This paper is concerned with quasi-linear parabolic equations driven by an additive forcing ξ ∈ C α−2 , in the full subcritical regime α ∈ (0, 1). We are inspired by Hairer's regularity structures, however we work with a more parsimonious model indexed by multi-indices rather than trees. This allows us to capture additional symmetries which play a crucial role in our analysis. Assuming bounds on this model, which is modified in agreement with the concept of algebraic renormalization, we prove local a priori estimates on solutions to the quasi-linear equations modified by the corresponding counter terms.1 A number of aspects of this paper also work for arbitrary α > 0, but the authors did not identify the renormalized PDE in the full sub-critical regime.2 Extending the linear theory developed in [23] to arbitrary α > 0 remains an interesting and challenging open problem.

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A broad class of possibly non-unique generalized kinetic solutions to hyperbolicparabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev spaces. The optimality of these results is shown by comparison to the non-local Barenblatt solution. The regularity results are used in order to obtain existence of generalized kinetic solutions.

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