Lie subgroups of symmetry groups of fluid dynamics and magnetohydro-dynamics equations

Abstract

We classify the subalgebras of the symmetry algebras of fluid dynamics and magnetohydrodynamics equations into conjugacy classes under their respective groups. Both systems of equations are invariant under a Galilean-similitude algebra. In the case of the fluid dynamics equations, when the adiabatic exponent γ = 5/3, the symmetry algebra widens to a Galilean-projective algebra. We extend our previous classification of the symmetry algebra in the case of a nonstationary and isentropic flow to the general case of fluid dynamics and magnetohydrodynamics equations in (3 + 1) dimensions. The representatives of these algebras are given in normalized lists and presented in tables. Examples of invariant and partially invariant solutions, for both systems, are computed from representatives of these classifications. The final part of this work contains an analysis of this classification in connection with a further classification of the symmetry algebras in the case of the equations describing the flow of perfect gases. An explicit solution, in terms of Riemann invariants, is constructed from infinite-dimensional subalgebras of the symmetry algebra of the magnetohydrodynamics equations in the (1 + 2)-dimensional case.