Local Dissipation of Energy for Continuous Incompressible Euler Flows - Phillip Isett

Explore a comprehensive analysis seminar on the construction of continuous solutions to incompressible Euler equations exhibiting local energy dissipation. Delve into the motivations behind this research, including weak solutions, energy conservation in smooth solutions, and hydrodynamic turbulence. Examine Onsager's Conjecture and its implications for ideal turbulence, as well as the K41 Folklore Conjecture for Navier-Stokes equations. Investigate the open problem of the Strong Onsager conjecture and recent theorems addressing it. Learn about the Euler-Reynolds equations, convex integration techniques, and the challenges in eliminating unresolved flux currents and stress. Gain insights into the speaker's approach to constructing solutions with local energy dissipation while maintaining the highest possible regularity.

Intro
Motivation: Weak Solutions to the Euler equations
Motivation: Sufficiently smooth solutions conserve energy
Motivation: Hydrodynamic turbulence
Onsager and Ideal Turbulence
Motivation: Onsager's Conjecture (1949)
K41 implies compactness
K41 Folklore Conjecture for Navier-Stokes
Zero viscosity limits dissipate energy locally
K41 Folklore Conjecture in the inviscid limit
Open Problem: Strong Onsager conjecture
Theorem: First result on the Strong Onsager Conjecture
Theorem: Improvement on the Strong Onsager Conjecture
Outline
Continuous Solutions: The Euler-Reynolds Equations
Continuous Solutions: Convex Integration for Euler
The High-Frequency Correction
Micralocal Lemma
The Main Error Terms
Dissipative Euler Reynolds flow
Plan of attack
The new terms: The Transport term
Getting rid of the unresolved flux density
Conflict: Eliminate the Unresolved Flux Current and Stress
Dangerous terms: algebraic cancellation saving the day