Fractional Evolution Equations and Applications

In recent years increasing interests and considerable researches have been given to the fractional differential equations both in time and space variables. These are due to the applications of the fractional differential operators to problems in a wide areas of physics and engineering science and a rapid development of the corresponding theory. Motivating examples include the so-called continuous time random walk process and the Levy process model for the mathematical finance. Basset integral is appearing in the equation of motion of a particle moving through a fluid. A fractional diffusion equation is derived as a homogenization of heterogeneous groundwater flow. In this lecture we develop solution methods based on the linear and nonlinear semigroup theory and apply it to solve the corresponding inverse and optimal control problems. The theory is applied to concrete examples including fractional diffusion equation, Navier-Stokes equations and conservation laws. For the linear case we develop the operator theoretic representation of solutions and the sectorial property of the fractional operator in time is used to establish the regularity and asymptotic of the solutions. The property and stability of the solutions as well as numerical integration methods are discussed. The lecture also covers the basic theory and application of the so-called Crandall-Ligget theory and the DS-approximation theory developed by Kobayashi-Kobayashi-Oharu for evolution operator and the semi-linear theory based on the sectorial estimates of the fractional equation.