Time periodic strong solutions to the Keller-Segel system coupled to Navier-Stokes equation

Abstract

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N ( N ≥ 4 ) . Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C ( R , L s , ∞ ( R N ) ) when S ( n ) = n − 1 2 . Next for the case S ( n ) = 0 , we study the mild solution and point out it can become strong solution in B C ( R , L s , ∞ ( R N ) ) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations.