Nagoya Mathematical Journal

Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

Huicheng Yin

Full-text: Open access


In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Article information

Nagoya Math. J., Volume 175 (2004), 125-164.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L67: Shocks and singularities [See also 58Kxx, 76L05]
Secondary: 35L65: Conservation laws 76L05: Shock waves and blast waves [See also 35L67] 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]


Yin, Huicheng. Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data. Nagoya Math. J. 175 (2004), 125--164.

Export citation


  • S. Alinhac, Temps de vie precise et explosion geometrique pour des systemes hyperboliques quasilineaires en dimension un d'espace , Ann. Scoula Norm. Sup. Pisa. Serie IV, XXII , no. 3, 493–515 (1995).
  • S. Alinhac, Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions , Ann. of Math., 149 , no. 1, 97–127 (1999).
  • A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for $n \times n$ systems of conservation laws, Mem. Amer. Math. Soc. 146, no. 694 (2000).
  • S. Chen and L. Dong, Formation of shock for the p-system with general smooth initial data , Sci. in China, Ser. A, 44 , no. 9, 1139–1147 (2001).
  • S. Chen, Z. Xin and H. Yin, Formation and construction of shock wave for quasilinear hyperbolic system and its application to 1-D inviscid compressible flow , preprint (1999).
  • S. Chen and Z. B. Zhang, On the generation of shock waves of first order quasilinear equations , Fudan Journal (Natural Science), 13–22 (1963).
  • G. Q. Chen and J. Glimm, Global solutions to the compressible Euler equations with geometrical structure , Comm. Math. Phys., 180 , 153–193 (1996).
  • C. M. Dafermos, Generalized characteristics in hyperbolic system of conservation laws , Arch. Rat. Mech. Anal., 107 , 127–155 (1989).
  • R. Diperna, Uniqueness of solutions to hyperbolic conservation laws , Indiana Univ. Math. J., 28 , 224–257 (1979).
  • J. Glimm, Solution in the large for nonlinear hyperbolic systems of equations , Comm. Pure Appl. Math., 18 , 697–715 (1965).
  • L. Hömander, Lectures on nonlinear hyperbolic differential equations, Mathematics and Applications 26, Springer-Verlag, Berlin (1997).
  • F. John, Formation of singularities in one-dimensional nonlinear wave propagation , Comm. Pure Appl. Math., 27 , 377–405 (1974).
  • F. John, Existence for large times of strict solutions of nonlinear wave equations in three space for small initial data , Comm. Pure Appl. Math., 40 , no. 1, 79–109 (1987).
  • F. John and S. Klainerman, Almost global existence to nonlinear wave equations in three space , Comm. Pure Appl. Math., 37 , no. 4, 443–455 (1984).
  • P. D. Lax, Hyperbolic systems of conservation laws II , Comm. Pure Appl. Math., X , 537–566 (1957).
  • P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shocks waves, Conf. Board Math. Sci. SIAM, 11 (1973).
  • M. P. Lebaud, Description de la formation d'un choc dans le $p$-systems , J. Math. Pures Appl., 73 , 523–565 (1994).
  • T. Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations , J. D. E., 33 , 92–111 (1979).
  • T. P. Liu and T. Yong, Well-posedness theory for hyperbolic conservation laws , Comm. Pure Appl. Math., 52 , no. 12, 1553–1586 (1999).
  • A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag (1984).
  • J. A. Smoller, Shock waves and reaction-diffusion equations, Berlin-Heiderberg-New York, Springer-Verlag, New York (1984).
  • H. Yin, The blowup mechanism of axisymmetric solutions for three dimensional quasilinear wave equations with small data , Science in China (Series A), 43 , no. 3, 252–266 (2000).
  • H. Yin, The blowup mechanism of small data solutions for the quasilinear wave equations in three space dimensions , Acta Math. Sinica, English Series, 17 , no. 1, 35–76 (2001).
  • H. Yin and Q. Qiu, The blowup of solutions for three dimensional spherically symmetric compressible Euler equations , Nagoya M. J., 154 , 157–169 (1999).