Fractional Partial Differential Equations

The following sections are included:

• Preface
• About the Author
• Introduction
• Contents

In this chapter, we introduce some notations and basic facts on fractional calculus, Mittag-Leffler functions, the Wright-type function, integral transforms and semigroups, which are needed throughout this monograph.

The following sections are included:

• Introduction
• Cauchy Problem in ℝⁿ
  • Definitions and Lemmas
  • Global/Local Existence
  • Local Existence
  • Regularity
• Existence and Hölder Continuity
  • Notations and Lemmas
  • Existence and Uniqueness
  • Hölder Continuous
• Approximations of Solutions
  • Preliminaries
  • Existence and Convergence of Approximate Solutions
  • Faedo-Galerkin Approximation
• Weak Solution and Optimal Control
  • Notations and Definitions
  • Existence and Uniqueness
  • Optimal Control
• Energy Methods
  • Notations and Lemmas
  • Local Existence
  • Global Existence
• Cauchy Problem in Sobolev Space
  • Definitions and Lemmas
  • Local Existence I
  • Local Existence II
• Notes and Remarks

The Rayleigh-Stokes problem has gained much attention with the further study of non-Newtonain fluids. In Section 3.1, we are interested in discussing the existence of solutions for nonlinear Rayleigh-Stokes problem for a generalized second grade fluid with the Riemann-Liouville fractional derivative. We firstly show that the solution operator of the problem is compact, and continuous in the uniform operator topology. Furthermore, we give an existence result of mild solutions for the nonlinear problem. In Section 3.2, we are devoted to the global/local well-posedness of mild solutions for a semilinear time-fractional Rayleigh-Stokes problem on ℝ^N. We are concerned with, the approaches rely on the Gagliardo-Nirenberg’s inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with the blow-up rate and the integrability in Lebesgue spaces. In Section 3.3, we consider the fractional Rayleigh-Stokes problem with the nonlinearity term satisfies certain critical conditions. The local existence, uniqueness and continuous dependence upon the initial data of ϵ-regular mild solutions are obtained. Furthermore, a unique continuation result and a blow-up alternative result of ϵ-regular mild solutions are given. In Section 3.4, we are devoted to the study of the existence, uniqueness and regularity of weak solutions in $L^{\infty }\left(0,b;L^2(\text{Ω})\right){{\displaystyle \cap }}^{\text{}}{L}^2\left(0,b;H_0^1(\text{Ω})\right)$ of the Rayleigh-Stokes problem. Furthermore, we prove an improved regularity result of weak solutions. In Section 3.5, we prove a long time existence result for fractional Rayleigh-Stokes equations. More precisely, we discuss the existence, uniqueness, continuous dependence on initial value and asymptotic behavior of global solutions in Besov-Morrey spaces. The results are formulated that allows for a larger class in initial value than the previous works and the approach is also suitable for fractional diffusion cases. In Section 3.6, we study a fractional nonlinear Rayleigh-Stokes problem with final value condition. By means of the finite dimensional approximation, we first obtain the compactness of solution operators. Moreover, we handle the problem in weighted continuous function spaces, and then the existence result of solutions is established. Finally, because of the ill-posedness of backward problem in the sense of Hadamard, the quasi-boundary value method is utilized to get the regularized solutions, and the corresponding convergence rate is obtained.

In this chapter, we study the time-fractional Fokker-Planck equations which can be used to describe the subdiffusion in an external time and space-dependent force field F (t, x). In Section 4.1, we present some results on existence and uniqueness of mild solutions allowing the “working space” that may have low regularity. Secondly, we analyze the relationship between “working space” and the value range of α when investigating the problem of classical solutions. Finally, by constructing a suitable weighted Hölder continuous function space, the existence of classical solutions is derived without the restriction on $\alpha \in \left(\frac{1}{2},1\right)$. In Section 4.2, a time-fractional Fokker-Planck initial-boundary value problem is considered, the spatial domain Ω ⊂ ℝ ^d, where d ≥ 1, has a smooth boundary. Existence, uniqueness and regularity of a mild solution u are proved under the hypothesis that the initial data u₀ lies in L²(Ω). For 1/2 < α < 1 and $u_0\in H^2(\text{Ω}){{\displaystyle \cap }}^{\text{}}{H}_0^1(\text{Ω})$, it is shown that u becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived.

In this chapter, we firstly study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order 0 < α < 1, and a self-adjoint generator A. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family {U_α(t)}_t≥0. Moreover, we prove that the solution family U_α(t) converges strongly to the family of unitary operators e^−itA, as α approaches to 1. Next, we apply the tools of harmonic analysis to study the Cauchy problem for time-fractional Schrödinger equation. Some fundamental properties of two solution operators are estimated. The existence and a sharp decay estimate for solutions of the given problem in two different spaces are addressed. Finally, we study fractional Schrödinger equations with potential and optimal controls. Existence, uniqueness, local stability and attractivity, and data continuous dependence of mild solutions are also presented respectively. Further, we study the optimal control problems for the controlled fractional Schödinger equations with potential. Existence and uniqueness of optimal pairs for the standard Lagrange problem are presented.

The following sections are included:

• Bibliography
• Index