The nonlinear Rayleigh‐Stokes problem with Riemann‐Liouville fractional derivative

Abstract

The Rayleigh‐Stokes problem has gained much attention with the further study of non‐Newtonain fluids. In this paper, we are interested in discussing the existence of solutions for nonlinear Rayleigh‐Stokes problem for a generalized second grade fluid with Riemann‐Liouville fractional derivative. We firstly show that the solution operator of the problem is compact and continuous in the uniform operator topology. Furtherly, we give an existence result of mild solutions for the nonlinear problem.

Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations

“…[1,2,3,4,12,16]. Regarding analytic representation for solution of this problem in linear form, we refer to [7,8,13,15,17,18]. Recently, the final value problem involving (1) has been addressed in [9,10,14], as an interesting supplement to qualitative investigation for this equation.…”

Existence and regularization of solutions for nonlinear fractional Rayleigh–Stokes problem with final condition

“…Lemma (Zhou and Wang 21 , Lemmas 3.1 and 3.2 ). The operator P α ( t ) is compact and continuous in the uniform operator topology for t > 0.…”

On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

“…This equation is employed to describe the behavior of non-Newtonian fluids and has been a subject of numerous studies (see, e.g. [2,13,14,19,21,24,25]). In the case γ = 0 and m(t) = m 0 g 1−α (t), we get ∂ t u + m 0 ∂ α t u − ∆u = f (u), that is the Basset equation mentioned in [1,15,23].…”