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

TL;DR

This paper develops a comprehensive well-posedness theory for the nonlinear KdV equation on star-graph structures, introducing the concept of s-compatibility and extending classical results to complex graph configurations.

Contribution

It introduces the notion of s-compatibility for boundary conditions and establishes sharp global well-posedness results for KdV equations on star graphs, extending classical analysis to complex network structures.

Findings

Established sharp global well-posedness for linear and nonlinear KdV on star graphs

Introduced the concept of s-compatibility for boundary conditions

Extended classical KdV analysis from single domain to complex graphs

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