Global existence and analyticity for the 2D Kuramoto-Sivashinsky equation

TL;DR

This paper investigates the long-term behavior and analyticity of solutions to the 2D Kuramoto-Sivashinsky equation on a torus, establishing conditions for global existence and growth of analyticity radius.

Contribution

It provides the first analytical results on global existence and analyticity growth for the 2D Kuramoto-Sivashinsky equation on a torus, depending on domain size and linear modes.

Findings

Small solutions exist globally if no linearly growing modes are present.

The radius of analyticity grows linearly in time for solutions without growing modes.

Estimates are provided for the analyticity radius in the presence of finitely many growing modes.

Abstract

There is little analytical theory for the behavior of solutions of the Kuramoto-Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus -- in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case (i.e., in the presence of a finite number of growing modes), we make estimates for how the radius of analyticity of solutions changes in time.