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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

L. D. Landau and E. M. Lifchitz, Fluid Mechanics,, 2$^{nd}$ edition, (1987), 505.

[13]

P. D. Lax, Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Appl. Math. Philadelphia, (1973).

[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.

[15]

T. T. Li and T. Qin, Physics and Parital Differential Equations,, 2$^{nd}$ edition, (2005).

[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.

[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.

[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.

[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.

[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.

[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.

[22]

T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation,, \emph{J. Math. Kyoto Univ}, 35 (1995), 105.

[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.

[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.

[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.

[26]

V. Pant, Global entropy solutions for isentropic relativistic fluid dynamics,, \emph{Commu. Partial. Diff. Eqs}, 21 (1996), 1609. doi: 10.1080/03605309608821240.

[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.

[28]

T. Sideris, Formation of sigularities of solution to nonlinear hyperbolic equations,, \emph{Arch. Ration. Mech. Anal}, 86 (1984), 369. doi: 10.1007/BF00280033.

[29]

T. Sideris, Formation of sigularities in three-dimensional compressible fluids,, \emph{Comm. Math. Phys}, 101 (1985), 475.

[30]

A. H. Taub, Relativistic Rankine-Hügoniot equations,, \emph{Phys. Rev}, 74 (1948), 328.

[31]

K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech}, 171 (1986), 365.

[32]

K. S. Thorne, Relativistic shocks: The Taub adiabatic,, \emph{Astrophys. J}, 179 (1973), 897.

[33]

S. Weinberg, Gravitation and cosmology: Applications of the General Theory of Relativity,, Wiley, (1972).

[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.

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).

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

L. D. Landau and E. M. Lifchitz, Fluid Mechanics,, 2$^{nd}$ edition, (1987), 505.

[13]

P. D. Lax, Systems of Conservation Laws and the Mathematical Theory of Shock Waves,, Society for Industrial and Appl. Math. Philadelphia, (1973).

[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.

[15]

T. T. Li and T. Qin, Physics and Parital Differential Equations,, 2$^{nd}$ edition, (2005).

[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.

[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.

[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.

[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.

[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.

[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.

[22]

T. Makino and S. Ukai, Local smooth solutions of the relativistic Euler equation,, \emph{J. Math. Kyoto Univ}, 35 (1995), 105.

[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.

[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.

[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.

[26]

V. Pant, Global entropy solutions for isentropic relativistic fluid dynamics,, \emph{Commu. Partial. Diff. Eqs}, 21 (1996), 1609. doi: 10.1080/03605309608821240.

[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.

[28]

T. Sideris, Formation of sigularities of solution to nonlinear hyperbolic equations,, \emph{Arch. Ration. Mech. Anal}, 86 (1984), 369. doi: 10.1007/BF00280033.

[29]

T. Sideris, Formation of sigularities in three-dimensional compressible fluids,, \emph{Comm. Math. Phys}, 101 (1985), 475.

[30]

A. H. Taub, Relativistic Rankine-Hügoniot equations,, \emph{Phys. Rev}, 74 (1948), 328.

[31]

K. Thompson, The special relativistic shock tube,, \emph{J. Fluid Mech}, 171 (1986), 365.

[32]

K. S. Thorne, Relativistic shocks: The Taub adiabatic,, \emph{Astrophys. J}, 179 (1973), 897.

[33]

S. Weinberg, Gravitation and cosmology: Applications of the General Theory of Relativity,, Wiley, (1972).

[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.

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