On the axisymmetric Navier-Stokes flow passing a cone with the total-slip boundary condition

Recently, [25] observed that, among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with the classical Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,\theta}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,\theta}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. We point out that there is no size restriction on other components of the initial velocity. Compared with [25], no parity symmetry assumption on $\boldsymbol{v}_0$ is required. There are four key ingredients in the proof. (1) In spherical coordinates, we introduce three new quantities \[ \mathcal{K}\buildrel\hbox{def}\over =\frac{\sin\phi}{\rho^2}\partial_\phi\Big(\frac{v_\theta}{\sin\phi}\Big) \,,\quad\quad\mathcal{F}\buildrel\hbox{def}\over =-\partial_\rho\Big(\frac{v_\theta}{\rho}\Big) \,,\quad\quad \mathcal{O}\buildrel\hbox{def}\over =\frac{1}{\rho\sin\phi}\Big(\omega_\theta-\frac{2v_\phi\eta(\rho)}{\rho}\Big) \,, \] and derive a self-closed energy estimate for them, where $\eta$ is a cut-off function which vanishes near the origin and equals $1$ away from the origin. (2) A boundary value problem of the pressure $P$ is proposed and an elliptic estimate for $P$ is established in order to control boundary terms arising from the Navier total-slip boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of the quantity $rv_\theta$ whose integral on $D$ vanishes for all the time. (4) A new anisotropic Hardy's inequality is derived for functions whose integral on $D$ vanish to overcome the lack of parity symmetry of $\boldsymbol{v}$.