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 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.

Vanishing viscosity for a $ 2\times 2 $ system modeling congested vehicular traffic

“…-see, e.g. [10,22,31,40] and references therein. The proofs rely on a compensated compactness argument: the key idea, introduced by Tartar and Murat (see, e.g., [16,Chapter 5] for a survey), is as follows: the invariant region method provides uniform L ∞ bounds on the sequence of viscous approximation, but the weak-star convergence does not allow to pass to limit in the nonlinear terms of the equations; however, the weak limit can be represented in terms of Young measures, which reduce to a Dirac mass (hence giving strong convergence) due to the entropy dissipation mechanism.…”