On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\varphi$ is merely continuous and nondecreasing and $\mathfrak{L}^{\sigma,\mu}$ is the generator of a general symmetric L\'evy process. This means that $\mathfrak{L}^{\sigma,\mu}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{\sigma,\mu}=\Delta-(-\Delta)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for $\mathfrak{L}^{\sigma,\mu}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.