Analysis in Mixed Norm Spaces and Application to Nonlinear Partial Differential Equations

The goal of this dissertation is to study partial differential equations in mixed norm and weighted mixed norm spaces. This entails the development of certain analytical tools, such as estimates of the heat semigroup in the weighted mixed norm space. Using these results we extend Masuda's uniqueness theorem for the Navier-Stokes equations to general mixed norm spaces, show the existence of mild solutions to the Keller-Segel model in weighted mixed norm spaces, and uniqueness of solutions the Keller-Segel model in a special case.