Partial Regularity for Stationary Navier-Stokes Systems by the Method of \(\mathcal{A}\)-Harmonic Approximation


In this article, we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition. The proof is based on the \(\mathcal{A}\)-harmonic approximation technique. In this article, we extend the result of Shuhong Chen and Zhong Tan [7] and Giaquinta and Modica [18] to the stationary Navier-Stokes system with subquadratic growth.

This is a preview of subscription content, access via your institution.


  1. [1]

    Acerbi E, Fusco N. Regularity for minimizers of nonquadratic functional: the case 1 < p < 2. J Math Anal Appl, 1989, 140(1): 115–135

    MathSciNet  Article  Google Scholar 

  2. [2]

    Allard W K. On the first variation of a varifold. Annals of Math, 1972, 95: 225–254

    MathSciNet  Article  Google Scholar 

  3. [3]

    Agmon S, Douglis A, Nirenberg L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm Pure Appl Math, 1964, 17: 35–92

    MathSciNet  Article  Google Scholar 

  4. [4]

    Campanato, S. Equazioni paraboliche del secondo ordine e spazi L2,θ(Ω). Ann Mat Pura Appl, 1966, 73(4): 55–102

    MathSciNet  Article  Google Scholar 

  5. [5]

    Carozza M, Fusco N, Mingione G. Partial regularity of minimizers of quasiconvex integrals with sub-quadratic growth. Ann Mat Pura Appl, 1998, 175(4): 141–164

    MathSciNet  Article  Google Scholar 

  6. [6]

    Chen S, Tan Z. The method of \(\mathcal{A}\)-harmonic approximation and optimal interior partial regularity for nonlinear elliptic systems under the controllable growth condition. J Math Anal Appl, 2007, 335(1): 20–42

    MathSciNet  Article  Google Scholar 

  7. [7]

    Chen S, Tan Z. Partial regularity for weak solutions of stationary Navier-Stokes systems. Acta Mathematica Scientia, 2008, 28(4): 877–894

    MathSciNet  Article  Google Scholar 

  8. [8]

    Chen S, Tan Z. Optimal partial regularity for nonlinear sub-elliptic systems. J Math Anal Appl, 2012, 387(1): 166–180

    MathSciNet  Article  Google Scholar 

  9. [9]

    Da Prato G. Spazi ℒp,υ(Ω,δ) e loro proprietà. Ann Mat Pura Appl, 1965, 69: 383–392

    MathSciNet  Article  Google Scholar 

  10. [10]

    Dai Y, Tan Z, Chen S. Partial regularity for subquadratic parabolic systems under controllable growth conditions. J Math Anal Appl, 2016, 439: 481–513

    MathSciNet  Article  Google Scholar 

  11. [11]

    De Giorgi E. Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61. Editrice Tecnico Scientifica, Pisa, 1961

    Google Scholar 

  12. [12]

    Douglis A, Nirenberg L. Interior estimates for elliptic systems of partial differential equations. Comm Pure Appl Math, 1955, 8: 503–538

    MathSciNet  Article  Google Scholar 

  13. [13]

    Duzaar F, Grotouski J F. Optimal interior partial regularity for nonlinear elliptic systems: The method of \(\mathcal{A}\)-harmonic approximation. Manuscripta Math, 2000, 103(3): 267–298

    MathSciNet  Article  Google Scholar 

  14. [14]

    Duzaar F, Grotowski J F, Kronz M. Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Ann Mat Pura Appl, 2005, 184(4): 421–448

    MathSciNet  Article  Google Scholar 

  15. [15]

    Duzaar F, Kristensen J, Mingione G. The existence of regular boundary points for non-linear elliptic systems. J Reine Angew Math, 2007, 602: 17–58

    MathSciNet  MATH  Google Scholar 

  16. [16]

    Duzaar F, Mingione, G. Harmonic type approximation lemmas. J Math Anal Appl, 2009, 352(1): 301–335

    MathSciNet  Article  Google Scholar 

  17. [17]

    Galdi G P. An introduction to the mathematical theory of the Navier-Stokes equations, Steady-state problems. Springer Monographs in Mathematics. New York: Springer, 2011

    Google Scholar 

  18. [18]

    Giaquinta M, Modica G. Nonlinear systems of the type of the stationary Navier-Stokes system. J Reine Angew Math, 1982, 330: 173–214

    MathSciNet  MATH  Google Scholar 

  19. [19]

    Giaquinta M. Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 1983, 105

    Google Scholar 

  20. [20]

    Heywood J G. The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ Math J, 1980, 29(5): 639–681

    MathSciNet  Article  Google Scholar 

  21. [21]

    Ladyzhenskay O. A. The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach, 1969

    Google Scholar 

  22. [22]

    Mingione G. The singular set of solutions to non-differentiable elliptic systems. Arch Rat Mech Anal, 2003, 166: 287–301

    MathSciNet  Article  Google Scholar 

  23. [23]

    Temam R. Navier-Stokes Equations and Nonlinear Functional Analysis. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics, 1995

    Google Scholar 

  24. [24]

    Simon L. Lectures on Geometric Measure Theory. Canberra: Australian National University Press, 1983

    Google Scholar 

  25. [25]

    Simon L. Theorems on Regularity and Singularity of Energy Minimizing Maps. Basel, Boston, Berlin: Birkhauser, 1996

    Google Scholar 

  26. [26]

    Schoen R, Uhlenbeck K. A regularity theorem for harmonic maps. J Differential Geom, 1982, 17(2): 307–335

    MathSciNet  Article  Google Scholar 

  27. [27]

    He L H, Tan Z. Partial regularity of stationary Naiver-Stokes systems under natural growth condition. Acta Mathematica Scientia, 2019, 39B(1): 94–110

    Article  Google Scholar 

Download references

Author information



Corresponding authors

Correspondence to Yichen Dai 戴神深 or Zhong Tan 谭忠.

Electronic supplementary material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dai, Y., Tan, Z. Partial Regularity for Stationary Navier-Stokes Systems by the Method of \(\mathcal{A}\)-Harmonic Approximation. Acta Math Sci 40, 835–854 (2020).

Download citation

Key words

  • Stationary Navier-Stokes systems
  • controllable growth condition
  • partial regularity
  • \(\mathcal{A}\)-harmonic approximation

2010 MR Subject Classification

  • 35J60
  • 35Q30
  • 76N10