Global Existence of Solutions for a Nonstrictly Hyperbolic System

Abstract

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded $L^{\mathrm{\infty }}$ estimates $z({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le B(x)$ and $w({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le \beta$ when $a(x)$ is increasing (similarly, $w({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le B(x)$ and $z({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le \beta$ when $a(x)$ is decreasing) for the $\epsilon$-viscosity and $\delta$-flux approximation solutions of nonhomogeneous, resonant system without the restriction $z_0(x)\le 0$ or $w_0(x)\le 0$ as given in Klingenberg and Lu (1997), where $z$ and $w$ are Riemann invariants of nonhomogeneous, resonant system; $B(x)>0$ is a uniformly bounded function of $x$ depending only on the function $a(x)$ given in nonhomogeneous, resonant system, and $\beta$ is the bound of $B(x)$. Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.