Nagoya Mathematical Journal

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

Huicheng Yin

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Abstract

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

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

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114632098

Mathematical Reviews number (MathSciNet)
MR2085314

Zentralblatt MATH identifier
1133.35374

Subjects
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]

Citation

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. https://projecteuclid.org/euclid.nmj/1114632098


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