On the existence and uniqueness of the solution to the Cauchy problem for a system of integral equations describing the motion of a rarefied mass of a self-gravitating gas

The Cauchy problem for a system of nonlinear Volterra-type integral equations that describes, in Lagrangian coordinates, the motion of a finite mass of a rarefied self-gravitating gas bounded by a free surface is studied. A theorem of the existence and uniqueness of a solution to the problem in the space of infinitely differentiable functions is proved. The solution is constructed in the form of a series with recursively calculated coefficients. The local convergence of the series is proved using the method of successive approximations.