Non-local conservation laws for the equations of the irrotational isentropic plane-parallel gas motion

Abstract

By introducing non-local variables, namely, the velocity potential and the stream function, and changing to the hodograph plane, the problem of finding the conservation laws for a non-linear system, describing the plane-parallel steady irrotational isentropic gas motion is reduced to the problem of finding the conservation laws for a linear Chaplygin system. The conservation laws of the zeroth and first orders for the Chaplygin system are obtained. It is established that the set of conservation laws of zeroth order that a Chaplygin system possesses consists of conservation laws that are linear in the velocity potential and the stream function, and a new non-linear conservation law. The linear conservation laws have functional arbitrariness. They produce linearity of this system and are defined by Green's operator formula. It turns out that all the conservation laws in the physical plane, obtained by Rylov, are generated by a linear combination of these linear conservation laws and trivial conservation laws. All the linear conservation laws of the first order for the Chaplygin system, generated by Green's operator formula, that are independent of the stream function, are obtained. It is shown that the Chaplygin system has no more than three first-order conservation laws, independent of the stream function, which are not a linear combination of these linear conservation laws and trivial conservation laws, and their components are quadratic in the velocity potential and its derivatives. All the Chaplygin functions for which the Chaplygin system has three such conservation laws are listed. These conservation laws are obtained.