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We investigate existence, uniqueness and regularity for solutions of roughparabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla+ \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a conceptof "differential rough driver", which comes with a counterpart of the usualcontrolled paths spaces in rough paths theory, built on the Sobolev spaces$W^{k,p}.$ We also define a natural notion of geometricity in this context, andshow how it relates to a product formula for controlled paths. In the case oftransport noise (i.e.\ when $Y=0$), we use this framework to prove an It\^oFormula (in the sense of a chain rule) for Nemytskii operations of the form$u\mapsto F(u),$ where $F$ is $C^2$ and vanishes at the origin. Our method isbased on energy estimates, and a generalization of the Moser Iteration argumentto prove boundedness of a dense class of solutions of parabolic problems asabove. In particular, we avoid the use of flow transformations and workdirectly at the level of the original equation. We also show the correspondingchain rule for $F(u)=|u|^p$ with $p\geq 2,$ but also when $Y\neq 0$ and $p\geq4.$ As an application of these results, we prove existence and uniqueness of asuitable class of $L^p$-solutions of parabolic equations with multiplicativenoise. Another related development is the homogeneous Dirichlet boundaryproblem on a smooth domain, for which a weak maximum principle is shown underappropriate assumptions on the coefficients.

An It\^o Formula for rough partial differential equations and some applications