Chapter 3: Fractional Rayleigh-Stokes Equations

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.