On the Loss and Propagation of Modulus of Continuity for the Two-Dimensional Incompressible Euler Equations

TL;DR

This paper investigates the propagation of various moduli of continuity for vorticity in 2D incompressible Euler equations, demonstrating that some rougher than Dini continuity are propagated while others are not.

Contribution

It introduces explicit families of moduli of continuity that are propagated and shows that not all moduli can be propagated, addressing a fundamental question in fluid dynamics.

Findings

Certain moduli of continuity rougher than Dini are propagated.

Some moduli of continuity cannot be propagated.

The paper provides explicit examples and counterexamples.

Abstract

It is known from the work of Koch that the two-dimensional incompressible Euler equations propagate Dini modulus of continuity for the vorticity. In this work, we consider the two-dimensional Euler equations with a modulus of continuity for vorticity rougher than Dini continuous. We first show that the two-dimensional Euler equations propagate an explicit family of moduli of continuity for the vorticity that are rougher than Dini continuity. The main goal of this work is to address the following question: Given a modulus of continuity for the 2D Euler equations, can we always propagate it? The answer to this question is \textit{No}. We construct a family of moduli of continuity for the 2D Euler equations that are \textit{not} propagated.