No 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 boundedL∞estimatesz(ρδ,ε,uδ,ε)≤B(x)andw(ρδ,ε,uδ,ε)≤βwhena(x)is increasing (similarly,w(ρδ,ε,uδ,ε)≤B(x)andz(ρδ,ε,uδ,ε)≤βwhena(x)is decreasing) for theε-viscosity andδ-flux approximation solutions of nonhomogeneous, resonant system without the restrictionz0(x)≤0orw0(x)≤0as given in Klingenberg and Lu (1997), wherezandware Riemann invariants of nonhomogeneous, resonant system;B(x)>0is a uniformly bounded function ofxdepending only on the functiona(x)given in nonhomogeneous, resonant system, andβis the bound ofB(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

In this paper, we apply the maximum principle and the theory of compensated compactness to establish an existence theorem for global weak solutions to the Cauchy problem of the non-strictly hyperbolic system-a special system of Euler equation with a general source term.