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

We establish the well-posedness of the 3D rotating incompressible Navier-Stokes equations with critical initial data $u_{0,\mathrm{\Omega }}\in X_{0,q,p}^{\mathrm{\Omega }}$ for $p<5$, where $X_{0,q,p}^{\mathrm{\Omega }}$ 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 $\mathrm{\Omega }\to 0^{+}$.