Singularities in finite time of the full compressible Euler equations in \(\mathbb{R}^d\). (English) Zbl 1532.35363
Summary: The present article is concerned with the study of blow-up phenomena of the smoooth solutions for the full compressible Euler equations in \(\mathbb{R}^d\), \(d \geq 1\), which have always been a great concern to physicists and mathematicians throughout history. The approach is to construct exact explicit function to study singularities in finite time of the spherically symmetric solutions, provided the initial data satisfy some conditions. Compared with the results obtained by T. C. Sideris [Commun. Math. Phys. 101, 475–485 (1985; Zbl 0606.76088)], in this article, the initial velocity field is not required to have a compact support and the initial density and entropy is not equal to a constant outside the support of the initial velocity field.
MSC:
| 35Q31 | Euler equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35L45 | Initial value problems for first-order hyperbolic systems |
| 35L60 | First-order nonlinear hyperbolic equations |
| 35L65 | Hyperbolic conservation laws |
| 35B44 | Blow-up in context of PDEs |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35A21 | Singularity in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
Keywords:
the full compressible Euler equations; polytropic ideal fluid; blow-up phenomena; spherically symmetric solutionsCitations:
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