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Dissipation concentration in two-dimensional fluids

Luigi De Rosa, Jaemin Park

Abstract

We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a ''quadratic'' space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Batchelor-Kraichnan dissipative scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated.

Dissipation concentration in two-dimensional fluids

Abstract

We study the dissipation measure arising in the inviscid limit of two-dimensional incompressible fluids. It is proved that the dissipation is Lebesgue in time and, for almost every time, it is absolutely continuous with respect to the defect measure of strong compactness of the solutions. When the initial vorticity is a measure, the dissipation is proved to be absolutely continuous with respect to a ''quadratic'' space-time vorticity measure. This results into the trivial measure if the initial vorticity has singular part of distinguished sign, or a spatially purely atomic measure if wild oscillations in time are ruled out. In fact, the dynamics at the Batchelor-Kraichnan dissipative scale is the only relevant one, in turn offering new criteria for anomalous dissipation. We provide kinematic examples highlighting the strengths and the limitations of our approach. Quantitative rates, dissipation life-span and steady fluids are also investigated.

Paper Structure

This paper contains 16 sections, 22 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. The measures of dissipation, defect and vorticity (Section \ref{['Measure_comparison']})
  3. The dissipative scale and anomalous dissipation criteria (Section \ref{['S:K41 scale']})
  4. Quantitative rates and dissipation life-span (Section \ref{['S:rates']})
  5. Steady fluids (Section \ref{['S:stationary']})
  6. Brief review of related literature
  7. Tools
  8. Mollification estimates
  9. Quantitative equi-integrability
  10. Curves of measures and absolute continuity
  11. Some remarks on Navier--Stokes
  12. Dissipation, defect and vorticity measures
  13. The dissipative scale
  14. Quantitative rates and dissipation life-span
  15. The stationary case
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{u_0^\nu\}_{\nu}\subset L^2(\mathbb{T}^2)$ be a strongly compact sequence of divergence-free vector fields and let $\{u^\nu\}_{\nu}$ be the corresponding sequence of Leray--Hopf solutions to NS. Assume that Then $D\in L^1([0,T];\mathcal{M}(\mathbb{T}^2))$. In addition, assume that $u^\nu \overset{*}{\rightharpoonup} u$ and $|u^\nu -u|^2 \overset{*}{\rightharpoonup} \Lambda$, respectively i

Theorems & Definitions (71)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 61 more