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
Abstract:We consider the existence of time periodic solutions for a nonlinear hyperbolic scalar conservation law with a time periodic outer force. The uniform asymptotic behavior of the Lax-Friedrichs difference approximation gives fixed points of the Poincare map and the convergence of the approximate periodic solution made from such fixed points is proved by the compensated compactness theory.
We consider the existence of the generalized solution for a free piston problem for isentropic gas dynamics. By the compensated compactness theory, we can show that an approximate solution converges to a generalized solution.
SynopsisFor piston problems for a system of isentropic gas dynamics, convergence theorems of a difference scheme are obtained by compensated compactness theory and by analysis of the difference scheme.
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