Periodic solutions to PDEs with fractional diffusion

Abstract

The aim of this Bachelor's Thesis is the study of periodic solutions to nonlinear equations involving the fractional Laplace operator. Our starting point is the Benjamin-Ono equation in water waves, a completely integrable nonlinear problem in one dimension. To better comprehend this equation, we introduce the notion of fractional operators, their probabilistic interpretation, fractional Sobolev spaces needed in their study, as well as the Hamiltonian structure. The main part of this work concerns a variational problem which leads to the existence of periodic solutions. Finally, we develop a numerical routine in order to strengthen the analytical results previously obtained and also find new properties (previously unknown) about similar equations to Benjamin-Ono which are not completely integrable.