Abstract
In a d-dimensional strip with d≥2, we study the non-stationary Stokes equation with no-slip boundary condition in the lower and upper plates and periodic boundary condition in the horizontal directions. In this paper we establish a new maximal regularity estimate in the real interpolation norm‖f‖(0,1)=inff=f0+f1{〈sup0<z<1|f0|〉+〈∫01|f1|dz(1−z)z〉}, where the brackets 〈⋅〉 denote the horizontal-space and time average. The norms involved in the definition of ‖⋅‖(0,1) are critical for two reasons: the exponents are borderline for the Calderón–Zygmund theory and the weight 1/z just fails to be Muckenhoupt. Therefore, the estimate is only true under horizontal bandedness condition (i.e. a restriction to a packet of wave numbers in Fourier space). The motivation to express the maximal regularity in such a norm comes from an application to the Rayleigh–Bénard problem (see [5]).