Formation and propagation of singularities in one-dimensional Chaplygin gas. (English) Zbl 1284.35260
Summary: In this paper, we investigate the formation and propagation of singularities for the system for one-dimensional Chaplygin gas, which is described by a quasilinear hyperbolic system with linearly degenerate characteristic fields. The phenomena of concentration and the formation of “\({\delta}\)-shock” waves are identified and analyzed systematically for this system under suitably large initial data. In contrast to the Rankine-Hugoniot conditions for classical shock, the generalized Rankine-Hugoniot conditions for “\({\delta}\)-shock” waves are established. Finally, it is shown that the total mass and momentum related to the solution are independent of time.
MSC:
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76N15 | Gas dynamics (general theory) |
Keywords:
system for Chaplygin gas; linearly degenerate characteristic; blowup; singularity; \(\delta\)-shockReferences:
| [1] | Alinhac, S., Blowup for nonlinear hyperbolic equations, (Progress in Nonlinear Differential Equations and Their Applications, Vol. 17 (1995), Birkhäuser) · Zbl 0820.35001 |
| [2] | Brenier, Y., Some geometric PDEs related to hydrodynamics and electrodynamics, Proc. ICM, 3, 761-772 (2002) · Zbl 1136.37355 |
| [4] | Christodoulou, D.; Miao, S., Compressible flow and Euler’s equations · Zbl 1329.76002 |
| [5] | Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26, 1097-1119 (1977) · Zbl 0377.35051 |
| [6] | John, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27, 377-405 (1974) · Zbl 0302.35064 |
| [7] | Kong, D.-X., Formation and propagation of singularities for 2×2 quasilinear hyperbolic systems, Trans. Amer. Math. Soc., 354, 3155-3179 (2002) · Zbl 1003.35079 |
| [8] | Kong, D.-X.; Zhang, Qiang, Solutions formula and time-periodicity for the motion of relativistic strings in the Minkowski space \(R^{1 + n}\), Physica D, 238, 902-922 (2009) · Zbl 1173.37068 |
| [9] | Luo, T.; Rauch, J.; Xie, C. J.; Xin, Z. P., Stability of transonic shock solutions for one-dimensional Euler-Poisson equations, Arch. Ration. Mech. Anal., 202, 787-827 (2011) · Zbl 1261.76055 |
| [10] | Nakane, S., Formation of shocks for a single conservation law, SIAM J. Math. Anal., 19, 1391-1408 (1988) · Zbl 0681.35057 |
| [11] | Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0537.76001 |
| [12] | Smoller, J., Shock Waves and Reaction-Diffusion Equations (1994), Springer-Verlag · Zbl 0807.35002 |
| [13] | Kruzhkov, S., First order quasilinear equations in several independent variables, Math. USSR Sb., 10(2), 217-243 (1970) · Zbl 0215.16203 |
| [14] | Lychagin, V., Singularities of multivalued solutions of nonlinear differential equations, and nonlinear phenomena, Acta Appl. Math., 3(2), 135-173 (1985) · Zbl 0562.35008 |
| [15] | Oleinik, O., Discontinuous solutions of non-linear differential equations, Uspekhi Mat. Nauk, 3(75), 3-73 (1957) · Zbl 0080.07701 |
| [16] | Sobolev, S., Applications of Functional Analysis in Mathematical Physics (1963), AMS · Zbl 0123.09003 |
| [17] | Chaplygin, S., On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21, 1-121 (1904) |
| [18] | Tsien, H. S., Two dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci., 6, 399-407 (1939) · JFM 65.1496.05 |
| [19] | von Kármán, Th., Compressibility effects in aerodynamics, J. Aeronaut. Sci., 8, 337-365 (1941) · JFM 67.0853.01 |
| [20] | Brenier, Y., Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations, J. Math. Fluid Mech., 7, 326-331 (2005) · Zbl 1085.35097 |
| [21] | Chen, G. Q.; Liu, H., Formation of delta-shocks and vacuum states in the vanishing pressure limit of slutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34, 925-938 (2003) · Zbl 1038.35035 |
| [22] | Danilov, V. G.; Shelkovich, V. M., Delta shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math., LXIII, 3, 401-427 (2005) |
| [23] | Freistühler, H., Linearly degeneracy and shock waves, Math. Z., 207, 583-596 (1991) · Zbl 0756.35048 |
| [24] | Li, J.; Zhang, Tong, On the initial-value problem for zero-pressure gas dynamics, (Hyperbolic problems: Theory, Numerics, Applications. Seventh International Conference in Zurich, February 1998 (1999), Birkhauser: Birkhauser Basel, Boston, Berlin), 629-640 · Zbl 0926.35120 |
| [25] | Nilsson, B.; Shelkovich, V. M., Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta shocks, Appl. Anal., 90, 1677-1689 (2011) · Zbl 1237.35111 |
| [26] | Shelkovich, V. M., Concepts of delta-shock type splutions to systems of conservation laws and the Rankine-Hugoniot conditions, Oper. Theory Adv. Appl., 231, 297-305 (2013) · Zbl 1263.35163 |
| [27] | Wang, G. D., The Riemann problem for one dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl., 403, 434-450 (2013) · Zbl 1426.76207 |
| [28] | Yang, H. C.; Zhang, Y. Y., New developements of delta shock waves and its applications in systems of conservation laws, J. Differential Equations, 252, 5951-5993 (2012) · Zbl 1248.35127 |
| [30] | Kong, D.-X.; Tusji, M., Global solutions for 2×2 hyperbolic systems with linearly degenerate characteristics, Funkcialaj Ekvacioj, 42, 129-155 (1999) · Zbl 1141.35416 |
| [31] | Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations, (Mathématiques and Applications, vol. 26 (1997), Springer) · Zbl 0881.35001 |
| [32] | Golubitsky, M., An introduction to catastrophy theorey and its aplications, SIAM Rev., 20, 352-387 (1987) · Zbl 0389.58007 |
| [33] | Eggers, J.; Hoppe, J., Singularity for time-like extremal hypersurfaces, Phys. Lett. B, 680, 274-278 (2009) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
