Global Existence for the Two-dimensional Kuramoto-Sivashinsky equation with a Shear Flow

TL;DR

This paper proves the global existence of solutions for a two-dimensional Kuramoto-Sivashinsky equation with shear flow, allowing for large initial data, under specific conditions on the shear's critical points and modes.

Contribution

It establishes the first global existence result for the 2D KSE with shear flow under broad conditions, using a bootstrap argument.

Findings

Global solutions exist for large initial data.

Conditions on shear flow critical points are sufficient for well-posedness.

The method applies to shear flows with finitely many critical points.

Abstract

We consider the Kuramoto-Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in $L^2$, using a bootstrap argument. The initial data can be taken arbitrarily large.