Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method

TL;DR

This paper develops a modified energy method to establish the existence of small, smooth solutions for a nonlinear Burgers-Hilbert equation over extended time scales, modeling vorticity discontinuities.

Contribution

It introduces a novel modified energy approach to prove long-time existence of solutions for a nonlinear inviscid Burgers-Hilbert equation.

Findings

Proves existence of solutions over cubically nonlinear time-scales

Demonstrates effectiveness of the modified energy method

Models vorticity discontinuities in fluid dynamics

Abstract

We consider an initial value problem for a quadratically nonlinear inviscid Burgers-Hilbert equation that models the motion of vorticity discontinuities. We use a modified energy method to prove the existence of small, smooth solutions over cubically nonlinear time-scales.