A nonhomogeneous boundary value problem for the Kuramoto-Sivashinsky equation in a quarter plane

TL;DR

This paper establishes local and global well-posedness results for the one-dimensional Kuramoto-Sivashinsky equation with nonhomogeneous boundary conditions in a quarter plane, expanding understanding of its boundary value problems.

Contribution

It provides the first analysis of the Kuramoto-Sivashinsky equation with nonhomogeneous boundary conditions, demonstrating well-posedness in Sobolev spaces.

Findings

Well-posedness in Sobolev space for s > -2

Local and global solutions established

Analysis of boundary integral operator used

Abstract

We study the initial boundary value problem for one-dimensional Kuramoto-Sivashinsky equation with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results on the Cauchy problem, we obtain both the local well-posedness and the global well-posedness for the nonhomogeneous initial boundary value problem. It is shown that the Kuramoto-Sivashinsky equation is well-posed in Sobolev space $C([0,T]; H^s (R^+)) \bigcap L^2(0,T; H^{s+2}(R^+))$ for $s>-2$.