Conservation laws and group properties of equations of isentropic gas motion

Abstract

A system of equations of isentropic gas motion with n 2 is classified in terms of zero-order conservation laws with the use of the method of A-operators. New conservation laws are found to be valid only for potential isentropic motion of the Chaplygin gas. In this case, the greatest number of nontrivial conservation laws is obtained, with n scalar conservation laws being nonlocal. Additional properties of symmetry of the considered equations associated with these conservation laws are indicated.Introduction. The search for conservation laws for systems of differential equations is discussed in many publications. In particular, Shmyglevskii [1] obtained a full system of zero-order conservation laws for equations of motion of a perfect gas in a three-dimensional formulation. Ibragimov [2] found additional conservation laws for equations of motion of a polytropic gas by applying point symmetry operators to the classical conservation laws. Ibragimov [2] also derived an additional conservation law for a system of equations of potential isentropic motion of a polytropic gas.In the present work, a system of equations of isentropic gas motion with n 2 is classified in terms of zero-order conservation laws with the use of the method of A-operators [3]. It is demonstrated that the condition of an isentropic character of the flow does not generate new conservation laws in this system (in contrast to the conventional system of equations of gas motion). A system of vortex-free isentropic gas motion and a system of equations of potential isentropic gas motion are classified in terms of the zero-order conservation laws by the method of A-operators. For the latter system of equations, the greatest number of nontrivial conservation laws is observed in the case of the Chaplygin gas, with n scalar conservation laws being nonlocal. A group classification is performed for the system of equations of potential isentropic gas motion, which allows the set of nontrivial zero-order conservation laws to be extended. It is found that the system for the Chaplygin gas admits the greatest Lie group of transformations. Isentropic gas motion is described by the equations

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

“…The conservation laws for the differential equations are important in the study of the solution set of the equation [16,[20][21][22][23][24]. The conservation laws can also be used in numerical calculations to improve the accuracy of the calculations.…”

Symmetry Properties and Solutions of Shallow Water Equations

“…To find all nontrivial conservation laws of zero-order, is used the method of A -operators proposed by Yu. A. Chirkunov [9][10][11][12][13]. The conservation law h A = 0 , uh A = 1 (5) is taken as a generating one.…”

“…is an evolutionary operator of generalized symmetry [8][9][10][11][12][13] follows, that the factor set of zero-order A -operators for the system (3) by the range of its trivial A -operators is generated by the operators: ...…”

“…is any solution of the system (12). Thus, the set of zero-order conservation laws for the system (3) and its main group of Lie transformations are determined by the same system of linear equations (12).…”

“…As the result of this transformation, the system ( 3) is equated with the linear system (12). The solutions of the system (3) and ( 12) are linked in the following way.…”