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Some special solutions of the multidimensional Euler equations in \(\mathbb R^N\). (English) Zbl 1083.35058

The multidimensional Euler equations for compressible gas are considered. The gas is polytropic, so the pressure \(p\) depends on the density \(\rho\): \(p(\rho)=\frac{\rho^\gamma}{\gamma}\). For the case of spherical symmetry the equations become simpler. The special solution for constant with respect to \(x\) density are obtained. Then the author finds out that the critical mass is infinite and thus no blow-up is possible for finite mass. For infinite mass the solution blows up everywhere in finite time for suitable initial velocity. Finally, using the total potential energy argument, the author proves that for \(\gamma>1\) finite total energy implies there is no \(\delta\)-function of bigger blow-up.

MSC:

35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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