Evolution Equations and Applications

Abstract

This project concerns Evolution Equations in Banach spaces and lies at the interface between Functional Analysis, Dynamical Systems, Modeling Theory and Natural Sciences. Evolution Equations include Partial Di erential Equations (PDEs) with time t as one of the independent variables and arise from many elds of Mathematics as well as Physics, Mechanics and Material Sciences (e.g., Systems of Conservation Law from Dynamics, Navier-Stokes and Eu- ler equations from Fluid Mechanics, Di usion equations from Heat transfer and Natural Sciences, Klein-Gordon and Schrödinger equations from Quantum Mechanics, Cahn-Hilliard equations and Porous media equations from Material Sciences, Evolution equations with memory from Pharma- cokinetics). In this project, we present the fundamental theory of abstract Evolution Equations by using the semigroup approach (which arises naturally from well-posed Cauchy problems: Theorem 2.2.6) and Fixed-point methods. More precisely, rst we review the basic notions of Functional Analysis and Di erential Analysis, secondly we study the theory of semigroups of bounded linear operators, and thirdly we consider Linear Evolution Equations (with emphasis on the di erence between the nite dimensional and the in nite dimensional case, that is due to domain restrictions) and moreover we give existence results (in appropriate sense) for Semilinear Evolution Equations of the form du = Au + f (t, u), dt t>0 ; u(0) = u0 where A is a linear operator that generates a C0 -semigroup and f satis es certain conditions. As applications we start with the evolution equation ∂t u + ∂x u = 0 in R and then after we show the existence of solutions to some Homogeneous Heat Equations, classical Wave equations, nonlinear Heat Equation, and to some nonlinear Wave equation.