A pathwise regularization by noise phenomenon for the evolutionary $p$-Laplace equation

TL;DR

This paper demonstrates that noise can regularize solutions to a nonlinear evolutionary p-Laplace equation, allowing solutions to exist for singular potentials where traditional methods fail, by leveraging the regularity of the noise's local time.

Contribution

It introduces a novel pathwise regularization by noise approach for the evolutionary p-Laplace equation with singular potentials, expanding the understanding of noise effects in nonlinear PDEs.

Findings

Existence of solutions for singular potentials under noise regularization

Pathwise regularization phenomenon established for nonlinear PDEs

Solution bounds achieved via local time regularity of the noise

Abstract

We study an evolutionary $p$-Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this non-linear problem.