Institute of Mathematics of NAS of Ukraine,
3 Tereshchenkivska Street, 01601 Kyiv-4, UKRAINE
e-mail: rop@imath.kiev.ua

Classical and Nonclassical Submodels of Euler Equations

Abstract:

The Euler equations describing motion of incompressible ideal fluid are investigated from the symmetry point of view. Exploiting Lie symmetry of the Euler equations, we describe all possible (inequivalent) Lie submodels of the latter. Namely, we find complete sets of inequivalent one-, two-, and three-dimensional subalgebras of the (infinite-dimensional) maximal Lie invariance algebra of the Euler equations. Then, we construct the corresponding ansatzes of codimension one, two, and three, as well
as, reduced systems of partial differential equations in three and two independent variables and reduced systems of ordinary differential equations. Lie symmetry properties of the reduced systems of partial differential equations are investigated. There exists a number of reduced systems admitting Lie symmetries which are not induced by Lie symmetries of the initial Euler equations. The reduced systems of ordinary differential equations are integrated or for them partial exact solutions are found. As a result, new large classes of exact solutions of the Euler equations, which contain, in particular, arbitrary functions, are constructed. We investigate a submodel of the Euler equations, which is partially invariant with the minimal defect under the group SO(3). It is proved that the Navier-Stokes equations do not have submodels of such type.
We have also obtained exhaustive description of non-classical symmetry operators admitted by the Euler equations. We prove that there exists only one class of nonclassical symmetries, which are inequivalent to Lie symmetry operators of the Euler equations.