Nonlinear PDEs with modulated dispersion II: Korteweg--de Vries equation

TL;DR

This paper investigates how irregular time-dependent modulations in dispersive PDEs like KdV, BO, and ILW can lead to regularization effects, improving well-posedness and smoothing properties even in regimes where the unmodulated equations are ill-posed.

Contribution

It demonstrates that irregular modulations can induce regularization by noise, establishing well-posedness and smoothing effects for modulated dispersive equations.

Findings

Modulated KdV is locally well-posed in all Sobolev spaces with irregular modulation.

Global well-posedness achieved for modulated KdV in negative Sobolev spaces.

Irregular modulations transform certain dispersive equations into semilinear forms.

Abstract

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. As primary models, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW), imposing certain irregularity conditions on the time non-homogeneous modulation. In this work, we establish phenomena called regularization by noise in three-folds: (i) When the modulation is sufficiently irregular, we show that the modulated KdV on both the circle and the real line is locally well-posed in the regime where the (unmodulated) KdV equation is known to be ill-posed. In particular, given any $s \in \mathbb R$, we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in $H^s(\mathbb T)$. Moreover, by adapting the $I$-method to the current modulated setting, we prove global well-posedness of the modulated KdV in negative Sobolev spaces. (ii) It is known that certain (semilinear) dispersive equations such as BO and ILW exhibit quasilinear nature. We show that sufficiently irregular modulations make the modulated versions of these equations semilinear by establishing their local well-posedness by a contraction argument, providing local Lipschitz continuity of the solution map. (iii) We also prove nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations. As applications of our approach, we also include further examples.