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

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Abstract

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

$$|v(x, t)| \le \frac{C |\ln r|^{1/2}}{r^2}, \qquad 0 < r \le 1/2,$$

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.

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Authors and Affiliations

  1. School of Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Zhen Lei

  2. Department of Mathematics, University of California, Riverside, CA, 92521, USA

    Esteban A. Navas & Qi S. Zhang

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  1. Zhen Lei
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  2. Esteban A. Navas
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  3. Qi S. Zhang
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Correspondence to Qi S. Zhang.

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Communicated by H.-T. Yau

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Lei, Z., Navas, E.A. & Zhang, Q.S. A Priori Bound on the Velocity in Axially Symmetric Navier–Stokes Equations. Commun. Math. Phys. 341, 289–307 (2016). https://doi.org/10.1007/s00220-015-2496-4

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  • DOI: https://doi.org/10.1007/s00220-015-2496-4

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