A para-differential renormalization technique for nonlinear dispersive equations

TL;DR

This paper introduces a para-differential renormalization method to establish global well-posedness for a class of nonlinear dispersive equations with fractional derivatives, addressing low-high frequency interactions.

Contribution

It presents a novel frequency-dependent renormalization technique to handle low-high frequency interactions in nonlinear dispersive PDEs with fractional derivatives.

Findings

Proves global well-posedness in L^2 for  <  equations.

Develops a new renormalization method for frequency interactions.

Addresses challenges of low-high frequency interactions in dispersive equations.

Abstract

For \alpha \in (1,2) we prove that the initial-value problem \partial_t u+D^\alpha\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t; u(0)=\phi, is globally well-posed in the space of real-valued L^2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.