Unsteady Solutions of Euler Equations Generated by Steady Solutions

Abstract

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H -invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.

On the construction of conservation laws: A mixed approach

“…(28) We notice that if γ = n+2 n , Euler equations admit an additional Lie point symmetry [19]- [21] spanned by…”

Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations

“…By requiring the invariance of the class E (p) in the augmented space A ≡ R 4 × R 5 × R 16 through the Lie's infinitesimal criterion [4], we determine 24 symmetry operators, whose expression is too long to be written here. In view of the results we want to achieve, we report the projections of the admitted operators on the space Z ≡ R 4 × R 5 :…”

“…In dealing with differential equations, Lie group theory [4,5,6,7,8,9,10,11] yields general algorithmic methods either for the determination of special (invariant) solutions [12,13,14,15,16] of initial and boundary value problems, or the derivation of conserved quantities, or the construction of relations between different differential equations that turn out to be equivalent [11,17,18,19,20,21,22,23,24].…”