We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an L p framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u ε , that are globally defined solutions to u ε t +A(u ε ) u ε x + g(u ε ) = εu ε xx , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L 1 with respect to the initial data. Finally, as ε → 0, u ε converges in L 1 loc to the admissible weak solution u of the system of balance laws u t + (f (u)) x + g(u) = 0 when A = Df .

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In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order t −4 in the Gauss curvature. Here we show that the decay of Hong [9] t −2−δ/2 where δ ∈ (0, 4) suffices.Mathematics Subject Classification (2010).

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