Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients.

Bruno Fornet (1, 2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

In this paper, two main results are proved. We consider a nonconservative linear Cauchy problem with discontinuous coefficients accross a noncharacteristic hypersurface. We introduce then a viscous perturbation of the problem; the viscous solution u^eps depends of the small positive parameter eps. This problem, obtained by small viscous perturbation, is parabolic for fixed positive eps. We prove then, under stability assumptions, the convergence, when eps vanishes, of u^eps towards the solution of a well-posed limit hyperbolic problem. Our first result is obtained, in the multi-D framework, for piecewise smooth coefficients and states the convergence of u^\eps towards an unique solution. Our second result shows that, in the expansive nonconservative scalar case, that is to say for sign(x a(x))>0, our viscous approach successfully singles out a solution. Even for scalar, piecewise constant 1-D nonconservative hyperbolic equations, this result is new and not treated during our analysis performed on systems. For both results, an asymptotic analysis of the convergenceis performed at any order, containing a boundary layer analysis.

Domaines: Equations aux dérivées partielles [math.AP]

Bruno Fornet. Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients.. 2007. ⟨hal-00172258v2⟩