Applications of differential invariants in hydrodynamics

Abstract:
The systematic use of group analysis methods to study concrete models of mathematical physics consists of the following three steps. These are: calculation of admitted group, construction of its optimal system of subgroups and obtaining of classes both of invariant and partially invariant solutions. The calculation of admitted group is performed according to a known and well-approved algorithm, based on infinitesimal transformations [1, 2]. The realization of the second step also does not have any particular difficulties when the admitted Lie group is finite-dimensional [3-5]. However, the study of subgroups set for the infinite-dimensional Lie group is more complicated problem. Meanwhile, the infinite-dimensional group are possessed by a lot of mathematical models (the Euler and the Navier-Stokes equations, stationary gas dynamics equations, boundary layer equations, etc).

One of the approaches to study group-invariant solutions for the equations, admitting infinite-dimensional transformations group, is related to its group stratification into resolving and automorphic parts [1]. The automorphic system describes the orbit of any solution under the group action. The resolving system distinguishes the orbits of essentially different solutions and thus describes the assemblage of orbits. At group stratification regarding infinite-dimensional part of admitted group the resolving system inherits only finite-dimensional part of the original group. That allows studying constructively the set of its group-invariant solutions. The solutions of the original system are restored then by means of integration of the automorphic system. The examples of group stratification are presented in [1, 6].

The construction of group stratification is based on the knowing of all differential invariants of the group being considered. For their description the classical theory of differential invariants is used (its presentation can be found in [1]). The central result of this theory is the theorem on the finiteness of differential invariants basis, which is valid both for finite- and infinite-dimensional groups. The theorem states the existence of such finite set of scalar differential invariants from which any other invari-ant can be obtained by means of functional operations and by application of the operators of invariant differentiation. The latter ones form Lie algebra over the field of differential invariants. The importance of differential invariants basis is stipulated by the fact that it completely determines the original transformations group, serves for mathematical description of models, admitting this group, and also gives new possibilities to form differentially invariant solutions of the equations being investigated.

 The purpose of present work is to calculate the basis of differential invariants for infinite-dimensional groups, admitted by the Navier-Stokes and Euiler equations, stationary gas dynamics and the Karman-Guderley equations. We demonstrate the utilization of the obtained bases to construct the differentially invariant solutions and the group stratification. The connection between solu-tions for original and resolving equations is shown by several examples.

References:

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