We study a class of degenerate convection diffusion equations with a
fractional nonlinear diffusion term. These equations are natural
generalizations of anomalous diffusion equations, fractional conservations
laws, local convection diffusion equations, and some fractional Porous medium
equations. In this paper we define weak entropy solutions for this class of
equations and prove well-posedness under weak regularity assumptions on the
solutions, e.g. uniqueness is obtained in the class of bounded integrable
functions. Then we introduce a monotone conservative numerical scheme and prove
convergence toward an Entropy solution in the class of bounded integrable
functions of bounded variation. We then extend the well-posedness results to
non-local terms based on general L\'evy type operators, and establish some
connections to fully non-linear HJB equations. Finally, we present some
numerical experiments to give the reader an idea about the qualitative behavior
of solutions of these equations
