Well-posedness and vanishing rotational limit for the rotating incompressible Navier-Stokes equations in hybird Besov space

Abstract

We establish the well-posedness of the 3D rotating incompressible Navier-Stokes equations with critical initial data $u_{0,Ω}\in X_{0,q,p}^Ω$ for $p<5$, where $X_{0,q,p}^Ω$ is defined by the norm \begin{equation*} \begin{aligned} &\|u_{0,Ω}\|_{X_{0,q,p}^Ω}:= Ω^{3- \frac{6}{q}}\|u_{0,Ω}\|_{\dot{B}_{q,\infty}^{-7+\frac{15}{q}}}^{\ell_Ω} +\|u_{0,Ω}\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}^{h_Ω}. \end{aligned} \end{equation*} This extends the previous results by Chen, Miao, and Zhang (\cite{CMZ2013}). The main ingredients are a new global-in-time dissipative-dispersive estimate for the Stokes--Coriolis semigroup and corresponding bilinear estimates. Furthermore, we establish the vanishing rotational limit for the 3D rotating Navier-Stokes equations as $Ω\rightarrow 0^{+}$.