Dissipation enhancement of planar helical flows and applications to three-dimensional Kuramoto-Sivashinsky and Keller-Segel equations

TL;DR

This paper introduces planar helical flows on a 3D torus to enhance dissipation and applies them to establish global well-posedness for 3D Kuramoto-Sivashinsky and Keller-Segel equations, regardless of initial data size.

Contribution

It presents a novel class of flows that improve dissipation in 3D PDEs and demonstrates their effectiveness in proving global solutions for complex equations.

Findings

Dissipation enhancement achieved by planar helical flows.

Global well-posedness of 3D Kuramoto-Sivashinsky without growing modes.

Global classical solutions for 3D Keller-Segel with arbitrary initial data.

Abstract

We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.