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A survey of geometric constraints on the blowup of solutions of the Navier–Stokes equation

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Abstract

In this survey article, we will discuss some regularity criteria for the Navier–Stokes equation that provide geometric constraints on any possible finite-time blowup. We will also discuss the physical significance of such regularity criteria.

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References

  1. Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934). (MR1555394)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fujita, H., Kato, T.: On the Navier–Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964). (MR0166499)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fefferman, C.L.: Existence and smoothness of the Navier–Stokes equation. Millennium Prize Problems. 57-67 (2006) (MR2238274)

  4. Kato, T.: Strong \(L^{p}\)-solutions of the Navier–Stokes equation in \({ R}^{m}\), with applications to weak solutions. Math. Z. 187(4), 471–480 (1984). (MR760047)

    Article  MathSciNet  MATH  Google Scholar 

  5. Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 (2001). (MR1808843)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ladyzhenskaya, O.A.: Uniqueness and smoothness of generalized solutions of Navier–Stokes equations. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5, 169–185 (1967). (MR0236541)

    MathSciNet  MATH  Google Scholar 

  7. Prodi, G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959). (MR0126088)

    Article  MathSciNet  MATH  Google Scholar 

  8. Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962). (MR0136885)

    Article  MathSciNet  MATH  Google Scholar 

  9. Escauriaza, L., Seregin, G.A., Šverák, V.: L3;1-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58(2), 3–44 (2003). (MR1992563)

    MathSciNet  Google Scholar 

  10. Seregin, G.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commun. Math. Phys. 312(3), 833–845 (2012). (MR2925135)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gallagher, I., Koch, G.S., Planchon, F.: Blow-up of critical Besov norms at a potential Navier–Stokes singularity. Commun. Math. Phys. 343(1), 39–82 (2016). (MR3475661)

    Article  MathSciNet  MATH  Google Scholar 

  12. Albritton, D.: Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces. Anal. PDE 11(6), 1415–1456 (2018). (MR3803715)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chae, D. and Choe, H.-J.: Regularity of solutions to the Navier–Stokes equation. Electron. J. Differ. Eqs. (05), 7 (1999). (MR1673067)

  14. Zhifei, Z., Qionglei, C.: Regularity criterion via two components of vorticity on weak solutions to the Navier–Stokes equations in \(\mathbb{R}^{3}\) 3. J. Differ. Eqs. 216(2), 470–481 (2005). (MR2165305)

    Article  MATH  Google Scholar 

  15. Guo, Z., Kučera, P., Skalák, Z.: Regularity criterion for solutions to the Navier–Stokes equations in the whole 3D space based on two vorticity components. J. Math. Anal. Appl. 458(1), 755–766 (2018). (MR3711930)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miller, E.: Global regularity for solutions of the three dimensional Navier–Stokes equation with almost two dimensional initial data. Nonlinearity 33(10), 5272–5323 (2020). (MR4143973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kukavica, I., Ziane, M.: Navier–Stokes equations with regularity in one direction. J. Math. Phys. 48(6), 065203 (2007). (MR2337002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kukavica, I., Rusin, W., Ziane, M.: Localized anisotropic regularity conditions for the Navier–Stokes equations. J. Nonlinear Sci. 27(6), 1725–1742 (2017). (MR3713929)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cao, C.: Sufficient conditions for the regularity to the 3D Navier–Stokes equations. Discr. Contin. Dyn. Syst. 26(4), 1141–1151 (2010). (MR2600739)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, Z.: An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field. Bull. Math. Sci. 8(1), 33–47 (2018). (MR3775266)

    Article  MathSciNet  MATH  Google Scholar 

  21. Namlyeyeva, Y., Skalak, Z.: The optimal regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity. ZAMM Z. Angew. Math. Mech. 100(1), e201800114 (2020). (MR4066958)

    Article  MathSciNet  Google Scholar 

  22. Skalak, Z.: The end-point regularity criterion for the Navier–Stokes equations in terms of \(\partial _3 u\). Nonlinear Anal. Real World Appl. 55, 103120 (2020) (MR4083827)

    Article  MathSciNet  MATH  Google Scholar 

  23. Skalak, Z.: An optimal regularity criterion for the Navier–Stokes equations proved by a blow-up argument. Nonlinear Anal. Real World Appl. 58, 103207 (2021). (MR4147644)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, H., Fang, D., Zhang, T.: Critical regularity criteria for Navier–Stokes equations in terms of one directional derivative of the velocity. Math. Methods Appl. Sci. 44, 5123–5132 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Giang, N.V., Khai, D.Q.: Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field. Nonlinear Anal. Real World Appl. 62, 103379 (2021). (MR4281948)

    Article  MathSciNet  Google Scholar 

  26. Liu, Y., Zhang, P.: Global solutions of 3-D Navier–Stokes system with small unidirectional derivative. Arch. Ration. Mech. Anal. 235(2), 1405–1444 (2020). (MR4064202)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Y., Paicu, M., Zhang, P.: Global well-posedness of 3-D anisotropic Navier–Stokes system with small unidirectional derivative. Arch. Ration. Mech. Anal. 238(2), 805–843 (2020). (MR4134152)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chemin, J.-Y., Zhang, P.: On the critical one component regularity for 3-D Navier–Stokes systems. Ann. Sci. Éc. Norm. Supér (4). 49(1), 131–167 (2016). (MR3465978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Chemin, J.-Y., Zhang, P., Zhang, Z.: On the critical one component regularity for 3-D Navier–Stokes system: general case. Arch. Ration. Mech. Anal. 224(3), 871–905 (2017). (MR3621812)

    Article  MathSciNet  MATH  Google Scholar 

  30. Han, B., Lei, Z., Li, D., Zhao, N.: Sharp one component regularity for Navier–Stokes. Arch. Ration. Mech. Anal. 231(2), 939–970 (2019). (MR3900817)

    Article  MathSciNet  MATH  Google Scholar 

  31. Neustupa, J., Novotný, A. and Penel, P. An interior regularity of a weak solution to the Navier–Stokes equations in dependence on one component of velocity. Topics Math. Fluid Mech. 163–183 (2002)(MR2051774)

  32. Guo, Z., Li, Y., Skalák, Z.: Regularity criteria of the incompressible Navier–Stokes equations via only one entry of velocity gradient. J. Math. Fluid Mech. 21(3), 35 (2019). (MR3962840)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, Z., Zhang, Y.: On regularity criteria for the Navier–Stokes equations based on one directional derivative of the velocity or one diagonal entry of the velocity gradient. Z. Angew. Math. Phys. 72(1), 24 (2021). (MR4198773)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Z., Zhong, D., Huang, X.: A refined regularity criterion for the Navier–Stokes equations involving one non-diagonal entry of the velocity gradient. J. Math. Anal. Appl. 453(2), 1145–1150 (2017). (MR3648279)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fang, D., Qian, C.: The regularity criterion for 3D Navier–Stokes equations involving one velocity gradient component. Nonlinear Anal. 78, 86–103 (2013). (MR2992988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhou, Y., Pokorný, M.: On the regularity of the solutions of the Navier–Stokes equations via one velocity component. Nonlinearity 23(5), 1097–1107 (2010). (MR2630092)

    Article  MathSciNet  MATH  Google Scholar 

  37. Cao, C., Titi, E.S.: Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202(3), 919–932 (2011). (MR2854673)

    Article  MathSciNet  MATH  Google Scholar 

  38. Penel, P., Pokorný, M.: Improvement of some anisotropic regularity criteria for the Navier–Stokes equations. Discr. Contin. Dyn. Syst. Ser. S 6(5), 1401–1407 (2013). (MR3039706)

    MathSciNet  MATH  Google Scholar 

  39. Qian, C.: The regularity criterion for the 3D Navier–Stokes equations involving end-point Prodi-Serrin type conditions. Appl. Math. Lett. 75, 37–42 (2018). (MR3692158)

    Article  MathSciNet  MATH  Google Scholar 

  40. Skalák, Z.: Regularity criteria for the Navier–Stokes equations based on one or two items of the velocity gradient. Nonlinear Anal. Real World Appl. 38, 131–145 (2017). (MR3670702)

    Article  MathSciNet  MATH  Google Scholar 

  41. Skalak, Z.: A regularity criterion for the Navier–Stokes equations via one diagonal entry of the velocity gradient. Commun. Math. Sci. 19(4), 1101–1112 (2021). (MR4278945)

    Article  MathSciNet  Google Scholar 

  42. Neustupa, J. and Penel, P. Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations. Math. Fluid Mech. Recent Results Open Questions 237–265 (2001) (MR1865056)

  43. Neustupa, J., Penel, P.: The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier–Stokes equations. C. R. Math. Acad. Sci. Paris 336(10), 805–810 (2003). (MR1990019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Miller, E.: A regularity criterion for the Navier–Stokes equation involving only the Middle Eigenvalue of the strain tensor. Arch. Ration. Mech. Anal. 235(1), 99–139 (2020). (MR4062474)

    Article  MathSciNet  MATH  Google Scholar 

  45. Neustupa, J. and Penel, P. Regularity of a weak solution to the Navier–Stokes equation in dependence on eigenvalues and eigenvectors of the rate of deformation tensor. Trends Partial Differ. Eqs. Math. Phys. 197-212 (2005) (MR2129619)

  46. Miller, E.: A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity. Proc. Am. Math. Soc. Ser. B 8, 60–74 (2021). (MR4214337)

    Article  MathSciNet  MATH  Google Scholar 

  47. 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). (MR1254117)

    Article  MathSciNet  MATH  Google Scholar 

  48. da Veiga, H.B., Berselli, L.C.: On the regularizing effect of the vorticity direction in incompressible viscous flows. Differ. Integral Eqs. 15(3), 345–356 (2002). (MR1870646)

    MathSciNet  MATH  Google Scholar 

  49. Skalak, Z.: Locally space–time anisotropic regularity criteria for the Navier–Stokes equations in terms of two vorticity components. J. Math. Fluid Mech. 23(2), 41 (2021). (MR4234292)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kemp, M.: Leonardo da Vinci’s laboratory: studies in flow. Nature 571, 322–323 (2019)

    Article  Google Scholar 

  51. Kolmogorov, A.: The local structure of turbulence in incompressible viscous fluid for very large Reynold’s numbers. C. R. (Doklady) Acad. Sci. URSS (N.S.) 30, 301–305 (1941). (MR0004146)

    MathSciNet  Google Scholar 

  52. Obukhov, A.: On the energy distribution in the spectrum of a turbulent flow. C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, 19–21 (1941). (MR0005852)

    MathSciNet  Google Scholar 

  53. Miller, E.: Finite-time blowup for a Navier–Stokes model equation for the self-amplification of strain, ar**v e-prints. ar**v:1910.05415. Available 1910.05415 (2021)

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Acknowledgements

This publication was supported in part by the Fields Institute for Research in the Mathematical Sciences while the author was in residence during the Fall 2020 semester. Its contents are solely the responsibility of the author and do not necessarily represent the official views of the Fields Institute. This material is based upon work supported by the National Science Foundation under Grant no. DMS-1440140 while the author participated in a program that was hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester. THE author would like to thank Raphaël Danchin, Reinhard Farwig, Sarka Necasova, Jiří Neustupa, and the staff of the Centre International de Rencontres Mathématiques (CIRM) for all of the work they did to make the conference Vorticity, Rotation and Symmetry (V)—Global Results and Nonlocal Phenomena a success in spite of the virtual format and the limitations due to the pandemic.

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    Evan Miller

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Miller, E. A survey of geometric constraints on the blowup of solutions of the Navier–Stokes equation. J Elliptic Parabol Equ 7, 589–599 (2021). https://doi.org/10.1007/s41808-021-00135-8

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