Initial boundary value problem for the spherically symmetric motion of isentropic gas

Abstract

We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is governed by the compressible Euler equation of isentropic gas dynamics. The associated initial boundary value problem is solved by using the compensated compactness method for initial data containing the vacuum. The constructed weak solutions are temporally local but the class of initial data includes the stationaxy solutions.Let us consider the system of equations (1)(2) p = Ap ~ on t-> 0 and 1 <= r < +c~. Here M , A and ~ are positive constants such t h a t 1 < "7 = 5/3. This system governs the isentropic and spherically symmetric motion of an atmosphere surrounding a solid star with radius 1 and mass M . The variable p means the density, u the velocity, p the pressure and -M / r 2 stands for the gravitational force. The problem is to find a solution of the system (1) (2) which satisfies the initial condition (3) p I~=0 = p~ u I~=0 = ~0(~) and the b o u n d a r y condition (4) ~ tr=l = 0.We are interested in the case where the support of p~ is compact in [1, +ex)). In the article [4] we established the local existence of "tame" solutions. However this

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“…From Taylor's expansion theorem Remember the following lemma proved in [8]. Therefore, we obtainĪ ≥ −C √ ∆x.…”

Spherically symmetric solutions to the compressible Euler equation with an asymptotic γ-law

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