Global existence of solutions for a nonstrictly hyperbolic system. (English) Zbl 1474.35458

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

Abstr. Appl. Anal. 2014, Article ID 691429, 7 p. (2014).

Summary: We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in [N. Tsuge, J. Math. Kyoto Univ. 46, No. 3, 457–524 (2006; Zbl 1145.35080)], we obtain the uniformly bounded \(L^{\infty}\) estimates \(z(\rho^{\delta, \varepsilon}, u^{\delta, \varepsilon}) \leq B(x)\) and \(w(\rho^{\delta, \varepsilon}, u^{\delta, \varepsilon}) \leq \beta\) when \(a(x)\) is increasing (similarly, \(w(\rho^{\delta, \varepsilon}, u^{\delta, \varepsilon}) \leq B(x)\) and \(z(\rho^{\delta, \varepsilon}, u^{\delta, \varepsilon}) \leq \beta\) when \(a(x)\) is decreasing) for the \(\varepsilon\)-viscosity and \(\delta\)-flux approximation solutions of nonhomogeneous, resonant system without the restriction \(z_0(x) \leq 0\) or \(w_0(x) \leq 0\) as given in [C. Klingenberg and Y.-g. Lu, Commun. Math. Phys. 187, No. 2, 327–340 (1997; Zbl 0883.35074)], 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, [F. Murat, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5, 489–507 (1978; Zbl 0399.46022)] and [L. Tartar, in: Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. 4. London: Pitman. 136–212 (1979; Zbl 0437.35004)], to prove the convergence of the approximation solutions.

MSC:

35L45	Initial value problems for first-order hyperbolic systems
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35D30	Weak solutions to PDEs
35L60	First-order nonlinear hyperbolic equations
35L65	Hyperbolic conservation laws

Citations:

Zbl 1145.35080; Zbl 0883.35074; Zbl 0399.46022; Zbl 0437.35004

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