Global Existence of Solutions for a Nonstrictly Hyperbolic System

Zheng De-yin, Yun-guang Lu, Guo-qiang Song and Xue-zhou Lu

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

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∞ estimates z(ρδ,ε, uδ,ε) ≤ B(x) and w(ρδ,ε, uδ,ε) ≤ β when a(x) is increasing (similarly, w(ρδ,ε, uδ,ε) ≤ B(x) and z(ρδ,ε, uδ,ε) ≤ β when a(x) is decreasing) for the ε‐viscosity and δ‐flux approximation solutions of nonhomogeneous, resonant system without the restriction z0(x) ≤ 0 or w0(x) ≤ 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 β 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.

Date: 2014
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https://doi.org/10.1155/2014/691429

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:691429

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