Note on loss of regularity for solutions of the 3D incompressible Euler and related equations
Abstract
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 Leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3D incompressible NavierStokes equations. The problem is still open. This report shows that breakdown of smooth solutions to the 3D incompressible slightly viscous (i.e., corresponding to high Reynolds numbers, or highly turbulent) NavierStokes equations cannot occur without breakdown in the corresponding solution of the incompressible Euler (ideal fluid) equation. It is proven then that solutions of distorted Euler equations, which are equations closely related to the Euler equations for short term intervals, do breakdown.
 Publication:

Technical Summary Report Wisconsin Univ
 Pub Date:
 November 1985
 Bibcode:
 1985wisc.reptS....C
 Keywords:

 Differential Equations;
 Euler Equations Of Motion;
 Incompressible Flow;
 NavierStokes Equation;
 Riesz Theorem;
 Turbulence;
 Ideal Fluids;
 Incompressible Fluids;
 Reynolds Number;
 Transformations (Mathematics);
 Viscous Fluids;
 Fluid Mechanics and Heat Transfer