BV solutions for a class of viscous hyperbolic systems

The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: (*) u_t + A(u)u_x = ε u_xx, u(0,x) = ū(x). We assume that the integral curves of the eigenvectors r_i of the matrix A are straight lines. On the other hand, we do not require the system (*) to be in conservation form, nor do we make any assumption on genuine linearity or linear degeneracy of the characteristic fields. In this setting we prove that, for some small constant η₀ > 0 the following holds. For every initial data ū ∈ L¹ with Tot. Var. {ū} < η₀, the solution u^ε of (*) is well defined for all t > 0. The total variation of u^ε(t, ·) satisfies a uniform bound, independent of t, ε. Moreover, as ε → 0+, the solutions u^ε(t, ·) converge to a unique limit u(t, ·). The map (t, ū) → S_tū; (approaches the limit) u(t, ·) is a Lipschitz continuous semigroup on a closed domain D ⊂ L¹ of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine. The results above 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 coefficient 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 different speeds.