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Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions

Abstract

This paper is concerned with geometric regularity criteria for the Navier–Stokes equations in \({\mathbb {R}}^3_{+}\times (0,T)\) with a no-slip boundary condition, with the assumption that the solution satisfies the ‘ODE blow-up rate’ Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets of

$$\begin{aligned} \bigcup _{t\in (T-1,T)} \big (B(0,R)\cap {\mathbb {R}}^3_{+}\big )\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t}), \end{aligned}$$

where the vorticity has large magnitude, then (0, T) is a regular point. This result is inspired by and improves the regularity criteria given by Giga et al.  [20]. We also obtain new local versions for suitable weak solutions near the flat boundary. Our method hinges on new scaled Morrey estimates, blow-up and compactness arguments and ‘persistence of singularites’ on the flat boundary. The scaled Morrey estimates seem to be of independent interest.

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Fig. 1

Notes

  1. 1.

    By scale-invariant quantities, we mean quantities which are invariant with respect to the Navier–Stokes rescaling \((u_{\lambda }(y,s),p_{\lambda }(y,s))=(\lambda u(\lambda y, \lambda ^2 s), \lambda ^2 p(\lambda y, \lambda ^2 s))\). The vast majority of regularity criteria for the Navier–Stokes equations are stated in terms of scale-invariant quantities since, heuristically at least, the diffusive effects and non-linear effects are ‘balanced’.

  2. 2.

    Here \(\angle (a,b)\) denotes the angle between the vectors a and b.

  3. 3.

    In [20] the authors assume that \(\eta \) is nondecreasing. However, this assumption can be removed. What matters is that \(\eta \) is continuous and \(\eta (0)=0\).

  4. 4.

    In particular, it satisfies the Duhamel formulation on any compact subinterval of \((-\infty ,0)\). We will sometimes refer to this property as being a ‘mild solution in \({\mathbb {R}}^3_{+}\times (-\infty ,0)\)’.

  5. 5.

    Arguments from [8] demonstrate that \({\bar{u}}\) is \(C^{\infty }\) in space-time on \(\overline{{\mathbb {R}}^{3}_{+}}\times (-\infty ,0)\), hence this pointwise condition on the vorticity is well defined.

  6. 6.

    Here, \(\vec {e}_{2}=(0,1,0)\) and \(\vec {e}_{3}=(0,0,1).\)

  7. 7.

    Technically, (28) must be assumed instead of (16). For the purpose of this discussion we overlook this point.

  8. 8.

    We use the definition of ‘suitable weak solution’ in Q(1) given in [43, Definition 1]. For the definition of suitable weak solutions in \(Q^{+}(1)\), we refer the reader to subsection 1.5.

  9. 9.

    This definition was given in [3] for local solutions defined in a ball. See also [39] for a related definition for local solutions with no-slip on the flat part of the boundary.

  10. 10.

    This definition is taken from [46] (Definition 1.3).

  11. 11.

    Recall that \(\gamma ^{(k)} \uparrow \infty \).

  12. 12.

    This can be inferred from arguments from pp. 555–556 of [8].

  13. 13.

    See the proof of Theorem 1.3 in [8].

  14. 14.

    Notice that our definition of \(\Omega _d\) here is slightly different from the one in the global setting. Indeed, in order to deduce that the blow-up flow is two-dimensional we need to use the no-slip boundary condition, as in Step 4 in Section 4. Hence, we require that the continuous alignment condition is true up to the boundary. In the global setting we do not need to assume this explicitly, since by (120) the flow is smooth up to the boundary.

  15. 15.

    See [9], as well as Chapter III of [19].

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Funding

The second author is partially supported by the project BORDS Grant ANR-16-CE40-0027-01 and by the project SingFlows Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). The second author also acknowledges financial support from the IDEX of the University of Bordeaux for the BOLIDE project.

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Correspondence to Christophe Prange.

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Barker, T., Prange, C. Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions. Arch Rational Mech Anal 235, 881–926 (2020). https://doi.org/10.1007/s00205-019-01435-z

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