Abstract
We point out some criteria that imply regularity of axisymmetric solutions to Navier–Stokes equations. We show that boundedness of Vert {v_{r}}/{sqrt{r^3}}Vert _{L_2({mathbb {R}}^3times (0,T))} as well as boundedness of Vert {omega _{varphi }}/{sqrt{r}} Vert _{L_2({mathbb {R}}^3times (0,T))}, where v_r is the radial component of velocity and omega _{varphi } is the angular component of vorticity, imply regularity of weak solutions.
Highlights
We consider the Cauchy problem to the three-dimensional axisymmetric Navier–Stokes equations: vt + v · ∇v − νΔv + ∇p = 0 (x, t) ∈ R3 × R+, div v = 0, (1.1)v t=0 = v(0), where x = (x1, x2, x3), v is the velocity of the fluid motion with v(x, t) = (v1(x, t), v2(x, t), v3(x, t)) ∈ R3, p = p(x, t) ∈ R1 denotes the pressure, ν is the viscosity coefficient and v0 is given initial velocity field
The first papers concerning regularity of axially symmetric solutions to the Navier–Stokes equations were independently proved by Ladyzhenskaya [1] and Yudovich–Ukhovskij [2] in 1968
There is a lot of criteria for regularity of axisymmetric solutions
Summary
The first papers concerning regularity of axially symmetric solutions to the Navier–Stokes equations were independently proved by Ladyzhenskaya [1] and Yudovich–Ukhovskij [2] in 1968. In these papers axisymmetric solutions without swirl were considered. In the period 1999–2002 arised many papers concerning sufficient conditions on regularity of axisymmetric solutions [3,4,5,6]. There is a lot of criteria for regularity of axisymmetric solutions (see [6,7,8,9,10,11,12] and the literature cited in these papers).
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