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

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 $\mathrm{Z}$-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 $\mathrm{Z}$-spaces and a self-improving property for super-linear reverse Hölder inequalities.