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Strict condition for the $L^{2}$-wellposedness of fifth and sixth order dispersive equations

Taehun Kim

Abstract

We provide a set of conditions that is necessary and sufficient for the $L^{2}$-wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of $L^{2}$-wellposedness.

Strict condition for the $L^{2}$-wellposedness of fifth and sixth order dispersive equations

Abstract

We provide a set of conditions that is necessary and sufficient for the -wellposedness of the Cauchy problem for fifth and sixth order variable-coefficient linear dispersive equations. The necessity of these conditions had been presented by Tarama, and we scrutinized their proof to split the conditions into several parts so that an inductive argument is applicable. This inductive argument simplifies the engineering process of the appropriate pseudodifferential operator needed for the proof of -wellposedness.

Paper Structure

This paper contains 21 sections, 8 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Variable-coefficient linear dispersive equations and known results
  3. Main results
  4. Classes of pseudodifferential operators
  5. Playing with toy case $k=4$
  6. Organization of the paper
  7. Acknowledgments
  8. Illposedness criteria
  9. Proof of Theorem \ref{['20241015thm2']} (Sufficiency in $k=5$)
  10. Proof of Theorem \ref{['20241015thm3']} (Sufficiency in $k=6$)
  11. $L^{2}$-wellposedness for the self-adjoint case
  12. Deferred calculations
  13. Calculations in the proof of Theorem \ref{['20241015thm2']} ($k=5$)
  14. $L \circ \Phi_{1} \equiv \Phi_{1} \circ L_{(1)}$
  15. $L_{(1)} \circ \Phi_{2} \equiv \Phi_{2} \circ L_{(2)}$
  16. ...and 6 more sections

Key Result

Theorem 1

Let $k \ge 3$ and $1 \le m \le k-2$ be given integers. Let $A$ be a differential operator of order $\le k-2$, which is self-adjoint. Let $b_{j} \in \overline{C}^{\infty}(\mathbb{R}) (1 \le j \le m)$ be given functions. Then, for the Cauchy problem to be $L^{2}$-wellposed, it is necessary that

Figures (1)

  • Figure 1: Proof scheme in case $k=4$

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Definition 6
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • ...and 4 more