Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state. (English) Zbl 1378.35234
Summary: We are concerned with the global existence and blowup of the classical solutions to the Cauchy problem of one-dimensional isentropic relativistic Euler equations (Chaplygin gas, pressureless perfect fluid and stiff matter) with linearly degenerate characteristics. We at first derive the exact representation formula for all the fluids by the property of linearly degenerate. Then for the Chaplygin gas and the pressureless perfect fluid, we give a classification of the initial data that leads to the global existence and the blowup of the classical solution, respectively. We construct, especially, a class of initial data that contributes to the formation of “cusp-type” singularity and study the evolution of the solution after blowup by introducing a weak solution called delta shock wave. At last, for the stiff matter, we show that this system is indeed a linear system and prove the global existence of the classical solution to this fluid.
MSC:
| 35Q31 | Euler equations |
| 35L40 | First-order hyperbolic systems |
| 35L65 | Hyperbolic conservation laws |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 76L05 | Shock waves and blast waves in fluid mechanics |
| 83F05 | Relativistic cosmology |
References:
| [1] | Alinhac, S., Blowup for Nonlinear Hyperbolic Equations, , Vol. 17 (Birkhäuser, 1995). · Zbl 0820.35001 |
| [2] | Brenier, Y., Some geometric PDEs related to hydrodynamics and electrodynamics, Proc. ICM3 (2002) 761-772. · Zbl 1136.37355 |
| [3] | Brenier, Y., Solutions with concentration to the Riemann problem for the one-dimensional Chaplygin gas equations, J. Math. Fluid Mech.7 (2005) 326-331. · Zbl 1085.35097 |
| [4] | N. Bilic, G. B. Tupper and R. Viollier, Dark matter, dark energy and Chaplygin gas, preprint (2002), arXiv:astro-hp/0207423. · Zbl 0995.83512 |
| [5] | Chaplygin, S., On gas jets, Sci. Mem. Moscow Univ. Math. Phys.21 (1904) 1-121. |
| [6] | Chen, G. Q. and Liu, H., Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal.34 (2003) 925-938. · Zbl 1038.35035 |
| [7] | Christoudoulou, D., The Formation of Shocks in Three-Dimensional Fluids, (European Math. Soc., Zürich, 2007). · Zbl 1117.35001 |
| [8] | Christoudoulou, D. and Miao, S., Compressible Flow and Euler’s Equations, , Vol. 9 (International Press, Somerville, MA; Higher Education Press, Beijing, 2014). · Zbl 1329.76002 |
| [9] | Dafermos, C. M., Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J.26 (1977) 1097-1119. · Zbl 0377.35051 |
| [10] | Danilov, V. G. and Shelkovich, V. M., Delta shock wave-type solution of hyperbolic systems of conservation laws, Quart. Appl. Math.63(3) (2005) 401-427. |
| [11] | Golubitsky, M., An introduction to catastrophy theory and its applications, SIAM Rev.20 (1987) 352-387. · Zbl 0389.58007 |
| [12] | Guckenheimer, J., Solving a single conservation law, in Dynamical Systems, , Vol. 468 (Springer, 1975), pp. 108-134. · Zbl 0306.35020 |
| [13] | Guo, L., Zhang, Y. and Yin, G., Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions, Math. Methods Appl. Sci.38 (2015) 2132-2148. · Zbl 1317.35141 |
| [14] | Freistühler, H., Linearly degeneracy and shock waves, Math. Z.207 (1991) 583-596. · Zbl 0756.35048 |
| [15] | Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations, , Vol. 26 (Springer, 1997). · Zbl 0881.35001 |
| [16] | John, F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math.27 (1974) 377-405. · Zbl 0302.35064 |
| [17] | Kong, D., Formation and propagation of singularities for \(2 \times 2\) quasilinear hyperbolic systems, Trans. Amer. Math. Soc.354 (2002) 3155-3179. · Zbl 1003.35079 |
| [18] | Kong, D., A nonlinear geometric equation related to electrodynamics, Europhys. Lett.66(5) (2004) 617-623. |
| [19] | Kong, D. and Tusji, M., Global solutions for \(2 \times 2\) hyperbolic systems with linearly degenerate characteristics, Funkcial. Ekvac.42 (1999) 129-155. · Zbl 1141.35416 |
| [20] | Kong, D. and Wei, C., Formation and propagation of singularities in one-dimensional Chaplygin gas, J. Geom. Phys.80 (2014) 58-70. · Zbl 1284.35260 |
| [21] | Kong, D., Wei, C. and Zhang, Q., Formation of singularities in one-dimensional Chaplygin gas, J. Hyperbolic Differ. Equations11 (2014) 521-561. · Zbl 1310.35154 |
| [22] | Kong, D. and Zhang, Q., Solutions formula and time-periodicity for the motion of relativistic strings in the Minkowski space \(R^{1 + n}\), Physica D238 (2009) 902-922. · Zbl 1173.37068 |
| [23] | Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math.10 (1957) 537-566. · Zbl 0081.08803 |
| [24] | LeFloch, P. G. and Xiang, S. Y., Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime, J. Math. Pures Appl.106 (2016) 1038-1090. · Zbl 1354.35105 |
| [25] | Lei, Z. and Wei, C., Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases, Math. Ann.367 (2017) 1363-1401. · Zbl 1372.35226 |
| [26] | Li, J. and Zhang, Tong, On the initial-value problem for zero-pressure gas dynamics, in Hyperbolic Problems: Theory, Numerics, Applications (Birkhäuser, Basel, 1999), pp. 629-640. · Zbl 0926.35120 |
| [27] | Luo, T., Rauch, J., Xie, C. J. and Xin, Z. P., Stability of transonic shock solutions for one-dimensional Euler-Poisson equations, Arch. Rational Mech. Anal.202 (2011) 787-827. · Zbl 1261.76055 |
| [28] | Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (Springer-Verlag, New York, 1984). · Zbl 0537.76001 |
| [29] | Nakane, S., Formation of shocks for a single conservation law, SIAM J. Math. Anal.19 (1988) 1391-1408. · Zbl 0681.35057 |
| [30] | Nilsson, B. and Shelkovich, V. M., Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta shocks, Appl. Anal.90 (2011) 1677-1689. · Zbl 1237.35111 |
| [31] | Shelkovich, V. M., Concepts of delta-shock-type solutions to systems of conservation laws and the Rankine-Hugoniot conditions, Oper. Theor. Adv. Appl.231 (2013) 297-305. · Zbl 1263.35163 |
| [32] | Smøller, J., Shock Waves and Reaction-Diffusion Equations (Springer, 1994). · Zbl 0807.35002 |
| [33] | Tsien, H. S., Two-dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci.6 (1939) 399-407. · JFM 65.1496.05 |
| [34] | von Kármán, Th., Compressibility effects in aerodynamics, J. Aeronaut. Sci.8 (1941) 337-365. · JFM 67.0853.01 |
| [35] | Wang, G. D., The Riemann problem for one-dimensional generalized Chaplygin gas dynamics, J. Math. Anal. Appl.403 (2013) 434-450. · Zbl 1426.76207 |
| [36] | C. Wei, The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations, submitted. · Zbl 1378.35190 |
| [37] | Yin, G. and Sheng, W., Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases, J. Math. Anal. Appl.355 (2009) 594-605. · Zbl 1167.35047 |
| [38] | Yang, H. C. and Zhang, Y. Y., New developments of delta shock waves and its applications in systems of conservation laws, J. Differ. Equations252 (2012) 5951-5993. · Zbl 1248.35127 |
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