Enhanced Lifespan of Smooth Solutions of a Burgers-Hilbert Equation

TL;DR

This paper proves the existence of small, smooth solutions for a Burgers-Hilbert equation over cubic nonlinear time-scales, indicating delayed filamentation in vorticity discontinuities through a normal form transformation.

Contribution

It introduces a normal form transformation via a near-identity coordinate change to establish long-time smooth solutions for the Burgers-Hilbert equation.

Findings

Existence of small, smooth solutions over cubic time-scales.

Normal form transformation effectively delays filamentation.

Provides insight into vorticity discontinuity 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 normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.