Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

TL;DR

This paper establishes local well-posedness for a class of dispersive equations with mildly regular nonlinearities in both localized and periodic settings, extending previous results to include non-smooth and exponentially growing nonlinearities.

Contribution

It introduces a new composition theorem for Besov spaces and extends well-posedness results to periodic cases with less regular nonlinearities.

Findings

Proves local existence in $H^s$, $s>3/2$, for dispersive equations with mild regularity nonlinearities.

Develops a new composition theorem for Besov spaces applicable to periodic and non-smooth nonlinearities.

Extends previous results on Nemytskii operators to the periodic setting using difference-derivative characterization.

Abstract

For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space $H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on $\mathbb{R}$ to the periodic setting by using the difference-derivative characterization of Besov spaces.