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Abstract:For the incompressible Navier-Stokes flows passing a certain type of cones $D$ with the 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, where $r$ means the distance to the $z$ axis, and $v_{0,\theta}$ represents the initial value of the azimuthal component of the velocity $\boldsymbol{v}$. Unlike in the previous conclusion in \cite{LPYZZZ24}, no parity assumption on $\boldsymbol{v}_0$ is needed. 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.

Submission history

From: Zijin Li [view email]
[v1] Sun, 24 May 2026 15:35:34 UTC (379 KB)