Well-posedness of a Class of Non-homogeneous Boundary Value Problems of the Korteweg-de Vries Equation on a Finite Domain

TL;DR

This paper establishes local well-posedness for a class of initial-boundary value problems of the Korteweg-de Vries equation on a finite domain in Sobolev spaces, addressing an open question in the field.

Contribution

It proves local well-posedness of the Korteweg-de Vries equation with non-homogeneous boundary conditions on a finite domain for Sobolev spaces with s > -3/4, resolving an open problem.

Findings

Well-posedness established for s > -3/4

Addresses open question by Colin and Ghidalia

Provides foundational results for boundary value problems

Abstract

In this paper, we study a class of initial-boundary value problems for the Korteweg-de Vries equation posed on a bounded domain $(0,L)$. We show that the initial-boundary value problem is locally well-posed in the classical Sobolev space $H^s(0,L)$ for $s>-\frac34$, which provides a positive answer to one of the open questions of Colin and Ghidalia .