Limit behavior of the approximate solutions to hyperbolic conservation laws with two linearly degenerate fields on some zero measure sets (Q5945669)

The author considers the \(2\times 2\) nonlinear hyperbolic system NEWLINE\[NEWLINEu_t+f(u,v)_x=0,\qquad v_t+g(u,v)_x=0NEWLINE\]NEWLINE subjected to \(L^{\infty}\) initial data \((u,v)(x,0)=(u_0(x),v_0(x)),\) where \((f,g):{\mathbb R}^2\to {\mathbb R}^2\) is a smooth nonlinear mapping. Assuming that two characteristic fields are linearly degenerate on a certain zero measure set, one proves that, if the system is hyperbolic, then each sequence of approximate solutions satisfying some extra technical conditions, has at least one pointwise almost everywhere convergent subsequence to an admissible \(L^{\infty}\) solution.