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On the Geometric Blow-Up Mechanism to Scalar Conservation Laws

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DOI: 10.4236/ijmnta.2013.23021    3,378 Downloads   5,016 Views  
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Shikuan Mao, Yongqin Liu

Affiliation(s)

School of Mathematics and Physics, North China Electric Power University, Beijing, China.

ABSTRACT

The focus of this article is on the geometric mechanism for the blow-up of solutions to the initial value problem for scalar conservation laws. We prove that the sufficient and necessary condition of blow-up is the formation of characteristics envelope. Whether the solution blows up or not relates to the topology structure of a set dominated by initial data. At last we take Burger’s equation as an example to verify our main theorem.

KEYWORDS

Blow-Up; Conservation Laws; Characteristics Envelope

Cite this paper

S. Mao and Y. Liu, "On the Geometric Blow-Up Mechanism to Scalar Conservation Laws," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 3, 2013, pp. 161-163. doi: 10.4236/ijmnta.2013.23021.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] S. Alinhac, “Blowup for Nonlinear Hyperbolic Equations,” Birkh?usera, Boston, 1995. doi:10.1007/978-1-4612-2578-2
[2] F. John, “Nonlinear Wave Equations, Formation of Singularities,” American Mathematical Society, Providence, 1990.
[3] H. A. Levine and M. H. Protter, “The Breakdown of Solutions of Quasilinear First Order Systems of Partial Differential Equations,” Archive for Rational Mechanics and Analysis, Vol. 95, No. 3, 1986, pp. 253-267. doi:10.1007/BF00251361
[4] J. Smoller, “Shock Waves and Reaction-Diffusion Equations,” Springer Verlag, New York, Heidelberg, Berlin, 1983. doi:10.1007/978-1-4684-0152-3

  
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