A note on the existence of L^2 valued solutions for a hyperbolic system with boundary conditions
Résumé
We prove existence of L 2-weak solutions of a p-system with boundary conditions. This is done using the vanishing viscosity with mixed Dirichlet-Neumann boundary conditions. Under these boundary conditions the free energy decreases and provides a uniform a priori estimate in L 2 , allowing us to use L 2 Young measures, together with the classical tools of compensated compactness. We then obtain that the viscous solutions converge to weak solutions of the p-system strongly in L p , for any p ∈ [1.2), that satisfy the boundary conditions in the sense given by Definition 2.1. Furthermore the free energy decreases along these solutions.
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