The Sobolev norms and localization on the Fourier side for the solutions to some evolution equations

TL;DR

This paper investigates the regularity, asymptotic behavior, and Fourier localization of solutions to certain one-dimensional evolution equations with rough, time-dependent potentials, revealing higher regularity for generic coupling constants.

Contribution

It introduces new results on the regularity and Fourier localization of solutions to evolution equations with rough potentials, highlighting generic coupling constant effects.

Findings

Solutions exhibit higher regularity for generic coupling constants

Asymptotic behavior of solutions analyzed for large time

Fourier localization properties of solutions studied

Abstract

Some evolution equations with rough time-dependent potential are studied in the case of one-dimensional torus. We show that the solution has higher regularity for the generic values of the coupling constant. The asymptotics for large time and the localization on the Fourier side are studied.