Non-linear parabolic PDEs with rough coefficients and critical data: existence, uniqueness and regularity of weak solutions

TL;DR

This paper establishes existence, uniqueness, and regularity of weak solutions for non-linear parabolic PDEs with rough coefficients and initial data in critical Besov spaces, using a novel analytical framework.

Contribution

It introduces a new theory of hypercontractive singular integral operators on weighted Z-spaces and applies it to prove well-posedness without smallness assumptions.

Findings

Proves well-posedness for rough reaction-diffusion equations.

Develops a self-improving property for super-linear reverse Hölder inequalities.

Introduces a novel approach to handle rough coefficients in non-linear PDEs.

Abstract

This article investigates the well-posedness of weak solutions to non-linear parabolic PDEs driven by rough coefficients with rough initial data in critical homogeneous Besov spaces. Well-posedness is understood in the sense of existence and uniqueness of maximal weak solutions in suitable weighted $Z$-spaces in the absence of smallness conditions. We showcase our theory with an application to rough reaction--diffusion equations. Subsequent articles will treat further classes of equations, including equations of Burgers-type and quasi-linear problems, using the same approach. Our toolkit includes a novel theory of hypercontractive singular integral operators (SIOs) on weighted $Z$-spaces and a self-improving property for super-linear reverse H\"older inequalities.