Single-scale wavelet representation of turbulence dynamics: Formulation and Navier-Stokes regularity

Jacques Lewalle

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

The 'triad interactions', arising from the spectral resolution of the nonlinear terms in the Navier-Stokes equations, have so far not been substantially modified in the wavelet representation. In this paper, the multiscale interactions are captured by exact expressions evaluated at a single scale of the Mexican hat wavelet coefficients: the larger-scale terms as a volume integral of nearby wavelet coefficients, and the smaller-scale contributions as iterated Laplacians of the coefficient at the point of interest. As a result, the Navier-Stokes equations are expressed exactly at a single scale. This facilitates the evaluation of the dominant Hölder exponent near singularities. From the scaling properties of wavelet coefficients, it is shown that Euler dynamics would generate stronger singularities for any h < 1, but that viscous dynamics would not unless h < -1 (a discontinuous case). We discuss how this conclusion could be affected by boundary conditions.

Original languageEnglish (US)
Pages (from-to)1232-1235
Number of pages4
JournalPhysica D: Nonlinear Phenomena
Volume239
Issue number14
DOIs
StatePublished - Aug 15 2010

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Keywords

  • Navier-Stokes
  • Regularity
  • Scaling
  • Turbulence
  • Wavelets

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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