Adapting the convex integration technique introduced in \cite{dLSz5} and subsequently developed in \cite{IsettVicol,IsettOh16,Buckmaster2013transporting}, we construct H\"{o}lder continuous weak solutions to the three dimensional Prandtl system and some other models with vertical viscosity.

We establish two nonlinear compactness theorems in Lp(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) [1]. The first theorem sharpens a result by Maitre (2003) [10] and is important in the study of doubly nonlinear elliptic–parabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskii˘ (1965) [5], which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type bead–spring chains model).

We prove that weak solutions to the obstacle problem for the porous medium equation are locally Hölder continuous, provided that the obstacle is Hölder continuous.

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