The Cauchy Problem for System of Volterra Integral Equations Describing the Motion of a Finite Mass of a Self-gravitating Gas

Abstract

In this article we investigate the Cauchy problem for a system of nonlinear integro-differential equations of gas dynamics that describes the motion of a finite mass of a self-gravitating gas bounded by a free boundary. It is assumed that gas moving is considered under the condition that at any time the free boundary consists of the same particles. This makes convenient the transition from Euler to Lagrangian coordinates. Initially, this system in Euler coordinates is transformed into a system of integro-differential equations in Lagrangian coordinates. A lemma on equivalence of these systems is proved. Then the system in Lagrange variables is transformed into a system consisting of Volterra integral equations and the equation continuity, for which the existence theorem for the solution of the Cauchy problem is proved with the help of the method of successive approximations. Based on the mathematical induction, the continuity of the solution and belonging of the solution to the space of infinitely differentiable functions are proved. The boundedness and the uniqueness of the solution are proved. The solution of a system of Volterra integral equations defines the mapping of the initial domain into the domain of moving gas, and also sets the law of motion of a free boundary, as a mapping of points of an initial boundary.

Keywords

• cauchy problem
• self-gravitating gas
• lagrangian coordinates
• system of nonlinear integro-differential equations
• method of successive approximations