Gradient flow solutions for porous medium equations with pressure driven by nonlocal symmetric Lévy operators

We examine a nonlocal porous medium equation, $\partial_t u + \operatorname{div}(u \nabla L^{-1} u) = 0$, where the pressure is governed by a general symmetric Lévy operator, $L$, which is widely recognized as the generator of a symmetric Lévy stochastic process. Namely, we consider a nonlocal operator of the form \begin{equation} Lu(x)=\operatorname{p.v.}\int_{\mathbb{R}^d} (u(x)-u(y))\nu(x-y)\mathrm{d} y (x\in\mathbb{R}^d), \end{equation} where $\min(1,|h|^2)\nu\in L^1(\mathbb{R}^d)$ and $\nu(h)=\nu(-h)$. A nonlocal symmetric Lévy operator generalizes the classical fractional Laplace operator $(-\Delta)^s $, with $s \in (0,1) $. We construct weak solutions in the context of the corresponding nonlocal Sobolev space using the Jordan-Kinderlehrer-Otto (JKO) minimizing movement scheme. The absence of interpolation and various tools from classical fractional Sobolev spaces renders our approach more challenging. Furthermore, we shall investigate the nonlocal-to-local convergence of the problem.