Gas dynamics applications of a characteristic Cauchy problem

Abstract

Some initial-boundary-value problems for a system of quasilinear partial differential equations of gas dynamics with the initial data prescribed on the characteristic surface (characteristic Cauchy problem) are considered. The following three-dimensional flow problems are investigated: the flow produced by a motion of an impermeable piston; the flow produced by a permeable piston with a given pressure; and the flow produced by the moving free boundary. In the first two problems, the piston motion is prescribed; in the last problem, the free boundary motion cannot be prescribed in advance and must be determined as a part of the problem. It is shown that those problems can be reduced to a characteristic Cauchy problem of a certain standard type that satisfies the analog of Cauchy–Kowalewski's existence theorem in the class of analytical functions (Differential Equations 12 (1977) 1438–1444). Thus, it is proved that, in the case of the analyticity of the input data, the considered problems have unique piecewise analytic solutions which may be expressed by infinite power series (the procedure of constructing the power series solution is described in Differential Equations 12 (1977) 1438–1444 as a part of the proof of the theorem).