We study the small initial data Cauchy problem for the three-dimensional Boussinesq equations with the Coriolis force in variable exponent Fourier-Besov spaces. By using the Fourier localization argument and Littlewood-Paley decomposition, we obtain the global well-posedness result for small initial data (u 0,θ 0) belonging to the critical variable exponent Fourier-Besov spaces $\mathcal{F}\mathcal{\dot{B}}_{p(\cdot),q}^{2-\frac{3}{p(\cdot)}}$.

We study the small initial date Cauchy problem for the generalized incompressible Navier-Stokes-Coriolis equations in critical hybrid-Besov space $\dot{\mathscr{B}}_{2,\, p}^{\frac{5}{2}-2\alpha, \frac{3}{p}-2\alpha+1}(\mathbb{R}^3)$ with $1/2<\alpha<2$ and $2\leq p\leq 4$. We prove that hybrid-Besov spaces norm of a class of highly osillating initial velocity can be arbitrarily small. and we prove the estimation of highly frequency $L^p$ smoothing effect for generalized Stokes-Coriolis semigroup with $1\leq p\leq\infty$, At the same time, we prove space-time norm estimation of hybrid-Besov spaces for Stokes-Coriolis semigroup. From this result we deduce bilinear estimation in our work space. Our method relies upon Bony’s high and low frequency decomposition technology and Banach fixed point theorem.

In this work, we demonstrate uniqueness of the weak solution to the fractional anisotropic Navier-Stokes system with only horizontal dissipation.