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The Korteweg-de Vries Equation on general star graphs

Márcio Cavalcante, José Marques

Abstract

In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.

The Korteweg-de Vries Equation on general star graphs

Abstract

In this paper, we establish local well-posedness for the Cauchy problem associated with the Korteweg-de Vries (KdV) equation on a general metric star graph. The graph comprises m + k semi-infinite edges: k negative half-lines and m positive half-lines, all joined at a common vertex. The choice of boundary conditions is compatible with the conditions determined by the semigroup theory. The crucial point in this work is to obtain the integral formula using the forcing operator method and the Fourier restriction method of Bourgain. This work extends the results obtained by Cavalcante for the specific case of the Y junction to a more general class of star graphs.

Paper Structure

This paper contains 17 sections, 13 theorems, 116 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Airy Equation on star graphs
  3. Formulation of the problem
  4. Statement of the main result
  5. Notations and function spaces
  6. Function Spaces
  7. Riemann-Liouville fractional integral
  8. Estimates associated of some operators
  9. Free popagator
  10. The Duhamel boundary forcing operator associated to the linear KdV equation
  11. The Duhamel Boundary Forcing Operator Classes associated to linear KdV equation
  12. The Duhamel Inhomogeneous Solution Operator
  13. Start of the Proof of Theorem \ref{['grandeteorema']}
  14. Truncated integral operator
  15. End of the proof: Obtaining a fixed point of $\Lambda$
  16. ...and 2 more sections

Key Result

Proposition 1.1

(Corolary 3.20 and Theorem 3.23 Noja or Proposition 3.2 solitons2). Let $Y \subset \mathbb K^{k} \oplus \mathbb K^m$ be a subspace and $B: \mathbb K^{m} \to \mathbb K^k$ linear. Then:

Figures (5)

  • Figure 1: Star graph with five negative half-lines (the blue ones) and five positive half-lines (the red ones)
  • Figure 2: A star graph with $m+k$ edges
  • Figure 3: $\mathcal{Y}$-junction
  • Figure 4: A star graph with five $(-\infty,0)$ edges (the blue ones) and eight $(0,\infty)$ edges (the red ones)
  • Figure 5: A balanced star graph with 20 edges

Theorems & Definitions (22)

  • Proposition 1.1
  • Theorem 2.1
  • Remark 2.1: About the range of s
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.1
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 4.1
  • ...and 12 more