General boundary value problems of the Korteweg-de Vries equation on a bounded domain

Roberto A. Capistrano-Filho; Shuming Sun; Bing-Yu Zhang

doi:10.3934/mcrf.2018024

Article Contents

2018, Volume 8, Issue 3&4: 583-605. Doi: 10.3934/mcrf.2018024

This issue Previous Article Numerical study of polynomial feedback laws for a bilinear control problem Next Article Analysis and optimal control of some quasilinear parabolic equations

General boundary value problems of the Korteweg-de Vries equation on a bounded domain

1.
Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-545, Brazil
2.
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
3.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA
4.
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Bing-Yu Zhang
* Corresponding author: Bing-Yu Zhang

The paper is dedicated to Jiongmin Yong for his 60th birthday.
The authors thank anonymous referees for many helpful comments, corrections and suggestions

Received: March 2017

Revised: October 2017

Published: September 2018

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Abstract

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval

$\begin{equation} u_t+u_x+u_{xxx}+uu_x = 0, ~~~~~ u(x, 0) = φ(x), ~~~~~ 0 < x < L, \ t>0~~~~~(0.1)\end{equation}$

subject to the nonhomogeneous boundary conditions,

$\begin{equation} B_1u = h_1(t), ~~~~~B_2 u = h_2 (t), ~~~~~ B_3 u = h_3 (t) ~~~~~t>0~~~~~(0.2) \end{equation} $

where

$ B_i u = \sum\limits_{j = 0}^2 \left(a_{ij} \partial ^j_x u(0, t) + b_{ij}\partial ^j_x u(L, t)\right), ~~~~~i = 1, 2, 3, $

and $a_{ij}, \ b_{ij}$ ($j = 0, 1, 2$ and $ i = 1, 2, 3$) are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, the IBVP (0.1)-(0.2) is shown to be locally well-posed in the space $H^s (0, L)$ for any $s \ge 0$ with $φ ∈ H^s (0, L)$ and boundary values $h_j$, $j = 1, 2, 3$, belonging to some appropriate spaces with optimal regularity.

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Mathematics Subject Classification: Primary: 35Q53, 35Q35; Secondary: 53C35.

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