Osaka Journal of Mathematics

On spherically symmetric motions of a gaseous star governed by the Euler--Poisson equations

Tetu Makino

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Abstract

We consider spherically symmetric motions of a polytropic gas under the self-gravitation governed by the Euler--Poisson equations. The adiabatic exponent ($=$ the ratio of the specific heats) $\gamma$ is assumed to satisfy $6/5 < \gamma \leq 2$. Then there are equilibria touching the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. We solve this problem by the Nash--Moser theorem. The result will realize the so-called physical vacuum boundary. But the present study restricts $\gamma$ to the case in which $\gamma/(\gamma-1)$ is an integer. Other cases are reserved to the future as an open problem. The time-local existence of smooth solutions to the Cauchy problems is also discussed.

Article information

Source
Osaka J. Math., Volume 52, Number 2 (2015), 545-581.

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1427202902

Mathematical Reviews number (MathSciNet)
MR3326626

Zentralblatt MATH identifier
1323.35180

Subjects
Primary: 35L05: Wave equation 35L52: Initial value problems for second-order hyperbolic systems 35L57: Initial-boundary value problems for higher-order hyperbolic systems 35L70: Nonlinear second-order hyperbolic equations 76L10

Citation

Makino, Tetu. On spherically symmetric motions of a gaseous star governed by the Euler--Poisson equations. Osaka J. Math. 52 (2015), no. 2, 545--581. https://projecteuclid.org/euclid.ojm/1427202902


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