Dragoş Manea
Simion Stoilow Mathematical Institute of the Romanian Academy, Bucharest, Romania

We study a non-local, non-linear convection-diffusion equation on the hyperbolic space $\mathbb{H}^N$, governed by two kernels, one for each of the diffusion and convection parts. One main novelty is the construction of the non-symmetric convection kernel defined on the tangent bundle and invariant under the geodesic flow. Next, we consider the relaxation of this model to a local problem, as the kernels get concentrated near the origin of each tangent space. Under some regularity and integrability conditions, we prove that the solution of the concentrated non-local problem converges to that of the local convection-diffusion equation. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brézis-Mironescu.