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A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs

Roberto de A. Capistrano Filho, Hugo Parada, Jandeilson Santos da Silva

TL;DR

The paper tackles the initial-boundary-value problem for the KdV equation on a star-shaped graph, introducing s-compatibility to harmonize initial data, boundary traces, and vertex coupling. It combines linear semigroup methods, a lifting technique, and a fixed-point approach to prove sharp local well-posedness in Sobolev spaces H^s for s in [0,3], and then derives global well-posedness using energy estimates and interpolation. The results extend the classical single-edge theory to networks of N edges, providing the first comprehensive well-posedness framework for KdV with coupled boundary conditions on graphs and setting the stage for future control and optimization analysis on networks.

Abstract

This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of $s$-compatibility, which generalizes classical compatibility conditions to star-shaped and more complex graph configurations, inspired by the works of Bona, Sun, and Zhang [14]. By combining analytical techniques with a fixed-point argument, we establish sharp global well-posedness for both the linear and nonlinear problems at the $H^s$ level. In this setting, our results extend the classical analysis for a single KdV equation [14] to star-shaped graphs composed of $N$ equations. These results provide the first comprehensive well-posedness theory for KdV equations with coupled boundary conditions on graphs. Although control issues are not treated in this article, the analytic results obtained here address several open problems, which will be addressed in a forthcoming

A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs

TL;DR

The paper tackles the initial-boundary-value problem for the KdV equation on a star-shaped graph, introducing s-compatibility to harmonize initial data, boundary traces, and vertex coupling. It combines linear semigroup methods, a lifting technique, and a fixed-point approach to prove sharp local well-posedness in Sobolev spaces H^s for s in [0,3], and then derives global well-posedness using energy estimates and interpolation. The results extend the classical single-edge theory to networks of N edges, providing the first comprehensive well-posedness framework for KdV with coupled boundary conditions on graphs and setting the stage for future control and optimization analysis on networks.

Abstract

This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of -compatibility, which generalizes classical compatibility conditions to star-shaped and more complex graph configurations, inspired by the works of Bona, Sun, and Zhang [14]. By combining analytical techniques with a fixed-point argument, we establish sharp global well-posedness for both the linear and nonlinear problems at the level. In this setting, our results extend the classical analysis for a single KdV equation [14] to star-shaped graphs composed of equations. These results provide the first comprehensive well-posedness theory for KdV equations with coupled boundary conditions on graphs. Although control issues are not treated in this article, the analytic results obtained here address several open problems, which will be addressed in a forthcoming

Paper Structure

This paper contains 19 sections, 23 theorems, 281 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dispersive systems on graph structure
  3. Setting of problem and functional framework
  4. Main results and structure of the article
  5. s-compatibility conditions
  6. Traces for $\phi_{0,j},\ \phi_{0,j}'$ and $\phi_{0,j}",\ j=1,...,N$, $s\in [0,3]$
  7. Traces for $p_j,\ g_0$ and $g_j$, $j=1,...,N$, $s \in [0,3]$
  8. Traces for $\phi_{k,j},\ \phi_{k,j}'$ and $\phi_{k,j}",\ j=1,...,N$, $s>3$
  9. Traces for $p_j^{(k)},\ g_0^{(k)}$ and $g_j^{(k)},\ j=1,...,N$, $s>3$
  10. The linear problem
  11. The case $s=0$
  12. The case $s=3$
  13. The case $s \in (0,3)$
  14. Local well-posedness of the nonlinear problem
  15. Global well-posedness of the nonlinear problem
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP kdV admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where

Figures (1)

  • Figure 1: Network with $N$ edges

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: $s$-compatibility for $0\leq s\leq 3$
  • Definition 2.2: $s$-compatibility for $s>3$
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3: Bona, Sun and Zhang, BSZ 2003
  • Proposition 3.4: Caicedo and Zhang, Caicedo and Zhang 2017
  • Lemma 3.1
  • ...and 33 more