Sharp local existence and nonlinear smoothing for dispersive equations with higher-order nonlinearities

TL;DR

This paper develops a rigorous theory showing that higher-order nonlinearities and dimensions lead to sharper local well-posedness results for dispersive equations, with explicit regularity gain bounds and applications to key PDEs.

Contribution

It establishes a sharp local well-posedness framework for nonlinear dispersive equations with higher-order nonlinearities and dimensions, confirming a conjecture on regularity gain.

Findings

Proves sharp local well-posedness in Sobolev spaces for equations with higher-order nonlinearities.

Provides explicit bounds on the regularity gain between linear and nonlinear solutions.

Applies the theory to generalized KdV, Zakharov-Kuznetsov, and nonlinear Schrödinger equations.

Abstract

We consider a general nonlinear dispersive equation with monomial nonlinearity of order $k$ over $\mathbb{R}^d$. We construct a rigorous theory which states that higher-order nonlinearities and higher dimensions induce sharper local well-posedness theories. More precisely, assuming that a certain positive multiplier estimate holds at order $k_0$ and in dimension $d_0$, we prove a sharp local well-posedness result in $H^s(\mathbb{R}^d)$ for any $k\ge k_0$ and $d\ge d_0$. Moreover, we give an explicit bound on the gain of regularity observed in the difference between the linear and nonlinear solutions, confirming the conjecture made in [CorreiaOliveiraSilva24] (doi.org/10.1137/23M156923X). The result is then applied to generalized Korteweg-de Vries, Zakharov-Kuznetsov and nonlinear Schr\"odinger equations.