An anisotropic Sobolev–Hardy inequality with application to 3D axisymmetric Navier–Stokes equations

ABSTRACT

In this paper, we first establish a new anisotropic Sobolev–Hardy inequality in which anisotropic singular weights are permitted. As an application, we study the three-dimensional axisymmetric Navier–Stokes system with non-zero swirl and obtain the global well-posedness result if the angular velocity $u_{\theta }$ satisfies $\underset{0<t<T}{sup}\parallel z{u}_{\theta }(\cdot ,t){\parallel }_{L^{\mathrm{\infty }}(r\le r_0)}\le ϵ,r_0\in (0,\frac{1}{2}).$

KEYWORDS:

• Navier–Stokes equations
• maximal functions
• anisotropic Sobolev–Hardy inequality

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q30
• 42B25
• 46E99

Disclosure statement

No potential conflict of interest was reported by the author.