Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 47–54 | Cite as

On the weak solutions to steady Navier-Stokes equations with mixed boundary conditions

  • Yanren HouEmail author
  • Shuaichao Pei
Article
  • 117 Downloads

Abstract

In this paper, for the Navier-Stokes equations in a bounded connected polygon or polyhedron \(\Omega \subset R^d\), \(d=2,3\), with a homogenous stress type mixed boundary condition, we establish an a priori estimate for the weak solutions and the existence result without small data and/or large viscosity restriction. And a global uniqueness result is obvious based on the a priori estimate obtained.

Keywords

Navier-Stokes equations Mixed boundary conditions A priori estimate Existence Global uniqueness 

References

  1. 1.
    Bernardi, C., Hecht, F., Verfürth, R.: A finite element discretization of the three-dimensional Navier-Stokes equations with mixed boundary conditions. ESAIM: Math. Model. Numer. Anal. 43, 1185–1201 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brenner, S., Scott, L.: The mathematical theory of finite element methods. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ciarlet, P.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  4. 4.
    Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations: theory and algorithms. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  5. 5.
    Girault, V., Rivière, B.: DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47, 2052–2089 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, K., An, R.: On the rotating Navier-Stokes equations with mixed boundary conditions. Acta Mathematica Sinica, English Series 24(4), 577–598 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Maz’ya, V., Rossmann, J.: Mixed boundary value problems for the stationary Navier-Stokes system in polyhedral domains. Arch. Rational Mech. Anal 194, 669–712 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Scott, L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Temam, R.: Navier-Stokes equations, theory and numerical analysis. North-Holland, Amsterdam (1977)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina

Personalised recommendations