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- Title
The Energy Measure for the Euler and Navier-Stokes Equations.
- Authors
Leslie, Trevor M.; Shvydkoy, Roman
- 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-∗<inline-graphic></inline-graphic> limit of the measures |u(t)|2dx<inline-graphic></inline-graphic> as t approaches the first possible blowup time. We show that membership of u in certain (weak or strong) LqLp<inline-graphic></inline-graphic> classes gives a uniform lower bound on the lower local dimension of E<inline-graphic></inline-graphic>; more precisely, it implies uniform boundedness of a certain upper s-density of E<inline-graphic></inline-graphic>. We also define and give lower bounds on the ‘concentration dimension’ associated to E<inline-graphic></inline-graphic>, which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of E<inline-graphic></inline-graphic> 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.
- Subjects
ENERGY measurement; NUMERICAL solutions to Navier-Stokes equations; FRACTAL dimensions; PARAMETER estimation; STEADY-state flow
- Publication
Archive for Rational Mechanics & Analysis, 2018, Vol 230, Issue 2, p459
- ISSN
0003-9527
- Publication type
Article
- DOI
10.1007/s00205-018-1250-4