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This article focuses on parabolic equations with rough diffusion coefficientswhich are ill-posed in the classical sense of distributions due to the presenceof a singular forcing. Inspired by the philosophy of rough paths and regularitystructures, we introduce a notion of modelled distribution which is suitable inthis context. We prove two general tools for reconstruction and integration, aswell as a product lemma which is tailor made for the reconstruction of therough diffusion operator. This yields a partially automated deterministictheory, which we apply to obtain an existence and uniqueness theory forparabolic equations with rough diffusion coefficients and a singular forcing inthe negative parabolic H\"{o}lder space of order larger than $-\frac{3}{2}$.

Parabolic equations with rough coefficients and singular forcing