Abstract and Applied Analysis

Global Existence of Solutions for a Nonstrictly Hyperbolic System

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

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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 ( ρ δ , ε , u δ , ε ) B ( x ) and w ( ρ δ , ε , u δ , ε ) β when a ( x ) is increasing (similarly, w ( ρ δ , ε , u δ , ε ) B ( x ) and z ( ρ δ , ε , u δ , ε ) β when a ( x ) is decreasing) for the ε -viscosity and δ -flux approximation solutions of nonhomogeneous, resonant system without the restriction z 0 ( x ) 0 or w 0 ( 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.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 691429, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279781

Digital Object Identifier
doi:10.1155/2014/691429

Mathematical Reviews number (MathSciNet)
MR3193536

Zentralblatt MATH identifier
07022888

Citation

Zheng, De-yin; Lu, Yun-guang; Song, Guo-qiang; Lu, Xue-zhou. Global Existence of Solutions for a Nonstrictly Hyperbolic System. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 691429, 7 pages. doi:10.1155/2014/691429. https://projecteuclid.org/euclid.aaa/1412279781


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