A Priori Bound on the Velocity in Axially Symmetric Navier-Stokes Equations

作者：Lei, Zhen^*; Navas, Esteban A.; Zhang, Qi S.

来源：Communications in Mathematical Physics, 2016, 341(1): 289-307.

DOI：10.1007/s00220-015-2496-4

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Let v be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound
vertical bar v(x,t)vertical bar <= C vertical bar ln r vertical bar(1/2)/r(2), 0 < r <= 1/2,
where r is the distance from x to the z axis, and C is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities, while the recent papers (Chiun-Chuan et al., Commun PDE 34(1-3):203-232, 2009; Koch et al., Acta Math 203(1):83-105, 2009) gave a lower bound. The gap is polynomial order 1 modulo a half log term.

• 出版日期2016-1
• 单位复旦大学