Existence and temporal decay of regular solutions to non-Newtonian fluids combined with Maxwell equations

Abstract We consider the Cauchy problem of a certain type of non-Newtonian fluids combined with Maxwell equations in three dimensions. We establish local existence of unique regular solutions for sufficiently smooth initial data. In addition, the regular solutions are globally extended in time, provided that the H 3 -norm of the initial data is small enough. Lastly, using the Fourier splitting method, we show that H l -norms of the global regular solution decay with the rate of ( 1 + t ) − ( 3 4 + l 2 ) for l ≥ 0 , as time tends to infinity.