Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions of generalized porous medium equations of the form $$\partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$},$$ where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, L\'evy operators. Our quantitative estimates take the form of precise $L^1$--$L^\infty$-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of $-\mathfrak{L}$ and $I-\mathfrak{L}$. In the linear case $m=1$, it is well-known that the $L^1$--$L^\infty$-smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting $m>1$. First, we can show that operators for which ultracontractivity holds, also provide $L^1$--$L^\infty$-smoothing effects in the nonlinear case. The converse implication is not true in general. A counterexample is given by $0$-order L\'evy operators like $-\mathfrak{L}=I-J\ast$. They do not regularize when $m=1$, but we show that surprisingly enough they do so when $m>1$, due to the convex nonlinearity. This reveals a striking property of nonlinear equations: the nonlinearity allows for better regularizing properties, almost independently of the linear operator. Finally, we show that smoothing effects, both linear and nonlinear, imply families of inequalities of Gagliardo-Nirenberg-Sobolev type, and we explore equivalences both in the linear and nonlinear settings through the application of the Moser iteration.