies the results above
mentioned.
