Integrable models of Rotationally Symmetric Motions of an Ideal Incompressible Fluid
We consider rotationally-symmetric motion of an ideal incompressible liquid in the absence of external forces. The system of Euler equations, which describes such motions, is of composite type. After transition to new unknown functions according to , the equations represent a coupled quasi-linear system, which consists of two inhomogeneous Cauchy-Riemann equations and two transport equations. The hyperbolic part of this system has multiple characteristics and it can’t be symmetrized, because the corresponding matrix has a Jordan structure. This is probably the reason for the absence of global theorems concerning the solvability of initial boundary value problems. Therefore, finding exact solutions to rotationally-symmetric motions of ideal incompressible liquid is an actual problem.
To find new exact solutions, we computed the group admitted by Euler equations in new variables. The corresponding basis of Lie algebra is formed by five operators, one of which contains an arbitrary function of time. For this algebra, we constructed an optimal system of one- and two-dimensional subalgebras, from which we have chosen subalgebras, which contain an operator, specific for rotationally-symmetric motions. This operator establishes a non-linear transformation of rotational velocity and the corresponding transformation of pressure, which preserve the initial system of Euler equations. Admissibility of such transformation was discovered by Kapitanskii .
There were given various examples of invariant and partially invariant solutions, obtained with the help of the operator mentioned above . One of them describes the process of displacement of plane vortex flow by axially symmetric flow in a ring with permeable walls. To this solution there corresponds a generalized solution to Euler equations, in which all the sought functions are continuous, while the peripheral velocity has an infinite derivative on the vortex domain boundary.
There was studied in detail a class of stationary solutions, which are determined from a system of ordinary differential equations of the fourth order, which admits an exact integration. There were given typical phase portraits of the flow, initiated by distributed along the axis of symmetry vorticity sources.
Also, we consider rotationally-symmetrical solutions to Euler equations with a linear dependence of axial component of velocity on axial coordinate. By methods of group analysis of differential equations these equations were reduced to one hyperbolic equation of the fourth order . For this equation a local in time unique solvability of initial boundary-value problem was proved. Also, for this equation a generalized Goursat problem was considered. There were formulated sufficient conditions of its solution blowing up within a finite time and conditions of classical solution existence in case it is defined for all values of the radial coordinate. It is established that in the class of considered solutions to Euler equations, setting up initial velocity field in whole space does not determine the solution to Cauchy problem uniquely .
We also consider solutions, built on the basis of partially invariant ones with the help of the method, proposed in . For this class of solutions there was studied a problem of evolution of the vortex domain, which is determined by condition of peripheral velocity circulation equality to zero. Boundary of the vortex domain is a compact material surface, on which the pressure and velocity are continuous, while the vorticity has a rupture.
The work is supported by the Council of the President of the Russian Federation for the Support of Leading Scientific Schools, project No. NSh–5873.2006.1.
 Meshcheryakova E. Yu., Pukhnachev V.V. // Dokl. Physics. 2007. Vol. 52, No. 1. pp. 47–51.
 Kapitanskii L.V. // Sov. Phys. Dokl. 1978. Vol. 23. No. 896.
 Pukhnachev V.V. // Appl. Mech. Tech. Phys. 2003. Vol.44. No.3. pp. 317–323.
 Meshcheryakova E. Yu. // Siberian Electr. Math. Reports. 2007.
Joint work with Vladislav Pukhnachev (Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia).