Vanishing Viscosity Solutions of Hyperbolic Systems on Manifolds

The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: u t + A(u)u x = " u xx ; u(0; x) = u(x): () We assume that the integral curves of the eigenvectors r i of the matrix A are straight lines. On the other hand, do not require the system () to be in conservation form, nor we make any assumption on genuine linearity or linear degeneracy of the characteristic elds. In this setting we prove that, for some small constant 0 > 0 the following holds. For every initial data u 2 L 1 with Tot.Var.f ug < 0 , the solution u " of (*) is well deened for all t > 0. The total variation of u " (t;) satisses a uniform bound, independent of t; ". Moreover, as " ! 0+, the solutions u " (t;) converge to a unique limit u(t;). The map (t; u) 7 ! S t u : = u(t;) is a Lipschitz continuous semigroup on a closed domain D L 1 of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine. The above results can also be applied to strictly hyperbolic systems on a Riemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz semigroup of \entropic" solutions, within a class of (possibly discontinuous) functions with small total variation. The semigroup trajectories can be obtained as the unique limits of solutions to a particular parabolic system, as the viscosity coeecient approaches zero. The proofs rely on some new a priori estimates on the total variation of solutions for a parabolic system whose components drift with strictly diierent speeds.