L1-Based Approach to Existence Theory for Stochastic Porous Media Equations
The existence theory developed in the previous chapter was based on energy estimates in the space H−1 obtained via Itô’s formula in approximating equations. This energetic approach leads to sharp existence results, but requires polynomial growth assumptions or strong coercivity for the nonlinear function β. The case of general maximal monotone functions β of arbitrary growth and in particular with exponential growth was beyond the limit of the previous theory. Here we develop a different approach based on sharp L1-estimates for the corresponding approximating equations which allows to treat these general situations.