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

• Published in: arXiv.org
• Main Authors: Capistrano-Filho, R. A, Sun, Shu-Ming, Zhang, Bing-Yu
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 23.03.2017
• Subjects: Boundary conditions; Boundary value problems; Mathematics - Analysis of PDEs; Well posed problems
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1703.08154

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.