Note on loss of regularity for solutions of the 3-D incompressible euler and related equations

One of the central problems in the mathematical theory of turbulence is that of breakdown of smooth (indefinitely differentiable) solutions to the equations of motion. In 1934 J. Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3-D incompressible Navier-Stokes equations. The problem is still open. We show in this report that breakdown of smooth solutions to the 3-D incompressible slightly viscous (i.e. corresponding to high Reynolds numbers, or "highly turbulent") Navier-Stokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. We prove then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.