On local well-posedness of nonlinear dispersive equations with partially regular data

TL;DR

This paper develops a new local well-posedness theory for nonlinear dispersive equations, showing that initial data with partial regularity in some variables suffices, using refined Strichartz estimates with variable-specific norms.

Contribution

It introduces a novel approach allowing partial regularity in initial data for dispersive equations, expanding the classical Sobolev space framework.

Findings

Partial regularity suffices for well-posedness

Refined Strichartz estimates with variable-specific norms

Broader class of initial data for dispersive equations

Abstract

We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.