Abstract: This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in a Hilbert or Banach space. The outline of the chapter is as follows. Section 10.1 deals with the theory of maximal monotone operators in Hilbert spaces. Section 10.2 introduces time discretizations and the concepts of mild solutions and the accretive operators in Banach spaces. Section 10.3 applies the theory of accretive operators to the filtration equation. The chapter ends with some new ideas of mass transportation and gradient flows and a review of different extensions to more general equations where new concepts of solution are needed.

Keywords: PME, GPME, maximal monotone operators, Hilbert spaces, time discretizations, Banach spaces, mass transfer,