Fractional Partial Differential Equations

Contents
Preface
About the Author
Introduction
1. Preliminaries
	1.1 Fractional Calculus
		1.1.1 Definitions
		1.1.2 Properties
	1.2 Some Results from Analysis
		1.2.1 Mittag-Leffler Function
		1.2.2 Laplace and Fourier Transforms
	1.3 Semigroups
		1.3.1 C0-Semigroup
		1.3.2 Analytic Semigroup
2. Fractional Navier-Stokes Equations
	2.1 Introduction
	2.2 Cauchy Problem in Rn
		2.2.1 Definitions and Lemmas
		2.2.2 Global/Local Existence
		2.2.3 Local Existence
		2.2.4 Regularity
	2.3 Existence and Hölder Continuity
		2.3.1 Notations and Lemmas
		2.3.2 Existence and Uniqueness
		2.3.3 Hölder Continuous
	2.4 Approximations of Solutions
		2.4.1 Preliminaries
		2.4.2 Existence and Convergence of Approximate Solutions
		2.4.3 Faedo-Galerkin Approximation
	2.5 Weak Solution and Optimal Control
		2.5.1 Notations and Definitions
		2.5.2 Existence and Uniqueness
		2.5.3 Optimal Control
	2.6 Energy Methods
		2.6.1 Notations and Lemmas
		2.6.2 Local Existence
		2.6.3 Global Existence
	2.7 Cauchy Problem in Sobolev Space
		2.7.1 Definitions and Lemmas
		2.7.2 Local Existence I
		2.7.3 Local Existence II
	2.8 Notes and Remarks
3. Fractional Rayleigh-Stokes Equations
	3.1 Initial Value Problem
		3.1.1 Introduction
		3.1.2 Existence
	3.2 Well-posedness on RN
		3.2.1 Introduction
		3.2.2 Preliminaries
		3.2.3 Well-posedness
		3.2.4 Integrability of Solution
	3.3 Well-posedness and Blow-up
		3.3.1 Introduction
		3.3.2 Preliminaries
		3.3.3 Existence of Mild Solutions
		3.3.4 Continuation and Blow-up Alternative
	3.4 Weak Solutions
		3.4.1 Introduction
		3.4.2 Preliminaries
		3.4.3 Existence, Uniqueness and Regularity
	3.5 Global Solutions in Besov-Morrey Spaces
		3.5.1 Introduction
		3.5.2 Notations and Function Spaces
		3.5.3 Properties of Solution Operators
		3.5.4 Well-posedness of Global Solutions
	3.6 Final Value Problem
		3.6.1 Introduction
		3.6.2 Preliminaries
		3.6.3 Existence of Solutions
		3.6.4 Quasi-boundary Value Method
	3.7 Notes and Remarks
4. Fractional Fokker-Planck Equations
	4.1 Operator Method
		4.1.1 Introduction
		4.1.2 Preliminaries
		4.1.3 The Solution Operators
		4.1.4 Mild Solutions
		4.1.5 Classical Solutions in Case of α ∈ (1/2; 1)
		4.1.6 Classical Solutions in Case of α ∈ (0; 1)
	4.2 Galerkin Approximation Method
		4.2.1 Introduction
		4.2.2 Notation and Definitions
		4.2.3 Technical Preliminaries
		4.2.4 Galerkin Approximation of Solution
		4.2.5 Existence and Uniqueness of Mild Solution
		4.2.6 Existence and Uniqueness of Classical Solution
			4.2.6.1 A priori estimates
			4.2.6.2 The classical solution
		4.2.7 Regularity of Classical Solution
	4.3 Notes and Remarks
5. Fractional Schrödinger Equations
	5.1 Linear Equations on Hilbert Space
		5.1.1 Introduction
		5.1.2 Preliminaries
		5.1.3 Existence of Dynamics
		5.1.4 Properties of the Solution Operator
		5.1.5 An Example
	5.2 Nonlinear Schrödinger Equation
		5.2.1 Introduction
		5.2.2 Solution Operators
		5.2.3 Existence
	5.3 Notes and Remarks
Bibliography
Index