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Local well-posedness for a fourth-order nonlinear dispersive system on the 1D torus

Eiji Onodera

Abstract

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives sufficient conditions on the coefficients of the system for the initial value problem to be time-locally well-posed in Sobolev spaces with high regularity. The proof is based on the energy method combined with the idea of a gauge transformation and the technique of Bona-Smith type parabolic regularization. The sufficient conditions can been found in connection with geometric analysis on a fourth-order geometric dispersive partial differential equation for curve flows on a compact locally Hermitian symmetric space.

Local well-posedness for a fourth-order nonlinear dispersive system on the 1D torus

Abstract

This paper is concerned with the initial value problem for a system of one-dimensional fourth-order dispersive partial differential equations on the torus with nonlinearity involving derivatives up to second order. This paper gives sufficient conditions on the coefficients of the system for the initial value problem to be time-locally well-posed in Sobolev spaces with high regularity. The proof is based on the energy method combined with the idea of a gauge transformation and the technique of Bona-Smith type parabolic regularization. The sufficient conditions can been found in connection with geometric analysis on a fourth-order geometric dispersive partial differential equation for curve flows on a compact locally Hermitian symmetric space.

Paper Structure

This paper contains 9 sections, 13 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Remark on the sufficient condition
  3. Uniform estimate for Bona-Smith regularized solutions in $L^{\infty}H^m$.
  4. Estimate for the difference of approximated solutions
  5. Proof of Theorem \ref{['theorem:lwp']} and Corollary \ref{['cor:cor930']}
  6. Example \ref{['ex:3']}
  7. Relevance to a geometric dispersive PDE
  8. PDE satisfied by $\nabla_x^mu_x$.
  9. Reduction to a system of PDEs.

Key Result

Theorem 1.1

Suppose that $M_a=aI_n$ with $a\neq 0$ where $I_n$ denotes the identity matrix of order $n$ and $F(Q,\partial_xQ,\partial_x^2Q)$ satisfies all the conditions (A1)-(A2) and (B1)-(B6). Let $m$ be an integer with $m \geqslant 4$. Then eq:apde-eq:adata is time-locally well-posed in $H^m(\mathbb{T};\math

Theorems & Definitions (30)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['proposition:GC']}
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['proposition:pF']}
  • ...and 20 more