Proceedings of the St. Petersburg Mathematical Society, Volume II (original) (raw)

Abstract

This is an expository paper on the theory of local regularity for weak solutions to the non-stationary 3D Navier-Stokes equations near the boundary of a domain.

Figures (1)

which holds in the sense of distributions and the boundary condition is understood in the sense of traces. Moreover, functions v’ and q” satisfy the local energy inequality

which holds in the sense of distributions and the boundary condition is understood in the sense of traces. Moreover, functions v’ and q” satisfy the local energy inequality

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (39)

  1. O.V. Besov, V.P. Il'in, S.M. Nikolskii, Integral representations of functions and imbedding theorems. Nauka, Moscow, 1975. Translation: Wiley&Sons, 1978.
  2. M.E. Bogovskii, On solution of some problems of vectoral analysis related to div and grad operators, Proc. of S.L. Sobolev Seminar 1 (1980), 5-40.
  3. S. Campanato, Proprieta di Hölderianita di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa 17 (3) (1963), 175-188.
  4. L. Caffarelli, R.V. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831.
  5. L. Escauriaza, G.A. Seregin, V. Sverak, On backward uniquness for parabolic equations, Arch. Rational Mech. Anal., 169 (2003), 145-157.
  6. L. Escauriaza, G.A. Seregin, V. Sverak, L 3,∞ -Solutions to the Navier-Stokes equations and backward uniqueness, Uspekhi Matematicheskih Nauk, 58 (2003) no. 2 (350), 3-44. English translation in Russian Mathematical Surveys, 58 (2003) no. 2, 211-250.
  7. L. Escauriaza, G.A. Seregin, V. Sverak, Backward uniquness for the heat operator in a half-space, St.-Petersburg Mathematical Journal, 15 (2004), №1, 139-148.
  8. N.D. Filonov, T.N. Shilkin, On the Stokes problem with non-zero divergence, Zap. Nauchn. Semin. POMI 370 (2009), 184-202.
  9. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear el- liptic systems, Princeton, 1983.
  10. S. Gustafson, K. Kang, T.-P. Tsai, Interiour regularity criteria for suitable weak solutions to he Navier-Stokes equations, Commun. Math. Phys., 273 (2007), 161-176.
  11. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichun- gen. Math. Nachr. 4, (1950), 213-231.
  12. K. Kang, On boundary regularity of the NavierЏStokes equations, Communica- tions in Partial Differential Equations, 29 (7-8) (2004), 955-987.
  13. K. Kang, Unbounded normal derivative for the Stokes system near boundary, Math- emtische Annalen 331 (2005), 87-109.
  14. T. Kato, Strong L p -solutions of the Navier-Stokes equations in R n , with applica- tions to weak solutions, Math. Z., 197 (1984), 471-480.
  15. A.A. Kiselev, O.A. Ladyzhenskaya, On the existence and uniqueness of the solution of the nonstationary problem for a viscous, incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat., 21:5 (1957), 655Џ680.
  16. O.A. Ladyzhenskaya, G.A. Seregin, On Partial regularity of suitable weak so- lutions to the three-dimensional Navier-Stokes equations, Journal of Mathematical Fluid Mechanics, 1 (1999), 356-387.
  17. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
  18. F.-H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), no. 3, 241-257.
  19. A.S. Mikhaylov, Local regularity for suitable weak solutions of the Navier-Stokes equations near the boundary, Zap. Nauchn. Semin. POMI 370 (2009), 73-93.
  20. A.S. Mikhaylov, On local regularity for suitable weak solutions of the NavierЏS- tokes equations near the boundary, Zap. Nauchn. Semin. POMI 385 (2010), 83-97.
  21. A.S. Mikhaylov, T.N Shilkin, L 3,∞ solutions to the Navier-Stokes equations near the boundary, Zap. Nauchn. Semin. POMI 336 (2006), 133-152.
  22. V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), 97-112.
  23. G.A. Seregin, Some estimates near the boundary for solutions to the non- stationary linearized Navier-Stokes equations, Zap. Nauchn. Semin. POMI 271 (2000), 204-223.
  24. G.A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, Journal of Mathematical Fluid Mechanics 4 (2002) no.1, 1-29.
  25. G.A. Seregin, Differentiability properties of weak solutions of the Navier-Stokes equations, St.-Petersburg Math. Journal 14 (2003), no.1, 147-178.
  26. G.A. Seregin, A note on local boundary regularity for the Stokes system, Zap. Nauchn. Semin. POMI 370 (2009), 151-159.
  27. G.A. Seregin, A certain necessary condition of potential blow up for Navier- Stokes equations, Communications in Math. Phys. 312 (3) (2012), 833-845.
  28. G.A. Seregin, Local regularity for suitable weak solutions of the Navier-Stokes equations, Russian Mathematical Surveys 62 (3) (2007), 595-614.
  29. G.A. Seregin, Local regularity theory of the Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, Vol. 4, (2007), 159-200.
  30. G.A. Seregin, Weak solutions to the Navier-Stokes equations with bounded scale- invariant quantities, Proceedings of the International Congress of Mathematics, 2010 (Hyderabad, India, 2010).
  31. G.A. Seregin, On smoothness of L 3,∞ -solutions to the Navier-Stokes equations up to boundary, Mathematische Annalen, 332 (2005), 219-238.
  32. G.A. Seregin, T.N. Shilkin, V.A. Solonnikov, Boundary patial regularity for the Navier-Stokes equations, Zap. Nauchn. Semin. POMI 310 (2004), 158-190.
  33. G.A. Seregin, V. Sverak, On a bounded shear flow in half-space, Zap. Nauchn. Semin. POMI, 385 (2010), 200-205.
  34. G. Seregin, V. Sverak, On type I singularities of the local axi-symmentric solutions of the Navier-Stokes equations, Communications in PDE's, 34 (2009), 171-201.
  35. J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equa- tions, Arch. Ration. Mech. Anal., 9 (1962), 187-195.
  36. T.N. Shilkin, V.A. Vyalov, Estimates of Solutions to the Perturbed Stokes System, Zap. Nauchn. Semin. POMI 410 (2013), to appear.
  37. V.A. Solonnikov, Estimates of solutions to the non-stationary Navier-Stokes system, Zap. Nauchn. Semin. POMI 28 (1973), 153-231.
  38. V.A. Solonnikov, Estimates of solutions of the Stokes equations in Sobolev spaces with a mixed norm, Zap. Nauchn. Semin. POMI 288 (2002), 204-231.
  39. V.A. Solonnikov, On the estimates of solutions of nonstationary Stokes problem in anisotropic S.L. Sobolev spaces and on the estimate of resolvent of the Stokes problem, Uspekhi Matematicheskih Nauk, 58 (2003) 2 (350) 123-156.