Dynamic Growth Estimates of Maximum Vorticity for 3D Incompressible Euler Equations and the SQG Model

TL;DR

This paper derives sharp estimates for the growth of maximum vorticity in 3D Euler and SQG models, showing under certain conditions that vorticity growth is at most double exponential, extending previous theoretical results.

Contribution

It provides a new dynamic growth estimate for maximum vorticity applicable to both 3D Euler and SQG equations, extending earlier work and incorporating geometric regularity assumptions.

Findings

Maximum vorticity growth is at most double exponential in time.

The estimates apply to specific initial vortex configurations like anti-parallel vortex tubes.

The results extend previous theoretical bounds by Córdoba-Fefferman and Deng-Hou-Yu.

Abstract

By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li. Under some additional assumption on the vorticity field, which seems to be consistent with the computational results of Hou-Li, we show that the maximum vorticity can not grow faster than double exponential in time. Our analysis extends the earlier results by Cordoba-Fefferman and Deng-Hou-Yu.