Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system

Abstract

We show that the weak solutions of the nonlinear hyperbolic system $$\left\{ \begin{gathered} \varepsilon u_t^\varepsilon + p(v^\varepsilon )_x = u^\varepsilon \hfill \\ v_t^\varepsilon - u_x^\varepsilon = 0 \hfill \\ \end{gathered} \right.$$ converge, as e tends to zero, to the solutions of the reduced problem $$\left\{ \begin{gathered} u + p(v)_x = 0 \hfill \\ v_t - u_x = 0 \hfill \\ \end{gathered} \right.$$ . Then they satisfy the nonlinear parabolic equation $$v_t + p(v)_{XX} = 0$$ . The limiting procedure is carried out using the techniques of “Compensated Compactness”. Some connections with the theory of nonlinear heat conduction and the theory of nonlinear diffusion in a porous medium are suggested. The main result is stated in th. (2.9).