Regularization by noise for some modulated dispersive PDEs
A talk in the Harmonic and Stochastic Analysis of Dispersive PDEs series by
Tristan Robert from Université de Lorraine
| Abstract: | It is known since pioneering works by Veretennikov and Krylov-Röckner that for ODEs driven
by a rough vector field, uniqueness of the solution can be recovered by adjunction of an additive
noise in the equation. Improvement on the behavior of an ODE or PDE by adding a noise term
is therefore referred to as a regularization by noise phenomenon, and is widely believed to hold
for a large class of ODEs/PDEs and perturbative noises. In this talk, I will consider nonlinear
dispersive PDEs where a deterministic noise is added as a distributional time coefficient in
front of the dispersion. Despite the roughness of the noise term, we will see that any semilinear
dispersive PDE with this noise term is well-posed at least in the same range of regularity
as its noiseless counterpart, as soon as well-posedness relies on linear space-time estimates.
Perhaps more surprisingly, provided that the noise is irregular enough, we will observe several
regularization by noise phenomena: large data global well-posedness for focusing mass-critical
equations, well-posedness at super-critical regularity for strongly non-resonant equations
through improved multilinear estimates, and improvement on the Cauchy theory for Kadomtsev-
Petviashvili equations through short-time multilinear estimates on longer time scales. |