On the local solvability in Morrey spaces of the Navier–Stokes equations in a rotating frame

Abstract

We prove local-in-time (non-uniform) solvability for the rotating Navier–Stokes equations in Morrey spaces Mp,μσ(R3). These spaces contain singular and nondecaying functions which are of interest in statistical turbulence. We give an algebraic relation between the size of existence time and angular velocity Ω. The evolution of velocity u is analyzed in suitable Kato–Fujita spaces based on Morrey spaces. We show the asymptotic behavior uΩ→w in L∞(0,T;Mp,μσ(R3)) as Ω→0, where w is the solution for the Navier–Stokes equations with the same data u0. Particularly, for μ=3−p, the solution is approximately self-similar for small |Ω|, when u0 is homogeneous of degree −1.