Stability of solutions to the Cauchy problem of symmetric hyperbolic systems

In this report we investigate the linear and nonlinear stability of stationary, constant solutions to the Cauchy problem of quasilinear, symmetric hyperbolic systems of equations. Writing the problem as u t = d X j=0 ? A 0j + "A 1j (x; t; u; ") with u(t = 0) = f(x); we say that the problem is non-linearly stable if, for " small enough, the solution u stays smooth for all t 0 and its maximum norm tends to zero for t ! 1. In this report we give suucient conditions for non-linear stability. This work generalizes a recent work by Kreiss, Kreiss and Lorenz where analogous stability results are shown for systems of conservation laws. This generalization is necessary for some applications such us the theories of relativistic dissipative uids. Research supported by a fellowship of the Consejo Nacional de Investigaciones Cien-t cas y T ecnicas (CONICET), Rep ublica Argentina.