Blow-up phenomena of solutions to the Euler equations for compressible fluid flow.(English)Zbl 1083.76051

Summary: The blow-up phenomena are investigated for Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blow-up behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form $$c(t)\mathbf x$$ are constructed to show the expanding and blow-up properties. The solution with velocity of the form $$\dot a(t)\mathbf x/a(t)$$ for $$\gamma \geqslant 1$$ and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form $$c(t)|\mathbf x|^{\alpha -1}\mathbf x$$ for $$\alpha\neq 1$$ and multi-space dimensions.

MSC:

 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q35 PDEs in connection with fluid mechanics

Keywords:

spherical symmetry
Full Text:

References:

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