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

We examine a nonlocal porous medium equation, ${\partial }_{t}u+\mathrm{div}(u\mathrm{\nabla }{L}^{-1}u)=0$, 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 $$Lu(x)=\mathrm{p}.\mathrm{v}.{\int }_{{\mathbb{R}}^d}(u(x)-u(y))\nu (x-y)\mathrm{d}y(x\in {\mathbb{R}}^d),$$ 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 $(-\mathrm{\Delta }{)}^s$, 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.