On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation

TL;DR

This paper introduces a novel reduced 2D Kuramoto-Sivashinsky model that is globally well-posed, providing new insights into the mathematical properties of these complex equations and their dynamics.

Contribution

The paper proposes a new reduced 2D KSE model that is globally well-posed, unlike the original, and explores its dynamics both analytically and computationally.

Findings

The reduced model is globally well-posed.

Solutions show qualitative similarities to the original 2D KSE.

The model exhibits 4th-order spatial derivatives and low-mode instability.

Abstract

The Kuramoto-Sivashinsky equations (KSE) arise in many diverse scientific areas, and are of much mathematical interest due in part to their chaotic behavior, and their similarity to the Navier-Stokes equations. However, very little is known about their global well-posedness in the 2D case. Moreover, regularizations of the system (e.g., adding large diffusion, etc.) do not seem to help, due to the lack of any control over the $L^2$ norm. In this work, we propose a new "reduced" 2D model that modifies only the linear part of (the vector form of) the 2D KSE in only one component. This new model shares much in common with the 2D KSE: it is 4th-order in space, it has an identical nonlinearity which does not vanish in energy estimates, it has low-mode instability, and it lacks a maximum principle. However, we prove that our reduced model is globally well-posed. We also examine its dynamics computationally. Moreover, while its solutions do not appear to be close approximations of solutions to the KSE, the solutions do seem to hold many qualitative similarities with those of the KSE. We examine these properties via computational simulations comparing solutions of the new model to solutions of the 2D KSE.