Remarks on nonlinear dispersive PDEs with rough dispersion management

In this work, we study the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. Under minimal assumptions on the occupation measure of this coefficient, we show that for the large class of semilinear dispersive PDEs whose well-posedness theory relies on linear estimates of Strichartz or local smoothing type, one has the same well-posedness theory with or without the modulation. We also show a regularization by noise type of phenomenon for rough modulations, namely, large data global well-posedness in the focusing mass-critical case for the modulated equation. Under rougher assumptions on the modulation, we show that one can also transfer the well-posedness theory based on multilinear Fourier analysis from the original dispersive PDE to the modulated one. In the case of the NLS equation on $\mathbb{R}^d$ and $\mathbb{T}^d$, this covers all the sub-critical and critical regularities, thus completing and extending the various results currently available in the literature. At last, in the case of the periodic Wick-ordered cubic NLS, we show an even stronger form of regularization by noise, namely well-posedness in the critical Fourier-Lebesgue space for rough modulations.