Oscillations and concentrations in weak solutions of the incompressible fluid equations

Ronald J. DiPerna, Andrew J. Majda

Research output: Contribution to journalArticle

Abstract

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Original languageEnglish (US)
Pages (from-to)667-689
Number of pages23
JournalCommunications in Mathematical Physics
Volume108
Issue number4
DOIs
StatePublished - Dec 1987

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Measure-valued Solutions
incompressible fluids
Incompressible Fluid
3D
Weak Solution
Oscillation
oscillations
Euler
Incompressible Euler Equations
Young Measures
Kinetic energy
Persistence
Reynolds number
Navier-Stokes Equations
Approximate Solution
Converge
high Reynolds number
Navier-Stokes equation
kinetic energy
Concepts

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Oscillations and concentrations in weak solutions of the incompressible fluid equations. / DiPerna, Ronald J.; Majda, Andrew J.

In: Communications in Mathematical Physics, Vol. 108, No. 4, 12.1987, p. 667-689.

Research output: Contribution to journalArticle

DiPerna, Ronald J. ; Majda, Andrew J. / Oscillations and concentrations in weak solutions of the incompressible fluid equations. In: Communications in Mathematical Physics. 1987 ; Vol. 108, No. 4. pp. 667-689.
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