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Archive for Rational Mechanics and Analysis

, Volume 230, Issue 2, pp 459–492 | Cite as

The Energy Measure for the Euler and Navier–Stokes Equations

  • Trevor M. Leslie
  • Roman Shvydkoy
Article

Abstract

The potential failure of energy equality for a solution u of the Euler or Navier–Stokes equations can be quantified using a so-called ‘energy measure’: the weak-\(*\) limit of the measures \({|u(t)|^2{\rm d}x}\) as t approaches the first possible blowup time. We show that membership of u in certain (weak or strong) \({L^q L^p}\) classes gives a uniform lower bound on the lower local dimension of \({\mathcal{E}}\); more precisely, it implies uniform boundedness of a certain upper s-density of \({\mathcal{E}}\). We also define and give lower bounds on the ‘concentration dimension’ associated to \({\mathcal{E}}\), which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of \({\mathcal{E}}\) measure the departure from energy equality. As an application of our estimates, we prove that any solution to the 3-dimensional Navier–Stokes Equations which is Type-I in time must satisfy the energy equality at the first blowup time.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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