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

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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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Notes

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

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

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

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

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

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

  13. See the proof of Theorem 1.3 in [8].

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

References

  1. Albritton, D., Barker, T.: Global weak Besov solutions of the Navier-Stokes equations and applications. Arch. Rational Mech. Anal. 232(1), 197–263, 2019

    ADS  MathSciNet  MATH  Google Scholar 

  2. Albritton, D., Barker, T.: Localised necessary conditions for singularity formation in the Navier–Stokes equations with curved boundary, 2018. arXiv:1811.00507

  3. Albritton, D., Barker, T.: On local type I singularities of the Navier–Stokes equations and Liouville theorems, 2018. arXiv:1811.00502

  4. Bae, H.-O., Jin, B.: Regularity for the Navier–Stokes equations with slip boundary condition. Proc. Am. Math. Soc. 136(7), 2439–2443, 2008

    MathSciNet  MATH  Google Scholar 

  5. Bae, H.-O., Jin, B.J.: Existence of strong mild solution of the Navier–Stokes equations in the half-space with nondecaying initial data. J. Korean Math. Soc. 49(1), 113–138, 2012

    MathSciNet  MATH  Google Scholar 

  6. Barker, T.: Uniqueness results for viscous incompressible fluids. PhD thesis, University of Oxford, 2017

  7. Barker, T., Prange, C.: Localized smoothing for the Navier–Stokes equations and concentration of critical norms near singularities, 2018. arXiv:1812.09115

  8. Barker, T., Seregin, G.: Ancient solutions to Navier–Stokes equations in half-space. J. Math. Fluid Mech. 17(3), 551–575, 2015

    ADS  MathSciNet  MATH  Google Scholar 

  9. Bogovskiĭ, M.E.: Solutions of some problems of vector analysis, associated with the operators \({\rm div}\) and \({\rm grad}\). Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, vol. 1980, pp. 5–40, 149. Trudy Sem. S. L. Soboleva, No. 1. Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk (1980)

  10. Chang, T., Kang, K.: Estimates of anisotropic Sobolev spaces with mixed norms for the Stokes system in a half-space. Ann. Univ. Ferrara Sez. VII Sci. Mat. 64(1), 47–82, 2018

    MathSciNet  MATH  Google Scholar 

  11. Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42(3), 775–789, 1993

    MathSciNet  MATH  Google Scholar 

  12. Constantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in MathematicsUniversity of Chicago Press, Chicago 1988

    MATH  Google Scholar 

  13. Da Veiga, H.B.: Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 5(4), 907, 2006

    MathSciNet  MATH  Google Scholar 

  14. da Veiga, H.B., Berselli, L.C., et al.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Differ. Integr. Equ. 15(3), 345–356, 2002

    MathSciNet  MATH  Google Scholar 

  15. Desch, W., Hieber, M., Prüss, J.: \(L^p\)-theory of the Stokes equation in a half-space. J. Evol. Equ. 1(1), 115–142, 2001

    MathSciNet  MATH  Google Scholar 

  16. Escauriaza, L., Seregin, G.A., Šverák, V.: \(L_{3,\infty }\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk58(2(350)), 3–44, 2003

    MathSciNet  Google Scholar 

  17. Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, revised edn. Textbooks in MathematicsCRC Press, Boca Raton 2015

    MATH  Google Scholar 

  18. Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equation. The Millennium Prize Problems, vol. 57, p. 67. American Mathematical Society, Providence 2006

    Google Scholar 

  19. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems, 2nd edn. Springer Monographs in MathematicsSpringer, New York 2011

    MATH  Google Scholar 

  20. Giga, Y., Hsu, P.-Y., Maekawa, Y.: A Liouville theorem for the planer Navier–Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion. Commun. Partial Differ. Equ. 39(10), 1906–1935, 2014

    MathSciNet  MATH  Google Scholar 

  21. Giga, Y., Kohn, R.: Asymptotically self-similar blow-up of semilinear heat equations. Commun. Pure Appl. Math. 38(3), 297–319, 1985

    MathSciNet  MATH  Google Scholar 

  22. Giga, Y., Miura, H.: On vorticity directions near singularities for the Navier–Stokes flows with infinite energy. Commun. Math. Phys. 303(2), 289–300, 2011

    ADS  MathSciNet  MATH  Google Scholar 

  23. Giga, Y., Sohr, H.: Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102(1), 72–94, 1991

    MathSciNet  MATH  Google Scholar 

  24. Grujić, Z.: Localization and geometric depletion of vortex-stretching in the 3D NSE. Commun. Math. Phys. 290(3), 861, 2009

    ADS  MathSciNet  MATH  Google Scholar 

  25. Grujić, Z., Ruzmaikina, A.: Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE. Indiana Univ. Math. J. 53, 1073–1080, 2004

    MathSciNet  MATH  Google Scholar 

  26. Grujić, Z., Zhang, Q.S.: Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE. Commun. Math. Phys. 262(3), 555–564, 2006

    ADS  MathSciNet  MATH  Google Scholar 

  27. Kang, K.: Unbounded normal derivative for the Stokes system near boundary. Math. Ann. 331(1), 87–109, 2005

    MathSciNet  MATH  Google Scholar 

  28. Koch, G., Nadirashvili, N., Seregin, G., Šverák, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203(1), 83–105, 2009

    MathSciNet  MATH  Google Scholar 

  29. Koch, H., Solonnikov, V.: \(L_q\)-estimates of the first-order derivatives of solutions to the nonstationary Stokes problem. Nonlinear Problems in Mathematical Physics and Related Topics, I, vol. 1, pp. 203–218. Int. Math. Ser. (N. Y.)Kluwer/Plenum, New York 2002

    Google Scholar 

  30. Ladyženskaja, O.A.: Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)7, 155–177, 1968

    MathSciNet  Google Scholar 

  31. Li, S.: On vortex alignment and boundedness of \({L}^{q} \) norm of vorticity, 2017. arXiv:1712.00551

  32. Maekawa, Y., Miura, H., Prange, C.: Estimates for the Navier–Stokes equations in the half-space for non localized data. ArXiv e-prints, Nov. 2017

  33. Maekawa, Y., Miura, H., Prange, C.: Local energy weak solutions for the Navier–Stokes equations in the half-space, 2017. arXiv:1711.04486

  34. Mikhaylov, A.: Local regularity for suitable weak solutions of the Navier–Stokes equations near the boundary. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370(Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 40):73–93, 220, 2009

  35. Neustupa, J., Penel, P.: Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component. Applied Nonlinear Analysis, pp. 391–402. Kluwer/Plenum, New York 1999

    Google Scholar 

  36. Pham, T.: Topics in the regularity theory of the Navier–Stokes equations. PhD thesis, University of Minnesota, 2018

  37. Rusin, W., Šverák, V.: Minimal initial data for potential Navier–Stokes singularities. J. Funct. Anal. 260(3), 879–891, 2011

    MathSciNet  MATH  Google Scholar 

  38. Seregin, G.: Liouville theorem for 2D Navier–Stokes equations in a half-space. J. Math. Sci. 210(6), 849–856, 2015

    MathSciNet  MATH  Google Scholar 

  39. Seregin, G.: On type I blowups of suitable weak solutions to Navier–Stokes equations near boundary, 2019. arXiv:1901.08842

  40. Seregin, G., Šverák, V.: On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations. Commun. Partial Differ. Equ. 34(2), 171–201, 2009

    MathSciNet  MATH  Google Scholar 

  41. Seregin, G., Šverák, V.: On a bounded shear flow in half-space. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 385(Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 41):200–205, 236, 2010

  42. Seregin, G., Šverák, V.: Rescalings at possible singularities of Navier–Stokes equations in half-space. St. Petersb. Math. J. 25(5), 815–833, 2014

    MathSciNet  MATH  Google Scholar 

  43. Seregin, G., Šverák, V.: Regularity criteria for Navier–Stokes solutions. In: Giga, Y., Novotný, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, pp. 829–867. Springer, Cham 2018

    Google Scholar 

  44. Seregin, G.A.: Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary. J. Math. Fluid Mech. 4(1), 1–29, 2002

    ADS  MathSciNet  MATH  Google Scholar 

  45. Seregin, G.A.: A new version of the Ladyzhenskaya–Prodi–Serrin condition. Algebra i Analiz18(1), 124–143, 2006

    MathSciNet  Google Scholar 

  46. Seregin, G.A.: A note on local boundary regularity for the Stokes system. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 370(Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 40):151–159, 221–222, 2009

  47. Seregin, G.A., Zajaczkowski, W.: A sufficient condition of local regularity for the Navier–Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336(Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 37):46–54, 274, 2006

  48. Simon, J.: Compact sets in the space \({L}^p(0,{T};{B})\). Ann. Mat. Pura Appl. 4(146), 65–96, 1987

    MATH  Google Scholar 

  49. Tsai, T.-P.: On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal. 143(1), 29–51, 1998

    ADS  MathSciNet  MATH  Google Scholar 

  50. Zhou, Y.: A new regularity criterion for the Navier–Stokes equations in terms of the direction of vorticity. Monatsh. Math. 144(3), 251–257, 2005

    MathSciNet  MATH  Google Scholar 

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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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  1. DMA, École Normale Supérieure, CNRS, PSL Research University, 75005, Paris, France

    Tobias Barker

  2. CNRS, UMR [5251], IMB, Université de Bordeaux, Bordeaux, France

    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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