Global Regular Solutions for the Multi-dimensional Kuramoto-Sivashinsky Equation posed on Smooth Domains

TL;DR

This paper proves the existence, uniqueness, and exponential decay of global regular solutions for the multi-dimensional Kuramoto-Sivashinsky equation on smooth bounded domains, extending understanding of this complex PDE in higher dimensions.

Contribution

It establishes the first comprehensive results on global regular solutions for the n-dimensional Kuramoto-Sivashinsky equation on smooth domains.

Findings

Existence of global regular solutions for 2 ≤ n ≤ 7

Uniqueness of solutions in the considered setting

Solutions exhibit exponential decay over time

Abstract

Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of global regular solutions as well as their exponential decay have been considered.