Propagation des oscillations dans les systemes hyperboliques non lineaires

Abstract

We study in this note the solutions of hyperbolic 2 × 2 systems of conservation laws with oscillating data, e.g. $u^\varepsilon (x,0) = a(x\tfrac{X}{\varepsilon }),a(x,.)$ being periodic. The analysis is done using the compensated compactness theory, following an idea of L. Tartar. The initial oscillations can be killed in a small time if the system is fully nonlinear. But they propagate in linearly degenerate systems. A general analysis gives rise to a relaxed problem, for which the unknown is a field U (x,t,y), y ε]0,1[. The resulting system is differential in (x,t), and y is a coupling parameter. When the initial problem is a system of balanced laws, then the relaxed problem is also integrodifferential in y.