The Hölder Regularity for Abstract Fractional Differential Equation with Applications to Rayleigh–Stokes Problems

In this paper, we studied the Hölder regularities of solutions to an abstract fractional differential equation, which is regarded as an abstract version of fractional Rayleigh–Stokes problems, rising up to describing a non-Newtonian fluid with a Riemann–Liouville fractional derivative. The purpose of this article was to establish the Hölder regularities of mild solutions, classical solutions, and strict solutions. We introduced an interpolation space in terms of an analytic resolvent to lower the spatial regularity of initial value data. By virtue of the properties of analytic resolvent and the interpolation space, the Hölder regularities were obtained. As applications, the main conclusions were applied to the regularities of fractional Rayleigh–Stokes problems.