Stationary solutions to Euler equations with spherical symmetry

2005 ◽  
Vol 61 (1-2) ◽  
pp. 261-267
Author(s):  
Yanhong Liu ◽  
Changjiang Zhu
Keyword(s):  
Euler Equations ◽  
Spherical Symmetry ◽  
Stationary Solutions

Existence and stability of static shells for the Vlasov–Poisson system

Analysis ◽  
2006 ◽  
Vol 26 (4) ◽  
Author(s):  
Achim Schulze
Keyword(s):  
Spherical Symmetry ◽  
Variational Approach ◽  
Stationary Solutions ◽  
Poisson System ◽  
Existence And Stability

We prove the existence and stability of stationary solutions to the Vlasov–Poisson System with spherical symmetry, which describe static shells, i.e., the support of their densities is bounded away from the origin. We use a variational approach which was established by Y. Guo and G. Rein.

scholarly journals Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Ka Luen Cheung ◽  
Sen Wong
Keyword(s):  
Positive Integer ◽  
Boundary Value Problem ◽  
Euler Equations ◽  
Spherical Symmetry ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Isothermal Case ◽  
Initial Boundary Value

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of theN-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the formc(t)xα-1x+b(t)(x/x)for any value ofα≠1or any positive integerN≠1. Then, we show that blowup phenomenon occurs whenα=N=1andc2(0)+c˙(0)<0. As a corollary, the blowup properties of solutions with velocity of the form(a˙t/at)x+b(t)(x/x)are obtained. Our analysis includes both the isentropic case(γ>1)and the isothermal case(γ=1).

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