Global well-posedness of the dynamical sine-Gordon model up to $6\pi$

TL;DR

This paper establishes the global well-posedness of the dynamical sine-Gordon model for parameters below 6π by introducing a deterministic resonant equation and controlling stochastic fluctuations with uniform estimates.

Contribution

It introduces a novel resonant equation that deterministically captures the solution's size, enabling analysis up to the third threshold of the model.

Findings

Proves global well-posedness for β² < 6π

Develops a deterministic resonant equation approach

Controls stochastic fluctuations with uniform estimates

Abstract

We prove the global well-posedness of the dynamical sine-Gordon model up to the third threshold, i.e., for parameters $\beta^2 < 6\pi$. The key novelty in our approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects.