On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

Abstract

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.Keywords: partial differential equations, quasilinear system, initial-boundary value problem, generalized Cauchy problem, unique existence theorems, locally analytic solution, series, convergence, majorant S. V. Kovalevskaya's theorem [1] ranks rightfully among the classical scientific results. One of the current problems of the theory of partial differential equations consists in proving some analogs and generalizations of Kovalevskaya's theorem.The article [2] by Riquier, a French mathematician, firstly addressed a generalization of the Cauchy problem to the case in which the initial data for different functions are given on different surfaces (also see [3]). Riquier's results were developed in the works of Russian mathematicians N. M. Gyunter [4] and S. L. Sobolev [5,6]. In [5] the problem with data on two surfaces was considered for a system of an arbitrary order in the case when the initial data for part of the sought functions are given on one surface and for all the remaining, on the other surface. The term "generalized Cauchy problem" (GCP) was first used in N. A. Lednev's article [7].The GCP for a quasilinear analytic system of the second order was studied in detail in S. P. Bautin's article [8]. Using the method of [8], we have managed to prove theorems of unique existence of a locally analytic solution to the GCP for a second-order quasilinear system in the case when the system has a singularity of the form u/x or x/u [9].The GCP arises in gas dynamics while describing gas flows with shock waves. V. M. Teshukov [10-13] studied the following problems: destruction of an arbitrary quasi-one-dimensional discontinuity; abrupt insertion of an impenetrable piston into a gas; reflection of a curvilinear shock wave from a rigid wall; and spatial interaction of strong discontinuities in a gas. In [14,15] the gas flows with shock waves were studied when the system of equations of gas dynamics has a singularity of the form u/r (u is the gas velocity and r is the distance to the axis or center of symmetry).We consider the following quasilinear system of partial differential equations in the case of three unknown functions depending on three independent variables:(1) Here U = {u, v, w} ...