Quasilinear generalized parabolic Anderson model equation

TL;DR

This paper proves local well-posedness for a 2D quasilinear generalized parabolic Anderson model using a transformation and paracontrolled calculus, simplifying the analysis of a singular stochastic PDE.

Contribution

It introduces a transformation approach to handle quasilinear singular SPDEs within paracontrolled calculus, advancing the understanding of such equations.

Findings

Established local well-posedness for the 2D quasilinear generalized parabolic Anderson model.

Simplified the analysis by transforming the quasilinear problem into a semilinear form.

Demonstrated the effectiveness of paracontrolled calculus in handling singular SPDEs.

Abstract

We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)\Delta u = g(u)\xi $$ The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting of paracontrolled calculus.