Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects

Abstract

Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order $\beta \in (0,1)$ are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In $H^{\delta ,r}$, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in $J_r$. Moreover, the regularity and existence of classical solutions to the equations in $J_r$. are established and presented.