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. Keywords: Riesz transforms. (Author)

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Document Details

Document Type
Technical Report
Publication Date
Nov 01, 1985
Accession Number
ADA163632

Entities

People

  • Petre Constantin

Organizations

  • University of Wisconsin–Madison

Tags

DTIC Thesaurus Topics

  • Computational Fluid Dynamics
  • Computational Science
  • Differential Equations
  • Equations
  • Equations Of Motion
  • Euler Equations
  • Formulas (Mathematics)
  • Incompressibility
  • Inequalities
  • Mathematics
  • Navier Stokes Equations
  • Notation
  • Reynolds Number
  • Scalar Functions
  • Three Dimensional
  • Turbulence
  • United States

Fields of Study

  • Mathematics

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  • Fluid Dynamics.
  • Linear Algebra
  • Theoretical Analysis.