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

Abstract

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 \[ \sup_{x\in D}r|v_{0,θ}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,θ}(x) \mathrm{d} x = 0, \] 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 \[ \mathcal{K}\buildrel\hbox{def}\over =\frac{\sinϕ}{ρ^2}\partial_ϕ\Big(\frac{v_θ}{\sinϕ}\Big) \,,\quad\quad\mathcal{F}\buildrel\hbox{def}\over =-\partial_ρ\Big(\frac{v_θ}ρ\Big) \,,\quad\quad \mathcal{O}\buildrel\hbox{def}\over =\frac{1}{ρ\sinϕ}\Big(ω_θ-\frac{2v_ϕη(ρ)}ρ\Big) \,, \] and derive a self-closed energy estimate for them, where $η$ 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 $rv_θ$ 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 $\boldsymbol{v}$.