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Dynamics of the Korteweg-de Vries equation on a balanced metric graph

Jaime Angulo Pava, Márcio Cavalcante

Abstract

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of $δ$-type at the {unique} graph-vertex. Also, we extend the linear instability result of Angulo and Cavalcante (2021) to one of nonlinear instability. For the proof of local well posedness theory the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class $C^2$.

Dynamics of the Korteweg-de Vries equation on a balanced metric graph

Abstract

In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of -type at the {unique} graph-vertex. Also, we extend the linear instability result of Angulo and Cavalcante (2021) to one of nonlinear instability. For the proof of local well posedness theory the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class .
Paper Structure (12 sections, 16 theorems, 174 equations, 4 figures)

This paper contains 12 sections, 16 theorems, 174 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sobolev spaces on $\mathbb R^{\pm}$ and $\mathcal{G}$
  4. Bourgain spaces
  5. Linear Group
  6. Duhamel operator
  7. A review concern potentials for a linearized KdV equation on the positive and negative half-lines.
  8. An integral equation that solves the problem on a balanced graph
  9. Energy estimates for the boundary functions $f$, $g$ and $h$
  10. Proof of Theorem \ref{['theorem1']}
  11. Proof of Theorem \ref{['dependence']}
  12. Nonlinear instability

Key Result

Theorem 1.2

Let $s\geqq 1$, $Z\neq 0$, and $\bold u_0 \in H^s(\mathcal{G})\cap \mathcal{N}_{0, Z}(s)$. Then there exists $T=T(\|\bold u_0\|_s)$ and a unique solution $\bold u$ of the IVP kdv3-Zcondition in the class $X_s(T)$ such that $\bold u(0)=\bold u_0$. Furthermore, for any $T_0\in (0, T)$ there exists a is Lipschitz.

Figures (4)

  • Figure 1: A $\mathcal{Y}$-junction graph
  • Figure 2: A balanced star graph with $6$ edges
  • Figure 3: $U_{Z}$, $Z>0$, bump profiles
  • Figure 4: $U_{Z}$, $Z<0$, tail profiles

Theorems & Definitions (40)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 30 more