Note on loss of regularity for solutions of the 3-D 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 3-D incompressible Navier-Stokes equations. The problem is still open. This report shows 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. 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;
- Navier-Stokes Equation;
- Riesz Theorem;
- Turbulence;
- Ideal Fluids;
- Incompressible Fluids;
- Reynolds Number;
- Transformations (Mathematics);
- Viscous Fluids;
- Fluid Mechanics and Heat Transfer