Volterra integral equation method in the study of dynamics of self-gravitating gas bounded by free surface

Abstract

The paper investigates dynamics of the finite mass of self-gravitating ideal gas with a variable flow bounded by free surface. Unsteady gas flows are described by a phenomenological mathematical model of gas dynamics, which is constructed using a system of nonlinear integral-differential equations written in Eulerian coordinates. The gas motion is considered under the condition that the free boundary at all times consists of the same particles. This circumstance makes it convenient to switch from Eulerian coordinates to Lagrangian coordinates, for which the domain of determining the solution of the gas motion problem will be fixed in advance. The transformation of the system to Lagrangian coordinates makes it possible to reduce it to an equivalent system consisting of Volterra-type integral equations and the continuity equation in Lagrangian form, and get rid of the unknown boundary. Using the method of successive approximations, properties of compact spaces, a series of original and specific estimates found for functions describing the flows of the self-gravitating gas, convergence of a sequence of approximate solutions is proved under assumptions about smoothness of the initial data. The theorem of existence and uniqueness of the solution of the Cauchy problem for the system of integral equations of the Volterra type is proved, and, consequently, the solution of the desired problem is found within the framework of the assumptions made about the motion in vacuum of the isolated mass of the self-gravitating ideal gas with a variable flow domain.