NONLOCAL FILTRATION EQUATIONS AND FRACTIONAL CURVATURE FLOWS
Johns Hopkins University
In this thesis, we record the joint work done by the author and Pak Tung Ho concerning the fractional Nirenberg problem with a symmetric assumption, which is addressed through fractional curvature flows on the Riemannian sphere Sn. Furthermore, we present the author’s work on nonlocal filtration equations related to the Heisenberg group Hn. The classical Nirenberg’s problem is to find a conformal metric of an n-dimensional Riemannian manifold such that its scalar curvature is a given function f. A geometric flow has been introduced to study Nirenberg’s problem by Struwe for n = 2 and has been generalized to n ≥ 3 by Chen and Xu on the sphere Sn. Using the scalar curvature flow, Leung and Zhou proved an existence result for prescribing scalar curvature when the given function f possesses certain reflection or rotation symmetry on the Riemannian sphere Sn in their paper. This led naturally to the study of the prescribing fractional order curvature problem with the same symmetric hypothesis on Sn, which has been proved by the author and Pak Tung Ho. Motivated by the extensive investigations of nonlocal filtration equations, utilizing the integral operators instead of the Laplacian operator on the Euclidean space Rn, we work on the same type of equations on the Heisenberg group Hn. We established the existence, uniqueness and large-time behavior of the corresponding solutions. Furthermore, an interesting result stated that a uniform Hölder regularity, as the function value tends to zero, holds for the porous medium type of equations, which can also be adapted to obtain uniform Hölder regularity on Rn.
Nonlocal Filtration Equations, Fractional Curvature Flows, Heisenberg Group