American Institute of Mathematical Sciences

March  2015, 14(2): 549-564. doi: 10.3934/cpaa.2015.14.549

Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force

 1 Department of Mathematics, Shanghai Institute of Technology, Shanghai 200235, China

Received  March 2014 Revised  September 2014 Published  December 2014

In this paper, we investigate the singularities formation for the relativistic Euler and Euler-Poisson equations with repulsive force, in spherical symmetry. We will show the non-trivial regular solution $(p^{\frac{\gamma-1}{2\gamma}}\in C^1$) with compact support in $[0,R]$, under a certain condition on initial data \begin{eqnarray} H_0:=\int_0^R rv_0dr>0, \end{eqnarray} will blow up, where $0 < R < \min\{\frac{8}{45},\frac{8(\gamma-1)}{5\gamma+23}\}$ is a constant. Since every term in the relativistic Euler-Poisson equations corresponding to $\rho, v$ in non-relativistic case [34] has a relativistic factor $\frac{1}{\sqrt{1-v^2/c^2}}$, we will separate variables and estimate some integration items instead of direct using integration method as [34].
Citation: Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549
References:
 [1] A. M. Anile, Relativistic Fluids and Magneto-Fluids: with Applications in Astrophysics and Plasma Physics,, Cambridge Monographs on Mathematical Physics, (1989). Google Scholar [2] D. H. Chae and E. Tadmor, On the finite time blow-up of the Euler-Poisson equations in $R^N$,, \emph{Commun. Math. Sci}, 6 (2008), 785. Google Scholar [3] M. Ding and Y. C. Li, Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations,, \emph{Z. angew. Math. Phys}, 64 (2013), 101. doi: 10.1007/s00033-012-0227-7. Google Scholar [4] M. Ding and Y. C. Li, An Overview of Piston Problems in Fluid Dynamics,, \emph{P. Edinburgh Math. Soc}, 49 (2014), 161. doi: 10.1007/978-3-642-39007-4_8. Google Scholar [5] S. Engelberg, Formation of singularities in the Euler and Euler-Poisson equations,, \emph{Phys. D}, 98 (1996), 67. doi: 10.1016/0167-2789(96)00087-5. Google Scholar [6] Y. C. Geng and Y. Li, Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations,, \emph{Z. Angew. Math. Phys}, 62 (2011), 281. doi: 10.1007/s00033-010-0093-0. Google Scholar [7] Y. C. Geng and Y. Li, Local smooth solutions to the 3-dimensional isentropic relativistic Euler equations,, \emph{Chin. Ann. Math}, 35B (2014), 301. doi: 10.1007/s11401-014-0820-5. Google Scholar [8] Y. C. Geng and L. Wang, Global smooth solutions to relativistic Euler-Poisson equations with repulsive force,, \emph{Acta. Math. Appl. Sini}, 30 (2014), 1025. Google Scholar [9] Y. Guo and S. Tahhvildar-Zadeh, Formation of singularities in relativistic fluid dynamics an in spherically symmetric plasma dynamics,, \emph{Cntemp. Math}, 238 (2009), 151. doi: 10.1090/conm/238/03545. Google Scholar [10] X. W. Hao and Y. C. Li, Non-relativistic global limits of entropy solutions to the Cauchy problem of the three dimensional relativistic Euler equations with spherical symmetry,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 365. doi: 10.3934/cpaa.2010.9.365. Google Scholar [11] C. H. Hsu, S. Lin and T. Makino, On spherically symmetric solutions of the relativistic Euler equation,, \emph{J. Diffrential. Equations}, 201 (2004), 1. doi: 10.1016/j.jde.2004.03.003. Google Scholar [12] L. D. Landau and E. M. Lifchitz, Fluid Mechanics,, 2$^{nd}$ edition, (1987), 505. Google Scholar [13] P. D. Lax, Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Appl. Math. Philadelphia, (1973). Google Scholar [14] P. Lefloch and S. Ukai, A symmetrization of the relativistic Euler equations in several spatial variables,, \emph{Kinet. Relat. Modles}, 2 (2009), 275. doi: 10.3934/krm.2009.2.275. Google Scholar [15] T. T. Li and T. Qin, Physics and Parital Differential Equations,, 2$^{nd}$ edition, (2005). Google Scholar [16] Y. C. Li and Y. C. Geng, Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations,, \emph{Z. angew. Math. Phys}, 57 (2006), 960. doi: 10.1007/s00033-006-0059-4. Google Scholar [17] L. S. Mai, J. Y. Li and K. J. Zhang, On the steady state relativistic Euler-Poisson equations,, \emph{Acta. Appl. Math}, 125 (2013), 135. doi: 10.1007/s10440-012-9784-1. Google Scholar [18] T. Makino, On a local existence theorem for the evolution of gaseous stars. Patterns and Waves,, eds. T. Nishida, (1986), 459. doi: 10.1016/S0168-2024(08)70142-5. Google Scholar [19] T. Makino, S. Ukai and S. Kawashima, Sur la solutions à support compact de l'équation d'Euler compressible,, \emph{Jappen J. Appl. Math}, 3 (1986), 249. doi: 10.1007/BF03167100. Google Scholar [20] T. Makino, B. Perthame, Sur les solutions à symmétrie sphérique de l'équation d'Euler-Poisson pour levolution d'étoiles gazeuses,, \emph{Japan J. Appl. Math}, 7 (1990), 165. doi: 10.1007/BF03167897. Google Scholar [21] T. Makino and S. Ukai, Sur l'existence des solutions locales de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses,, \emph{J. Math. Kyoto Univ}, 27 (1987), 387. Google Scholar [22] T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation,, \emph{J. Math. Kyoto Univ}, 35 (1995), 105. Google Scholar [23] T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation. II,, \emph{Kodai Math. J}, 18 (1995), 365. doi: 10.2996/kmj/1138043432. Google Scholar [24] K. Mizohata, Global solutions to the relativistic Euler equation with spherical symmetry,, \emph{Jappen J. Indust. Appl. Math}, 14 (1997), 125. doi: 10.1007/BF03167315. Google Scholar [25] R. Pan and J. Smoller, Blowup of smooth solutions for relativistic Euler equations,, \emph{Commun. Math. Phys}, 262 (2006), 729. doi: 10.1007/s00220-005-1464-9. Google Scholar [26] V. Pant, Global entropy solutions for isentropic relativistic fluid dynamics,, \emph{Commu. Partial. Diff. Eqs}, 21 (1996), 1609. doi: 10.1080/03605309608821240. Google Scholar [27] B. Perthame, Nonexistence of global solutions to Euler-Poisson equations for repulsive forces,, \emph{Japan J. Appl. Math}, 7 (1990), 363. doi: 10.1007/BF03167849. Google Scholar [28] T. Sideris, Formation of sigularities of solution to nonlinear hyperbolic equations,, \emph{Arch. Ration. Mech. Anal}, 86 (1984), 369. doi: 10.1007/BF00280033. Google Scholar [29] T. Sideris, Formation of sigularities in three-dimensional compressible fluids,, \emph{Comm. Math. Phys}, 101 (1985), 475. Google Scholar [30] A. H. Taub, Relativistic Rankine-Hügoniot equations,, \emph{Phys. Rev}, 74 (1948), 328. Google Scholar [31] K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech}, 171 (1986), 365. Google Scholar [32] K. S. Thorne, Relativistic shocks: The Taub adiabatic,, \emph{Astrophys. J}, 179 (1973), 897. Google Scholar [33] S. Weinberg, Gravitation and cosmology: Applications of the General Theory of Relativity,, Wiley, (1972). Google Scholar [34] M. W. Yuen, Blowup for the Euler and Euler-Poisson equations with repulsive forces,, \emph{Nonl. Anal}, 74 (2011), 1465. doi: 10.1016/j.na.2010.10.019. Google Scholar

show all references

References:
 [1] A. M. Anile, Relativistic Fluids and Magneto-Fluids: with Applications in Astrophysics and Plasma Physics,, Cambridge Monographs on Mathematical Physics, (1989). Google Scholar [2] D. H. Chae and E. Tadmor, On the finite time blow-up of the Euler-Poisson equations in $R^N$,, \emph{Commun. Math. Sci}, 6 (2008), 785. Google Scholar [3] M. Ding and Y. C. Li, Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations,, \emph{Z. angew. Math. Phys}, 64 (2013), 101. doi: 10.1007/s00033-012-0227-7. Google Scholar [4] M. Ding and Y. C. Li, An Overview of Piston Problems in Fluid Dynamics,, \emph{P. Edinburgh Math. Soc}, 49 (2014), 161. doi: 10.1007/978-3-642-39007-4_8. Google Scholar [5] S. Engelberg, Formation of singularities in the Euler and Euler-Poisson equations,, \emph{Phys. D}, 98 (1996), 67. doi: 10.1016/0167-2789(96)00087-5. Google Scholar [6] Y. C. Geng and Y. Li, Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations,, \emph{Z. Angew. Math. Phys}, 62 (2011), 281. doi: 10.1007/s00033-010-0093-0. Google Scholar [7] Y. C. Geng and Y. Li, Local smooth solutions to the 3-dimensional isentropic relativistic Euler equations,, \emph{Chin. Ann. Math}, 35B (2014), 301. doi: 10.1007/s11401-014-0820-5. Google Scholar [8] Y. C. Geng and L. Wang, Global smooth solutions to relativistic Euler-Poisson equations with repulsive force,, \emph{Acta. Math. Appl. Sini}, 30 (2014), 1025. Google Scholar [9] Y. Guo and S. Tahhvildar-Zadeh, Formation of singularities in relativistic fluid dynamics an in spherically symmetric plasma dynamics,, \emph{Cntemp. Math}, 238 (2009), 151. doi: 10.1090/conm/238/03545. Google Scholar [10] X. W. Hao and Y. C. Li, Non-relativistic global limits of entropy solutions to the Cauchy problem of the three dimensional relativistic Euler equations with spherical symmetry,, \emph{Commun. Pure Appl. Anal}, 9 (2010), 365. doi: 10.3934/cpaa.2010.9.365. Google Scholar [11] C. H. Hsu, S. Lin and T. Makino, On spherically symmetric solutions of the relativistic Euler equation,, \emph{J. Diffrential. Equations}, 201 (2004), 1. doi: 10.1016/j.jde.2004.03.003. Google Scholar [12] L. D. Landau and E. M. Lifchitz, Fluid Mechanics,, 2$^{nd}$ edition, (1987), 505. Google Scholar [13] P. D. Lax, Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Appl. Math. Philadelphia, (1973). Google Scholar [14] P. Lefloch and S. Ukai, A symmetrization of the relativistic Euler equations in several spatial variables,, \emph{Kinet. Relat. Modles}, 2 (2009), 275. doi: 10.3934/krm.2009.2.275. Google Scholar [15] T. T. Li and T. Qin, Physics and Parital Differential Equations,, 2$^{nd}$ edition, (2005). Google Scholar [16] Y. C. Li and Y. C. Geng, Non-relativistic global limits of entropy solutions to the isentropic relativistic Euler equations,, \emph{Z. angew. Math. Phys}, 57 (2006), 960. doi: 10.1007/s00033-006-0059-4. Google Scholar [17] L. S. Mai, J. Y. Li and K. J. Zhang, On the steady state relativistic Euler-Poisson equations,, \emph{Acta. Appl. Math}, 125 (2013), 135. doi: 10.1007/s10440-012-9784-1. Google Scholar [18] T. Makino, On a local existence theorem for the evolution of gaseous stars. Patterns and Waves,, eds. T. Nishida, (1986), 459. doi: 10.1016/S0168-2024(08)70142-5. Google Scholar [19] T. Makino, S. Ukai and S. Kawashima, Sur la solutions à support compact de l'équation d'Euler compressible,, \emph{Jappen J. Appl. Math}, 3 (1986), 249. doi: 10.1007/BF03167100. Google Scholar [20] T. Makino, B. Perthame, Sur les solutions à symmétrie sphérique de l'équation d'Euler-Poisson pour levolution d'étoiles gazeuses,, \emph{Japan J. Appl. Math}, 7 (1990), 165. doi: 10.1007/BF03167897. Google Scholar [21] T. Makino and S. Ukai, Sur l'existence des solutions locales de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses,, \emph{J. Math. Kyoto Univ}, 27 (1987), 387. Google Scholar [22] T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation,, \emph{J. Math. Kyoto Univ}, 35 (1995), 105. Google Scholar [23] T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation. II,, \emph{Kodai Math. J}, 18 (1995), 365. doi: 10.2996/kmj/1138043432. Google Scholar [24] K. Mizohata, Global solutions to the relativistic Euler equation with spherical symmetry,, \emph{Jappen J. Indust. Appl. Math}, 14 (1997), 125. doi: 10.1007/BF03167315. Google Scholar [25] R. Pan and J. Smoller, Blowup of smooth solutions for relativistic Euler equations,, \emph{Commun. Math. Phys}, 262 (2006), 729. doi: 10.1007/s00220-005-1464-9. Google Scholar [26] V. Pant, Global entropy solutions for isentropic relativistic fluid dynamics,, \emph{Commu. Partial. Diff. Eqs}, 21 (1996), 1609. doi: 10.1080/03605309608821240. Google Scholar [27] B. Perthame, Nonexistence of global solutions to Euler-Poisson equations for repulsive forces,, \emph{Japan J. Appl. Math}, 7 (1990), 363. doi: 10.1007/BF03167849. Google Scholar [28] T. Sideris, Formation of sigularities of solution to nonlinear hyperbolic equations,, \emph{Arch. Ration. Mech. Anal}, 86 (1984), 369. doi: 10.1007/BF00280033. Google Scholar [29] T. Sideris, Formation of sigularities in three-dimensional compressible fluids,, \emph{Comm. Math. Phys}, 101 (1985), 475. Google Scholar [30] A. H. Taub, Relativistic Rankine-Hügoniot equations,, \emph{Phys. Rev}, 74 (1948), 328. Google Scholar [31] K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech}, 171 (1986), 365. Google Scholar [32] K. S. Thorne, Relativistic shocks: The Taub adiabatic,, \emph{Astrophys. J}, 179 (1973), 897. Google Scholar [33] S. Weinberg, Gravitation and cosmology: Applications of the General Theory of Relativity,, Wiley, (1972). Google Scholar [34] M. W. Yuen, Blowup for the Euler and Euler-Poisson equations with repulsive forces,, \emph{Nonl. Anal}, 74 (2011), 1465. doi: 10.1016/j.na.2010.10.019. Google Scholar
 [1] La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 [2] Yachun Li, Qiufang Shi. Global existence of the entropy solutions to the isentropic relativistic Euler equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 763-778. doi: 10.3934/cpaa.2005.4.763 [3] Philippe G. LeFloch, Seiji Ukai. A symmetrization of the relativistic Euler equations with several spatial variables. Kinetic & Related Models, 2009, 2 (2) : 275-292. doi: 10.3934/krm.2009.2.275 [4] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [5] Yachun Li, Xucai Ren. Non-relativistic global limits of the entropy solutions to the relativistic Euler equations with $\gamma$-law. Communications on Pure & Applied Analysis, 2006, 5 (4) : 963-979. doi: 10.3934/cpaa.2006.5.963 [6] Hong Cai, Zhong Tan. Stability of stationary solutions to the compressible bipolar Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4677-4696. doi: 10.3934/dcds.2017201 [7] Manwai Yuen. Cylindrical blowup solutions to the isothermal Euler-Poisson equations. Conference Publications, 2011, 2011 (Special) : 1448-1456. doi: 10.3934/proc.2011.2011.1448 [8] A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515 [9] Huahui Li, Zhiqiang Shao. Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2373-2400. doi: 10.3934/cpaa.2016041 [10] Xingwen Hao, Yachun Li, Zejun Wang. Non-relativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure & Applied Analysis, 2010, 9 (2) : 365-386. doi: 10.3934/cpaa.2010.9.365 [11] Jiang Xu, Ting Zhang. Zero-electron-mass limit of Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4743-4768. doi: 10.3934/dcds.2013.33.4743 [12] Haigang Li, Jiguang Bao. Euler-Poisson equations related to general compressible rotating fluids. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1085-1096. doi: 10.3934/dcds.2011.29.1085 [13] Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011 [14] Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569 [15] Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101 [16] Myoungjean Bae, Yong Park. Radial transonic shock solutions of Euler-Poisson system in convergent nozzles. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 773-791. doi: 10.3934/dcdss.2018049 [17] Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029 [18] Yonggeun Cho, Tohru Ozawa. On radial solutions of semi-relativistic Hartree equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 71-82. doi: 10.3934/dcdss.2008.1.71 [19] Dongho Chae. On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1139-1150. doi: 10.3934/dcdss.2013.6.1139 [20] Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

2018 Impact Factor: 0.925