Differential Equations on Measures and Functional Spaces

Switzerland : Birkhauser/ Springer, 2019.

Колокольцов В. Н.

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be 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 meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

Ключевые слова: метод параметрикса, Fractional Equations, дробные уравнения, kinetic equations, кинетические уравнения, Nonlinear Schrodinger equation, нелинейное уравнение Шредингера, parametrix method, Дифференциальные уравнения в локально выпуклых пространствах, Markov semigroups, differential equations in locally convex spaces, operator semigroups, полугруппы операторов, Hamilton-Jacobi-Bellman equation, уравнения Гамильтона-Якоби-Беллмана

ПУБЛИКАЦИЯ ПОДГОТОВЛЕНА ПО РЕЗУЛЬТАТАМ ПРОЕКТА: Анализ стохастических явлений, описываемых дробными стохастическими уравнениями, динамикой Ланжевена с упругими соударениями, анализ экстремальных событий (2019)