Global solvability of the Navier-Stokes equations in spaces based on sum-closed frequency sets

Abstract

We prove existence of global regular solutions for the $3$D Navier-Stokes equations with (or without) Coriolis force for a class of initial data $u_0$ in the space ${{{\mathrm{FM}}}_{\sigma,\delta}}$, i.e., for functions whose Fourier image ${\widehat{u}}_0$ is a vector-valued Radon measure and that are supported in sum-closed frequency sets with distance $\delta$ from the origin. In our main result we establish an upper bound for admissible initial data in terms of the Reynolds number, uniform on the Coriolis parameter $\Omega$. In particular this means that this upper bound is linearly growing in $\delta$. This implies that we obtain global-in-time regular solutions for large (in norm) initial data $u_0$ which may not decay at space infinity, provided that the distance $\delta$ of the sum-closed frequency set from the origin is sufficiently large.