Large field problem in coercive singular PDEs

TL;DR

This paper develops a framework for deriving a priori estimates for singular PDEs with irregular noise, using rough path theory and regularity structures, focusing on local coercivity and scaling arguments.

Contribution

It introduces an abstract estimate that extends local coercivity to global estimates via scaling, applicable to a broad class of singular PDEs.

Findings

Established local a priori estimates independent of boundary conditions.

Derived a method to extend local coercivity to global estimates using scaling.

Reduced complex singular PDE analysis to the case of small irregular noise.

Abstract

We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi \] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $\xi$ is an irregular distribution such that the equation is subcritical. The differential operator $\mathcal L$ is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on $P$ is that solutions with $\xi=0$ exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when $\xi$ is small.