Existence and asymptotic behavior of measure-valued solutions for degenerate conservation laws (Q1913668)

A one-dimensional system of \(n\) conservation laws \(\partial U/\partial t+ \partial F(U)/\partial x= 0\) with a nonlinearity \(F: \mathbb{R}^n\to \mathbb{R}^n\) is considered. The hyperbolicity, consisting in linear independence of \(n\) eigenvectors of the \(n\times n\)-matrix \(\nabla F(U)\) with real eigenvalues, is allowed to degenerate. This enables to treat also symmetrical systems where \(F(U)= \varphi(U) U\). A Young-measure valued solution is defined, and its existence is proved. Besides, the asymptotic behaviour of this solution is investigated, showing its convergence a.e. to Dirac measures, indicating also the uniqueness of the asymptotic equilibrium. Systems of Maxwell equations for electromagnetic planar fields are investigated, too.