This advanced book focuses on ordinary differential equations (ODEs) in
Banach and more general locally convex spaces, most notably the ODEs on
measures and various function spaces. It briefly discusses the
fundamentals before moving on to the cutting edge research in linear and
nonlinear partial and pseudo-differential equations, general kinetic
equations and fractional evolutions. The level of generality chosen is
suitable for the study of the most important nonlinear equations of
mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov,
Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear
Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions,
mass-action-law kinetics from chemistry. It also covers nonlinear
evolutions arising in evolutionary biology and mean-field games,
optimization theory, epidemics and system biology, in general models of
interacting particles or agents describing splitting and merging,
collisions and breakage, mutations and the preferential-attachment
growth on networks.
The book is intended mainly for upper undergraduate and graduate
students, but is also of use to researchers in differential equations
and their applications. It particularly highlights the interconnections
between various topics revealing where and how a particular result is
used in other chapters or may be used in other contexts, and also
clarifies the links between the languages of pseudo-differential
operators, generalized functions, operator theory, abstract linear
spaces, fractional calculus and path integrals.
