SpringerLink
Log in

Zero-Viscosity Limit of the Navier–Stokes Equations in the Analytic Setting

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper, we consider the zero-viscosity limit of the Navier–Stokes equations in a half space with non-slip boundary condition. Based on the vorticity formulation and the use of conormal derivative, we develop an energy method to justify the zero-viscosity limit for the analytic data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abidi H., Danchin R.: Optimal bounds for the inviscid limit of Navier–Stokes equations. Asymptot. Anal. 38, 35–46 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Alexandre R., Wang Y., Xu C.-J., Yang T.: Well-posedness of the Prandtl Equation in Sobolev Spaces. J. Amer. Math. Soc. 28, 745–784 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bahouri, H., Chemin, J.Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften 343. Springer, Berlin Heidelberg, 2011

  4. Beale J.T., Majda A.: Rates of convergence for viscous splitting of the Navier–Stokes. Math. Comp. 37, 243–259 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chemin J.-Y., Gallagher I., Paicu M.: Global regularity for some classes of large solutions to the Navier–Stokes equations. Ann. Math. 173, 983–1012 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Constantin, P., Kukavica, I., Vicol, V.: On the inviscid limit of the Navier–Stokes equations. ar**v:1403.5748v2

  7. Gerard-Varet D., Dormy E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23, 591–609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gérard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. ar**v:1305.0221

  9. Grard-Varet D., Nguyen T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal., 77, 71–88 (2012)

    MathSciNet  MATH  Google Scholar 

  10. Guo Y., Nguyen T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64, 1416–1438 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, Y., Nguyen, T.: Prandtl boundary layer expansions of steady Navier–Stokes flows over a moving plate. ar**v:1411.6984v1

  12. Iftimie D., Planas G.: Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity 19, 899–918 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Iftimie D., Sueur F.: Viscous boundary layer for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199, 145–175 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kato T.: Nonstationary flows of viscous and ideal fluids in R 3. J. Funct. Anal. 9, 296–305 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), pp. 85–98. Mathematical Sciences Research Institute Publications, 2. Springer, New York, 1984

  16. Kelliher J.P.: On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J. 56, 1711–1721 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kelliher J.P.: On the vanishing viscosity limit in a disk. Math. Ann. 343, 701–726 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kukavica I., Masmoudi N., Vicol V., Wong T.: On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46, 3865–3890 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, C., Wang, Y., Yang, T.: On the ill-posedness of the Prandtl equations in three space dimensions, ar**v:1412.2843

  20. Liu, C., Wang, Y., Yang, T.: Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. ar**v:1509.03856

  21. Lombardo M. C., Cannone M., Sammartino M.: Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35, 987–1004 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maekawa Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67, 1045–1128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Masmoudi N., Rousset F.: Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal. 203, 529–575 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Masmoudi, N., Rousset, F.: Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations. ar**v:1202.0657

  25. Masmoudi N., Wong T.K.: Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Commun. Pure Appl. Math. 68, 1683–1741 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation 15. Chapman & Hall/CRC, Boca Raton, 1999

  27. Paicu M., Zhang Z.: Global regularity for the Navier–Stokes equations with some classes of large initial data. Anal. PDE 4, 95–113 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Prandtl, L.: Uber flüssigkeits-bewegung bei sehr kleiner reibung. Actes du 3me Congrés international dse Mathématiciens. Teubner, Leipzig, Heidelberg, pp. 484–491, 1904

  29. Sammartino M., Caflisch R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Commun. Math. Phys. 192, 433–461 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Sammartino M., Caflisch R.E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. II. Construction of the Navier–Stokes solution. Commun. Math. Phys. 192, 463–491 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Swann H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in R 3. Trans. Am. Math. Soc. 157, 373–397 (1971)

    MathSciNet  MATH  Google Scholar 

  32. Temam R., Wang X.: On the behavior of the solutions of the Navier–Stokes equations at vanishing viscosity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 807–828 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Wang X.: A Kato type theorem on zero viscosity limit of Navier–Stokes flows. Indiana Univ. Math. J. 50, 223–241 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang L., **n Z., Zang A.: Vanishing viscous limits for 3D Navier–Stokes equations with a Navier slip boundary condition. J. Math. Fluid Mech. 14, 791–825 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Wang Y., **n Z., Yong Y.: Uniform regularity and vanishing viscosity limit for the compressible Navier–Stokes with general Navier-Slip boundary conditions in three-dimensional domains. SIAM J. Math. Anal. 47, 4123–4191 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. **ao Y., **n Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60, 1027–1055 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. **n Z., Zhang L.: On the global existence of solutions to the Prandtl system. Adv. Math. 181, 88–133 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang P., Zhang Z.: Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal. 270, 2591–2615 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Bei**g, 100871, China

    Chao Wang, Yuxi Wang & Zhifei Zhang

Authors
  1. Chao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuxi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhifei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhifei Zhang.

Additional information

Communicated by N. Masmoudi

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, C., Wang, Y. & Zhang, Z. Zero-Viscosity Limit of the Navier–Stokes Equations in the Analytic Setting. Arch Rational Mech Anal 224, 555–595 (2017). https://doi.org/10.1007/s00205-017-1083-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1083-6

Search

Navigation