Geometric-Analytic Framework for Energy Dissipation in Incompressible Euler Equations

Overview

This study introduces a geometric-analytic framework designed to define and analyze properties of energy dissipation within weak solutions of the incompressible Euler equations. The framework addresses the regularity of the Duchon-Robert distribution and its relationship to Hausdorff measure, which in turn imposes quantitative constraints on the dissipative set and structure function exponents associated with intermittency in turbulence. The approach also provides a means to recover established results related to Onsager singularity.

Research Context

The research is situated within the study of turbulence, particularly focusing on the phenomenon of "intermittency" and the dissipation of energy in fluid dynamics. The incompressible Euler equations serve as the mathematical model for describing inviscid fluid flow. A key concept explored is the Duchon-Robert distribution, which quantifies energy dissipation in weak solutions. The study examines how the spatial regularity of solutions, specifically in Besov spaces, influences the regularity properties of this distribution. The concept of Onsager singularity, a known aspect of turbulence theory, is also referenced as a by-product of the presented framework.

Approach

The methodology involves the development of a geometric-analytic framework. This framework is applied to weak solutions of the incompressible Euler equations that possess spatial Besov regularity. Key elements of the approach include an analysis of the regularity of the Duchon-Robert distribution and its characteristics when it constitutes a Radon measure.

Findings

• For weak solutions of the incompressible Euler equations with spatial Besov regularity, the Duchon-Robert distribution exhibits optimal improved regularity within a negative Besov space.
• If the Duchon-Robert distribution is a Radon measure, it is absolutely continuous with respect to a suitable Hausdorff measure.
• These findings impose quantitative constraints on two specific aspects:
  • The dimension of the dissipative set, which can be fractal.
  • The admissible structure functions exponents.
  These constraints relate to the phenomenon of "intermittency" in turbulence.
• The developed framework also enables the recovery of many known results associated with "Onsager singularity".