## Abstract and Applied Analysis

### 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 bounded ${L}^{\mathrm{\infty }}$ estimates $z({\rho }^{\delta ,\epsilon },{u}^{\delta ,\epsilon })\le B(x)$ and $w({\rho }^{\delta ,\epsilon },{u}^{\delta ,\epsilon })\le \beta$ when $a(x)$ is increasing (similarly, $w({\rho }^{\delta ,\epsilon },{u}^{\delta ,\epsilon })\le B(x)$ and $z({\rho }^{\delta ,\epsilon },{u}^{\delta ,\epsilon })\le \beta$ when $a(x)$ is decreasing) for the $\epsilon$-viscosity and $\delta$-flux approximation solutions of nonhomogeneous, resonant system without the restriction ${z}_{0}(x)\le 0$ or ${w}_{0}(x)\le 0$ 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 $\beta$ 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

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

#### References

• P. G. LeFloch and M. Westdickenberg, “Finite energy solutions to the isentropic Euler equations with geometric effects,” Journal de Mathématiques Pures et Appliquées, vol. 88, no. 5, pp. 389–429, 2007.
• N. Tsuge, “Global ${L}^{\infty }$ solutions of the compressible Euler equations with spherical symmetry,” Journal of Mathematics of Kyoto University, vol. 46, no. 3, pp. 457–524, 2006.
• G. Q. Chen and J. Glimm, “Global solutions to the compressible Euler equations with geometrical structure,” Communications in Mathematical Physics, vol. 180, no. 1, pp. 153–193, 1996.
• J. M. Hong and B. Temple, “A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law,” SIAM Journal on Applied Mathematics, vol. 64, no. 3, pp. 819–857, 2004.
• J. M. Hong, “An extension of Glimms method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the Riemann problem,” Journal of Differential Equations, vol. 222, no. 2, pp. 515–549, 2006.
• Y. C. Su, J. M. Hong, and S. W. Chou, “An extension of Glimm's method to the gas dynamical model of transonic flows,” Nonlinearity, vol. 26, no. 6, pp. 1581–1597, 2013.
• E. Isaacson and B. Temple, “Nonlinear resonance in systems of conservation laws,” SIAM Journal on Applied Mathematics, vol. 52, no. 5, pp. 1260–1278, 1992.
• K. Oelschläger, “On the connection between Hamiltonian many-particle systems and the hydrodynamical equations,” Archive for Rational Mechanics and Analysis, vol. 115, no. 4, pp. 297–310, 1991.
• K. Oelschläger, “An integro-differential equation modelling a Newtonian dynamics and its scaling limit,” Archive for Rational Mechanics and Analysis, vol. 137, no. 2, pp. 99–134, 1997.
• S. Caprino, R. Esposito, R. Marra, and M. Pulvirenti, “Hydrodynamic limits of the Vlasov equation,” Communications in Partial Differential Equations, vol. 18, no. 5-6, pp. 805–820, 1993.
• R. J. DiPerna, “Global solutions to a class of nonlinear hyperbolic systems of equations,” Communications on Pure and Applied Mathematics, vol. 26, pp. 1–28, 1973.
• C. Klingenberg and Y. G. Lu, “Existence of solutions to hyperbolic conservation laws with a source,” Communications in Mathematical Physics, vol. 187, no. 2, pp. 327–340, 1997.
• Y. G. Lu, “Existence of global entropy solutions of a nonstrictly hyperbolic system,” Archive for Rational Mechanics and Analysis, vol. 178, no. 2, pp. 287–299, 2005.
• Y. G. Lu, “Global existence of solutions to resonant system of isentropic gas dynamics,” Nonlinear Analysis, vol. 12, no. 5, pp. 2802–2810, 2011.
• Y. G. Lu, “Some results for general systems of isentropic gas dynamics,” Differential Equations, vol. 43, no. 1, pp. 130–138, 2007.
• F. Murat, “Compacité par compensation,” Annali della Scuola Normale Superiore di Pisa, vol. 5, no. 3, pp. 489–507, 1978.
• L. Tartar, “Compensated compactness and applications to partial differential equations,” in Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, R. J. Knops, Ed., vol. 4, Pitman Press, London, UK, 1979.
• P. L. Lions, B. Perthame, and P. E. Souganidis, “Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates,” Communications on Pure and Applied Mathematics, vol. 49, no. 6, pp. 599–638, 1996.
• P. L. Lions, B. Perthame, and E. Tadmor, “Kinetic formulation of the isentropic gas dynamics and $p$-systems,” Communications in Mathematical Physics, vol. 163, no. 2, pp. 415–431, 1994. \endinput