# General Boundary Value Problems of the Korteweg-de Vries Equation on a Bounded Domain

## TL;DR

This paper establishes the local well-posedness of the initial boundary value problem for the Korteweg-de Vries equation on a finite interval with general boundary conditions, under certain coefficient assumptions.

## Contribution

It provides the first comprehensive analysis of well-posedness for the KdV equation with general boundary conditions on a bounded domain.

## Key findings

- Proves local well-posedness in H^s for s ≥ 0.
- Handles nonhomogeneous boundary conditions with optimal regularity.
- Provides conditions on boundary coefficients for well-posedness.

## Abstract

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 0<x<L, \ t>0 \qquad (1) \end{equation} subject to the nonhomogeneous boundary conditions, \begin{equation} B_1u=h_1(t), \qquad B_2 u= h_2 (t), \qquad B_3 u= h_3 (t) \qquad t>0 \qquad (2) \end{equation} where \[ B_i u =\sum _{j=0}^2 \left(a_{ij} \partial ^j_x u(0,t) + b_{ij} \partial ^j_x u(L,t)\right), \qquad i=1,2,3,\] and $a_{ij}, \ b_{ij}$ $ (j,i=0, 1,2,3)$ are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, $ j,i=0, 1,2,3$, the IBVPs (1)-(2) is shown to be locally well-posed in the space $H^s (0,L)$ for any $s\geq 0$ with $\phi \in H^s (0,L)$ and boundary values $h_j, j=1,2,3$ belonging to some appropriate spaces with optimal regularity.