Advertisement

The lifespan of 3D radial solutions to the non-isentropic relativistic Euler equations

  • Changhua Wei
Article

Abstract

This paper investigates the lower bound of the lifespan of three-dimensional spherically symmetric solutions to the non-isentropic relativistic Euler equations, when the initial data are prescribed as a small perturbation with compact support to a constant state. Based on the structure of the hyperbolic system, we show the almost global existence of the smooth solutions to Eulerian flows (polytropic gases and generalized Chaplygin gases) with genuinely nonlinear characteristics. While for the Eulerian flows (Chaplygin gas and stiff matter) with mild linearly degenerate characteristics, we show the global existence of the radial solutions, moreover, for the non-strictly hyperbolic system (pressureless perfect fluid) satisfying the mild linearly degenerate condition, we prove the blowup phenomenon of the radial solutions and show that the lifespan of the solutions is of order \(O(\epsilon ^{-1})\), where \(\epsilon \) denotes the width of the perturbation. This work can be seen as a complement of our work (Lei and Wei in Math Ann 367:1363–1401, 2017) for relativistic Chaplygin gas and can also be seen as a generalization of the classical Eulerian fluids (Godin in Arch Ration Mech Anal 177:497–511, 2005, J Math Pures Appl 87:91–117, 2007) to the relativistic Eulerian fluids.

Keywords

Relativistic Euler Lifespan Spherically symmetric solution Null condition 

Mathematics Subject Classification

35L50 35Q31 35Q35 

References

  1. 1.
    Alinhac, S.: The null condition for quasilinear wave equations in two space dimensions I. Invent. Math. 145, 597–618 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Courant, R., Fridriches, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers, New York (1948)Google Scholar
  3. 3.
    Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics. EMS Publishing House, Z\(\ddot{u}\)rich (2007)Google Scholar
  4. 4.
    Christodoulou, D., Lisibach, A.: Shock development in spherical symmetry. Ann. PDE 2, 1–246 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Christodoulou, D., Miao, S.: Compressible flow and Euler’s Equations, Surveys of Modern Mathematics, vol. 9. International Press, Somerville (2014)MATHGoogle Scholar
  6. 6.
    Ding, B., Witt, I., Yin, H.: The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases. J. Differ. Equ. 258, 445–482 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Godin, P.: The lifespan of a class of smooth spherically symmetric solutions of the compressible Euler equations with variable entropy in three space dimensions. Arch. Ration. Mech. Anal. 177, 497–511 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Godin, P.: Global existence of a class of smooth 3D spherically symmetric flows of Chaplygin gases with variable entropy. J. Math. Pures Appl. 87, 91–117 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Guo, Y., Tahvildar-Zedeh, S.: Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics. Contemp. Math. 238, 151–161 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Holzegel, G., Klainerman, S., Speck, J., Wong, Y.: Small-data shock formation in solutions to 3D quasilinear wave equations: an overview. J. Hyperbolic Differ. Equ. 13, 1–105 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    John, F.: Nonlinear wave equations, formation of singularities. In: Pitcher Lectures Delivered at Lehigh University (1989)Google Scholar
  12. 12.
    Kato, T.: The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)CrossRefMATHGoogle Scholar
  13. 13.
    Kong, D.X., Tsuji, M.: Global solutions for \(2\times 2\) hyperbolic systems with linearly degenerate characteristics. Funkc. Ekvac. 42, 129–155 (1999)MATHGoogle Scholar
  14. 14.
    Kong, D.X., Liu, K., Wang, Y.: Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases. Sci. China Math. 53, 719–738 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lax, P.: Development of singularity of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)CrossRefMATHGoogle Scholar
  16. 16.
    Lax, P.: Hyperbolic systems of conservation laws in several space variables. In: Current Topics in Partial Differential Equations, pp. 327–341 (1986)Google Scholar
  17. 17.
    LeFloch, P.G., Ukai, S.: A symmetrization of the relativistic Euler equations in several spatial variables. Kinet. Relat. Models 2, 275–292 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, T.: The development of singularity in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Equ. 33, 92–111 (1979)CrossRefMATHGoogle Scholar
  19. 19.
    Lei, Z., Du, Y., Zhang, Q.: Singularities of solutions to compressible Euler equations with vacuum. Math. Res. Lett. 20, 41–50 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lei, Z., Lin, F., Zhou, Y.: Global solutions of the evolutionary Faddeev model with small initial data. Acta Math. Sin. (Engl. Ser.) 27, 309–328 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lei, Z., Wei, C.H.: Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases. Math. Ann. 367, 1363–1401 (2017)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    LeFloch, P.G., Wei, C.H.: The global nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FRW geometry. arXiv: 1512.03754v1
  23. 23.
    Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables. Commun. Pure Appl. Math. 28, 607–676 (1975)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Makino, T., Ukai, S.: Local smooth solutions of the relativistic Euler equation. J. Math. Kyoto Univ. 35, 105–114 (1995)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Miao, S., Yu, P.: On the formation of shocks for quasilinear wave equations. Invent. Math. 207, 697–831 (2017)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pan, R., Smoller, J.: Blowup of smooth solutions for relativistic Euler equations. Commun. Math. Phys. 262, 729–755 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Perthame, B.: Non-existence of global solutions to Euler–Poisson equations for repulsive forces. Jpn. J. Appl. Math. 7, 363–367 (1990)CrossRefMATHGoogle Scholar
  28. 28.
    Smoller, J., Temple, B.: Global solutions of the relativistic Euler equations. Commun. Math. Phys. 156, 67–99 (1993)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Smoller, J.: Shock Waves and Reaction Diffusion Equations, 2nd edn. Springer, Berlin (1993)MATHGoogle Scholar
  30. 30.
    Sideris, T.: Formation of singularity in three-dimensional compressible fluids. Commun. Math. Phys. 101, 475–485 (1985)CrossRefMATHGoogle Scholar
  31. 31.
    Sideris, T.: Formation of singularities of solutions to nonlinear hyperbolic equations. Arch. Ration. Mech. Anal. 86, 369–381 (1984)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sideris, T.: Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. Math. 151, 849–874 (2000)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sideris, T.: Delayed singularity formation in 2D compressible flow. Am. J. Math. 119, 371–422 (1997)CrossRefMATHGoogle Scholar
  34. 34.
    Sideris, T.: Spreading of the free boundary of an ideal fluid in a vacuum. J. Differ. Equ. 257, 1–14 (2014)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sideris, T., Thomases, B., Wang, D.: Long time behavior of solutions to the 3D compressible Euler equations with damping. Commun. Partial Differ. Equ. 28, 795–816 (2003)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Speck, J.: The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Sel. Math. 18, 633–715 (2012)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Tahvildar-Zadeh, A.S.: Relativistic and nonrelativistic elastodynamics with small shear strains. Ann. Inst. Henri Poincar\(\acute{e}\) Phys. Th\(\acute{e}\)or. 69, 275–307 (1998)Google Scholar
  38. 38.
    Wei, C.H.: Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state. J. Hyperbolic Differ. Equ. 14, 535–563 (2017)Google Scholar
  39. 39.
    Wei, C.H., Han, B.: Spreading of the free boundary of relativistic Euler equations in a vacuum. Math. Res. Lett. (to appear) Google Scholar
  40. 40.
    Zhou, Y., Han, W.: Blow up for some semilinear wave equations in multi-space dimensions. Commun. Partial Differ. Equ. 39, 652–665 (2014)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Zhou, Y., Han, W.: Life-span of solutions to critical semilinear wave equations. Commun. Partial Differ. Equ. 39, 439–451 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouChina

Personalised recommendations