Skip to main content
SpringerLink
Log in

Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at a certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental problem is whether concentration could be formed at the origin. In this paper, we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions, and establish the convergence of the approximate solutions to a global finite-energy entropy solution of the isentropic Euler equations with spherical symmetry and large initial data. This indicates that concentration is not formed in the vanishing viscosity limit, even though the density may blow up at a certain time. To achieve this, we first construct global smooth solutions of appropriate initial-boundary value problems for the Euler equations with designed viscosity terms, approximate pressure function, and boundary conditions, and then we establish the strong convergence of the viscosity approximate solutions to a finite-energy entropy solution of the Euler equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bianchini S., Bressan A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2) 161, 223–342 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta. Math. Sci. 6B, 75–120 (1986) (in English); 8A, 243–276 (1988) (in Chinese)

  3. Chen, G.-Q.: The compensated compactness method and the system of isentropic gas dynamics. In: Proceedings of Lecture Notes, preprint MSRI-00527-91, Berkeley, October (1990)

  4. Chen G.-Q.: Remarks on R J. DiPerna’s paper: “Convergence of the viscosity method for isentropic gas dynamics” [Comm. Math. Phys. 91 (1983), 1–30]. Proc. Am. Math. Soc. 125, 2981–2986 (1997)

    Article  MATH  Google Scholar 

  5. Chen G.-Q.: Remarks on spherically symmetric solutions of the compressible Euler equations. Proc. Roy. Soc. Edinb. 127, 243–259 (1997)

    Article  MATH  Google Scholar 

  6. Chen G.-Q., Glimm J.: Global solutions to the compressible Euler equations with geometrical structure. Commun. Math. Phys. 180, 153–193 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Chen G.-Q., Li T.-H.: Global entropy solutions in L ∞ to the Euler equations and Euler-Poisson equations for isothermal fluids with spherical symmetry. Methods Appl. Anal. 10, 215–243 (2003)

    MATH  MathSciNet  Google Scholar 

  8. Chen G.-Q., Perepelitsa M.: Vanishing viscosity limit of the Navier–Stokes equations to the Euler equations for compressible fluid flow. Comm. Pure Appl. Math. 63, 1469–1504 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Courant R., Friedrichs K.O.: Supersonic Flow and Shock Waves. Springer, New York (1948)

    MATH  Google Scholar 

  10. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  11. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax–Friedrichs scheme for the isentropic gas dynamics (I)–(II). Acta. Math. Sci. 5B, 483–500, 501–540 (1985) (in English); 7A, 467–480 (1987); 8A, 61–94 (1989) (in Chinese)

  12. Ding, X., Chen, G.-Q., Luo, P.: Convergence of the fractional step Lax–Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics. Commun. Math. Phys. 121, 63–84 (1989)

  13. DiPerna R.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Guderley G.: Starke kugelige und zylindrische Verdichtungsstosse inder Nahe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19(9), 302–311 (1942)

    MathSciNet  Google Scholar 

  15. Ladyzhenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, LOMI–AMS (1968)

  16. Lax P.D.: Shock wave and entropy. In: Zarantonello, E.A. (ed.) Contributions to Functional Analysis, pp. 603–634. Academic Press, New York (1971)

    Chapter  Google Scholar 

  17. Liu T.-P.: Quasilinear hyperbolic system. Commun. Math. Phys. 68, 141–572 (1979)

    Article  ADS  MATH  Google Scholar 

  18. LeFloch Ph.G., Westdickenberg M.: Finite energy solutions to the isentropic Euler equations with geometric effects. J. Math. Pures Appl. 88, 386–429 (2007)

    Article  MathSciNet  Google Scholar 

  19. Lions P.-L., Perthame B., Souganidis P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49, 599–638 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lions P.-L., Perthame B., Tadmor E.: Kinetic formulation of the isentorpic gas dynamics and p-systems. Comm. Math. Phys. 163, 415–431 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Makino T., Mizohata K., Ukai S.: Global weak solutions of the compressible Euler equations with spherical symmetry I, II. Jpn. J. Ind. Appl. Math. 9, 431–449 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Makino T., Takeno S.: Initial boundary value problem for the spherically symmetric motion of isentropic gas. Jpn. J. Ind. Appl. Math. 11, 171–183 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 5, 489–507 (1978)

    MATH  MathSciNet  Google Scholar 

  24. Rosseland S.: The Pulsation Theory of Variable Stars. Dover Publications Inc., New York (1964)

    Google Scholar 

  25. Slemrod M.: Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit. Proc. Roy. Soc. Edinb. 126, 1309–1340 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tartar, L.: Compensated compactness and applications to partial differential equations, In: Knops, R.J. (ed.) Proceedings of Research Notes in Mathematics, Nonlinear Analysis and Mechanics, Herriot–Watt Symposium, vol. 4, Pitman Press, (1979)

  27. Whitham G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  28. Yang T.: A functional integral approach to shock wave solutions of Euler equations with spherical symmetry I. Commun. Math. Phys. 171, 607–638 (1995)

    Article  ADS  MATH  Google Scholar 

  29. Yang T.: A functional integral approach to shock wave solutions of Euler equations with spherical symmetry II. J. Diff. Eqs. 130, 162–178 (1996)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Gui-Qiang G. Chen

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Gui-Qiang G. Chen

  3. Department of Mathematics, University of Houston, 651 PGH Building, Houston, TX, 77204–3008, USA

    Mikhail Perepelitsa

Authors
  1. Gui-Qiang G. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Perepelitsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gui-Qiang G. Chen.

Additional information

Communicated by W. Schlag

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, GQ.G., Perepelitsa, M. Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data. Commun. Math. Phys. 338, 771–800 (2015). https://doi.org/10.1007/s00220-015-2376-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-015-2376-y

Keywords

Search

Navigation