A priori bounds for stochastic porous media equations via regularity structures

We prove a priori bounds for solutions of singular stochastic porous media equations with multiplicative noise in their natural $L^1$-based regularity class. We consider the first singular regime, i.e.~noise of space-time regularity $\alpha-2$ for $\alpha\in(2/3,1)$, and prove modelledness of the solution in the sense of regularity structures with respect to the solution of the corresponding linear stochastic heat equation. The proof relies on the kinetic formulation of the equation and a novel renormalized energy inequality. A careful analysis allows to balance the degeneracy of the diffusion coefficient against sufficiently strong damping of the multiplicative noise for small values of the solution.