The Cauchy problem for 1-D linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient: A viscous approach

Abstract

In this paper, we consider linear nonconservative Cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface { x = 0 } . For the sake of simplicity, we restrict ourselves to piecewise constant hyperbolic operators of the form ∂ t + A ( x ) ∂ x with A ( x ) = A + 1 x > 0 + A − 1 x < 0 , where A ± ∈ M N ( R ) . Under assumptions, incorporating a sharp spectral stability assumption, we prove that a unique solution is successfully singled out by a vanishing viscosity approach. Due to our framework, which includes systems with expansive discontinuities of the coefficient, the selected small viscosity solution satisfies an unusual hyperbolic problem, which is well-posed even though it does not satisfy, in general, a Uniform Lopatinski Condition. In addition, based on a detailed analysis of our stability assumption, explicit examples of 2 × 2 systems checking our assumptions are given. Our result is new and contains both a stability result and a description of the boundary layers forming, at any order. Two kinds of boundary layers form, each polarized on specific linear subspaces in direct sum.