Existence of Weak Solutions for a Family of Fractional Diffusion Models with Aggregation

In this thesis we study a family of fractional diffusion models with aggregation of formal order $(2 + 2s)$ for any $s > 0$. In particular, we present conditions to ensure the existence of weak solutions for the aforementioned problems on every open, bounded, smooth and convex domain $\Omega \subset \mathbb{R}^n$ in dimension $n \geq 1$. To achieve such a result, we interpret the problem as a 2-Wasserstein gradient flow for a functional given by the difference between a fractional Dirichlet energy and a suitable power-law functional. Employing a JKO scheme we construct then a sequence of discrete (in time) solutions and we show that, up to a subsequence, it converges to a weak solution for the considered problem.