Well-posedness for a Class of Pseudodifferential Diffusion Equations on the Torus

In this work we establish the well-posedness of the Cauchy problem for a class of pseudodifferential diffusion equations on the torus. The class considered includes fractional diffusion equations and as a special case we consider fractional diffusion equations with drift. By applying toroidal pseudodifferential calculus we establish regularity estimates, existence and uniqueness with respect to the usual Sobolev spaces on the torus.