On existence and uniqueness of classical solutions to Euler equations in a rotating cylinder

Abstract

In this paper we consider the existence and uniqueness of classical solutions of the non-stationary Euler equations for an ideal incompressible flow in a cylinder rotating with a constant angular velocity about an axis orthogonal to the axis of the cylinder. If we consider, say, one of the most-studied three-dimensional cases of fluid flow, axially-symmetric flow in a pipe, it can be easily seen that the (Navier–Stokes or Euler) equations are essentially two-dimensional because they split into a two-dimensional analog and a one-dimensional equation. The two-dimensional equations are known to have global classical solutions. If we add, as is standard in the literature, rotation along the axis of symmetry, the same conclusions are valid. However the picture completely changes if the axis of rotation does not coincide with the symmetry axis. The flow is then “stirred” and the splitting we mentioned above is coupled. Therefore in spite of the symmetry, i.e., the dependence on two spatial variables, the flow is essentially not two-dimensional. The flows of this type are called in physical literature 2D–3C flows (two-dimensional, three-components flows). In the present paper we show that in this case for any T > 0 the unique classical solution exists on the interval [ 0 , T ] for a sufficiently small angular velocity of rotation.