Global solutions for the ultra-relativistic Euler equations. (English) Zbl 1369.35046
Summary: A front tracking scheme for the ultra-relativistic Euler equations is introduced. This scheme is based on piecewise constant approximations to the front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. We study the interaction estimates of the generalized shocks (entropy and nonentropy shocks) of the ultra-relativistic Euler equations and the outcoming asymptotic Riemann solution. Moreover we use a new function to measure the strengths of the waves of the ultra-relativistic Euler equations. This function has the important implication that the strength is non increasing for the interactions of the generalized shocks. This enables us to define a new kind of total variation of a solution. The main application of this scheme, is proving the global existence of weak solutions for the ultra relativistic Euler equations.
MSC:
| 35Q31 | Euler equations |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35D30 | Weak solutions to PDEs |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
relativistic Euler equations; conservation laws; hyperbolic system; shock waves; entropy conditions; non-entropy shocks; Riemann solutions; front tracking; existence theoremReferences:
| [1] | Abdelrahman, M. A.E., Analytical and Numerical Investigation of the Ultra-Relativistic Euler Equations (2013), Otto-von-Guericke University: Otto-von-Guericke University Magdeburg, (Ph.D. thesis) · Zbl 1315.35001 |
| [2] | Abdelrahman, M. A.E.; Kunik, M., The interaction of waves for the ultra-relativistic Euler equations, J. Math. Anal. Appl., 409, 1140-1158 (2014) · Zbl 1309.35062 |
| [3] | Abdelrahman, M. A.E.; Kunik, M., A new front tracking scheme for the ultra-relativistic euler equations, J. Comput. Phys., 275, 213-235 (2014) · Zbl 1349.76579 |
| [4] | Abdelrahman, M. A.E.; Kunik, M., The ultra-relativistic Euler equations, Math. Methods Appl. Sci., 38, 1247-1264 (2015) · Zbl 1319.35170 |
| [5] | Alber, H. D., Local existence of weak solutions to the quasilinear wave equation for large initial values, Math. Z., 190, 249-276 (1985) · Zbl 0581.35049 |
| [6] | Amadori, D.; Guerra, G., Global BV solutions and relaxation limit for a system of conservation laws, Proc. Roy. Soc. Edinburgh Sect. A, 131, 1, 1-26 (2001) · Zbl 0971.35050 |
| [7] | Ancona, F.; Marson, A., A wavefront tracking algorithm for \(N \times N\) nongenuinely nonlinear conservation laws, J. Differential Equations, 177, 454-493 (2001) · Zbl 0993.35062 |
| [8] | Ancona, F.; Marson, A., Existence theory by front tracking for general nonlinear hyperbolic system, J. Differential Equations, 185, 287-340 (2007) · Zbl 1128.35068 |
| [9] | Baiti, P.; Jenssen, H. K., On the front tracking algorithm, J. Math. Anal. Appl., 217, 395-404 (1998) · Zbl 0966.35078 |
| [10] | Bressan, A., Global solutions to systems of conservation laws by wavefront tracking, J. Math. Anal. Appl., 170, 414-432 (1992) · Zbl 0779.35067 |
| [11] | Bressan, A., Lecture Notes on Conservation Laws (1995), S.I.S.S.A: S.I.S.S.A Trieste |
| [12] | Bressan, A., Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0997.35002 |
| [13] | Dafermos, C. M., Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38, 33-41 (1972) · Zbl 0233.35014 |
| [14] | Diperna, R. J., Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Comm. Differential Equations, 20, 187-212 (1976) · Zbl 0314.58010 |
| [15] | Kunik, M., Selected Initial and Boundary Value Problems for Hyperbolic Systems and Kinetic Equations (2005), Otto-von-Guericke University, (Habilitation thesis) |
| [16] | Risebro, N. H., A front tracking alternative to the random choice method, Proc. Amer. Math. Soc., 117, 1125-1139 (1993) · Zbl 0799.35153 |
| [17] | Risebro, N. H.; Tveito, A., A front tracking method for conservation laws in one dimension, J. Comput. Phys., 101, 130-139 (1992) · Zbl 0756.65120 |
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