## Abstract and Applied Analysis

### Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

#### Abstract

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 296361, 15 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512210

Digital Object Identifier
doi:10.1155/2013/296361

Mathematical Reviews number (MathSciNet)
MR3143557

Zentralblatt MATH identifier
1294.35082

#### Citation

Yin, Gan; Song, Kyungwoo. Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes. Abstr. Appl. Anal. 2013 (2013), Article ID 296361, 15 pages. doi:10.1155/2013/296361. https://projecteuclid.org/euclid.aaa/1393512210

#### References

• S. Wein, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, NY, USA, 1972.
• T. Chang and L. Hsiao, “The riemann problem and interaction of waves in gas dynamics,” in Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, Longman Scientific and Technical, Essex, England, UK, 1989.
• R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Wiley, New York, NY, USA, 1948.
• J. Glimm, “Solutions in the large for nonlinear hyperbolic systems of equations,” Communications on Pure and Applied Mathematics, vol. 18, no. 4, pp. 697–715, 1965.
• J. L. Johnson, “Global continuous solutions of hyperbolic systems of quasi-linear equations,” Bulletin of the American Mathematical Society, vol. 73, pp. 639–641, 1967.
• P. D. Lax, “Development of singularities of solutions of nonlinear hyperbolic partial differential equations,” Journal of Mathematical Physics, vol. 5, pp. 611–613, 1964.
• B. L. Rozdestvenskii and N. N. Yanenko, Systems of Quasilinear Hyperbolic Equations, Izd Nauka, Moskva, Russia, 1968.
• R. Smith, “The Riemann problem in gas dynamics,” Transactions of the American Mathematical Society, vol. 249, no. 1, pp. 1–50, 1979.
• A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford University Press, Oxford, UK, 2000.
• P. G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lectures in Mathematics, ETH Zrich, Birkhuser, 2002.
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, NY, USA, 1983.
• Y. Brenier, “Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations,” Journal of Mathematical Fluid Mechanics, vol. 7, no. supplement 3, pp. S326–S331, 2005.
• D. Serre, “Multidimensional shock interaction for a Chaplygin gas,” Archive for Rational Mechanics and Analysis, vol. 191, no. 3, pp. 539–577, 2009.
• Z. Wang and Q. Zhang, “The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations,” Acta Mathematica Scientia B, vol. 32, no. 3, pp. 825–841, 2012.
• E. Weinan, Yu. G. Rykov, and Ya. G. Sinai, “Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics,” Communications in Mathematical Physics, vol. 177, no. 2, pp. 349–380, 1996.
• S. F. Shandarin and Ya. B. Zel'dovich, “The large-scale structure of the universe: turbulence, intermittency, structures in a self-gravitating medium,” Reviews of Modern Physics, vol. 61, no. 2, pp. 185–220, 1989.
• W. Sheng and T. Zhang, “The Riemann problem for the transportation equations in gas dynamics,” Memoirs of the American Mathematical Society, vol. 137, no. 654, 1999.
• Y. Brenier and E. Grenier, “Sticky particles and scalar conservation laws,” SIAM Journal on Numerical Analysis, vol. 35, no. 6, pp. 2317–2328, 1998.
• H. Kalisch and D. Mitrović, “Singular solutions of a fully nonlinear $2\times 2$ system of conservation laws,” Proceedings of the Edinburgh Mathematical Society, vol. 55, no. 3, pp. 711–729, 2012.
• V. M. Shelkovich, “Singular solutions of $\delta$- and ${\delta }^{'}$-shock wave type of systems of conservation laws, and transport and concentration processes,” Rossiĭskaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo, vol. 63, no. 3, pp. 73–146, 2008.
• A. Anile, Relativistic Fluids and Magnetouids, Cambridge University Press, London, UK, 1989.
• J. Ma. Martí and E. Müller, “The analytical solution of the Riemann problem in relativistic hydrodynamics,” Journal of Fluid Mechanics, vol. 258, pp. 317–333, 1994.
• A. Taub, “Relativistic uid mechanics,” Annual Review of Fluid Mechanics, vol. 10, pp. 301–332, 1978.
• J. Smoller and B. Temple, “Global solutions of the relativistic Euler equations,” Communications in Mathematical Physics, vol. 156, no. 1, pp. 67–99, 1993.
• J. Chen, “Conservation laws for the relativistic $p$-system,” Communications in Partial Differential Equations, vol. 20, no. 9-10, pp. 1605–1646, 1995.
• C.-H. Hsu, S.-S. Lin, and T. Makino, “On the relativistic Euler equation,” Methods and Applications of Analysis, vol. 8, no. 1, pp. 159–207, 2001.
• P. G. LeFloch and M. Yamazaki, “Entropy solutions of the Euler equations for isothermal relativistic fluids,” International Journal of Dynamical Systems and Differential Equations, vol. 1, no. 1, pp. 20–37, 2007.
• G.-Q. Chen and Y. Li, “Stability of Riemann solutions with large oscillation for the relativistic Euler equations,” Journal of Differential Equations, vol. 202, no. 2, pp. 332–353, 2004.
• Y. Li, D. Feng, and Z. Wang, “Global entropy solutions to the relativistic Euler equations for a class of large initial data,” Zeitschrift für Angewandte Mathematik und Physik, vol. 56, no. 2, pp. 239–253, 2005.
• L. Min and S. Ukai, “Non-relativistic global limits of weak solutions of the relativistic Euler equation,” Journal of Mathematics of Kyoto University, vol. 38, no. 3, pp. 525–537, 1998.
• L. Ruan and C. Zhu, “Existence of global smooth solution to the relativistic Euler equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 60, no. 6, pp. 993–1001, 2005.
• M. Ding and Y. Li, “Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations,” Zeitschrift für Angewandte Mathematik und Physik, vol. 64, no. 1, pp. 101–121, 2013.
• R. Pan and J. A. Smoller, “Blowup of smooth solutions for relativistic Euler equations,” Communications in Mathematical Physics, vol. 262, no. 3, pp. 729–755, 2006.
• H. Cheng and H. Yang, “Riemann problem for the relativistic Chaplygin Euler equations,” Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 17–26, 2011.
• S. Chaplygin, “On gas jets,” Scientific Memoirs, Moscow University Mathematic Physics, vol. 21, pp. 1–121, 1904.
• J. Li, “Note on the compressible Euler equations with zero temperature,” Applied Mathematics Letters, vol. 14, no. 4, pp. 519–523, 2001.
• G.-Q. Chen and H. Liu, “Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,” SIAM Journal on Mathematical Analysis, vol. 34, no. 4, pp. 925–938, 2003.
• G. Yin and W. Sheng, “Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 594–605, 2009.
• G. Yin and W. Sheng, “Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations,” Chinese Annals of Mathematics B, vol. 29, no. 6, pp. 611–622, 2008.
• D. Mitrović and M. Nedeljkov, “Delta shock waves as a limit of shock waves,” Journal of Hyperbolic Differential Equations, vol. 4, no. 4, pp. 629–653, 2007.
• G.-Q. Chen and H. Liu, “Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,” Physica D, vol. 189, no. 1-2, pp. 141–165, 2004.
• C. Shen and M. Sun, “Formation of delta shocks and vacuum states in the vanishing pressure limit of Riemann solutions to the perturbed Aw-Rascle model,” Journal of Differential Equations, vol. 249, no. 12, pp. 3024–3051, 2010.
• V. G. Danilov and V. M. Shelkovich, “Dynamics of propagation and interaction of $\delta$-shock waves in conservation law systems,” Journal of Differential Equations, vol. 211, no. 2, pp. 333–381, 2005.