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

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+1x>0+A−1x<0, where A±∈MN(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.