On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid

摘要

Certain classes of flows of an ideal incompressible liquid which with time gradually lose their smoothness are studied. The loss of smoothness is expressed as infinite growth of the vorticity as t → ∞ for three-dimensional flows and an increase of the gradient of the vorticity for planar and axisymmetric flows. Examples of such flows in the planar and axisymmetric cases are flows with a rectilinear streamline; this can be established using a special local Lyapunov function. Incompressible flows of a dusty medium are another example (it turns out that collapse is impossible for such flows, but the vorticity and the rate of deformation, as a rule, grow with no limit). Other examples can be constructed by composition of shear flows. It is shown that in the vorticity metric almost all stationary planar flows are unstable with respect to three-dimensional disturbances and in the vorticity gradient metric planar and axisymmetric flows with a rectilinear streamline are unstable.