The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1490–1519 | Cite as

N–S Systems via \(\mathcal {Q}\)\(\mathcal {Q}^{-1}\) Spaces

  • Jie XiaoEmail author
  • Junjie Zhang


Under \((\alpha ,p,n-1)\in (-\infty ,1)\times (2,\infty )\times {\mathbb {N}}\), this paper uses \(\mathcal {Q}_\alpha ({\mathbb {R}}^n)\) and \(\mathcal {Q}^{-1}_{\alpha }(\mathbb R^n):=\hbox {div}\big (\mathcal {Q}_{\alpha }({\mathbb {R}}^{n})\big )^n\) (covering \(\mathrm{BMO}({\mathbb {R}}^n)\) and \(\mathrm{BMO}^{-1}({\mathbb {R}}^n)\)) where \( f\in \mathcal {Q}_\alpha ({\mathbb {R}}^{n}) \Leftrightarrow \int \nolimits _{\mathbb R^n}\frac{|f(x)|}{1+|x|^{n+1}}\,\mathrm{d}x<\infty \ \ \mathrm{and} \ \ \underset{\mathrm{coordinate}\, \mathrm{cubes}\, I}{\sup }\left( \iint \nolimits _{I\times (0,\ell (I))}|\nabla e^{t^2\Delta }f(x)|^2\right. \left. \frac{\omega _{\alpha }\left( \frac{t}{\ell (I)}\right) }{t^{n-1}}{\mathrm{d}x\mathrm{d}t}\right) ^\frac{1}{2}<\infty \) with \( (0,1]\ni s\mapsto \omega _\alpha (s)={\left\{ \begin{array}{ll} s^n\quad &{}\hbox {as}\quad \alpha \in (-\infty ,0);\\ s^n\big (\ln \frac{e}{s}\Big )^2\quad &{}\hbox {as}\quad \alpha =0;\\ s^{n-2\alpha }\quad &{}\hbox {as}\quad \alpha \in (0,1), \end{array}\right. }\) to demonstrate that the incompressible Navier–Stokes system \( {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=\partial _t u\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{1+n}_+;\\ u(0,x)=a(x) &{} \text {as } x\in {\mathbb {R}}^{n} \end{array}\right. }\) has a unique mild solution under \(\Vert a\Vert _{\big (\mathcal {Q}_\alpha ^{-1}({\mathbb {R}}^n)\big )^n}\) being sufficiently small; however, its steady state\( {\left\{ \begin{array}{ll} \Delta u-(u \cdot \nabla ) u+\nabla \mathrm{p}=0\ \ \mathrm{and}\ \ \hbox {div}\,u=0 &{} \text {in } {\mathbb {R}}^{n};\\ u(x)\rightarrow 0 &{} \text {as}\ \infty \leftarrow x\in {\mathbb {R}}^{n} \end{array}\right. }\) has only zero solution under \(u\in \big (\mathrm{BMO}^{-1}(\mathbb R^n)\cap \mathscr {L}^{p,\frac{p(n-2)}{2}}({\mathbb {R}}^n)\big )^n\).


N–S systems \(\mathcal {Q}\)\(\mathcal {Q}^{-1}\) spaces Mild–null solutions 

Mathematics Subject Classification

30H25 35Q30 42B37 46E35 


  1. 1.
    Bao, G., Wulan, H.: \(Q_{K}\) spaces of several real variables. Abstr. Appl. Anal. 2014.
  2. 2.
    Bourgain, J., Pavlović, N.: Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J. Funct. Anal. 255, 2233–2247 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cannone, M., Meyer, Y., Planchon, F.: Solutions auto-similaires des équations de Navier-Stokes in \({\mathbb{R}}^3\), Exposé n. VIII, Séminaire X-EDP. Ecole Polytechnique, Palaiseau (1963)Google Scholar
  4. 4.
    Chae, D., Wolf, J.: On Liouville type theorems for steady Navier-Stokes equations in \({\mathbb{R}}^3\). J. Differ. Equ. 261, 5541–5560 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dafni, G., Xiao, J.: Some new tent spaces and duality theorems for fractional Carleson measures and \(\cal{Q}_\alpha ({\mathbb{R}}^n)\). J. Funct. Anal. 208, 377–422 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dafni, G., Xiao, J.: The dyadic structure and atomic decomposition of \(Q\) spaces in several variables. Tohoku Math. J. 57, 119–145 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Escauriaza, L., Seregin, G., Shverak, V.: \(L_{3,\infty }\)-solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk. 58, 3 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Essén, M., Janson, S., Peng, L., Xiao, J.: \(Q\) spaces of several real variables. Indiana Univ. Math. J. 49, 575–615 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Farhat, A., Grujić, Z., Leitmeyer, K.: The space \(B^{-1}_{\infty ,\infty }\), volumetric sparseness, and 3D NSE. arXiv:1603.08763v5 [math.AP] 16 Nov 2016
  10. 10.
    Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer Tracts in Natural Philosophy 39, vol. 2. Springer, New York (1994)CrossRefGoogle Scholar
  12. 12.
    Giaquinta, M.: Multiple Inegrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematical Studies. Princeton University Press, Princeton (1983)Google Scholar
  13. 13.
    Gogatishvili, A., Mustafayev, RCh.: A note on boundedness of the Hardy-Littlewood maximal operator on Morrey spaces. Mediterr. J. Math. 13, 1885–1891 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., London (2004)zbMATHGoogle Scholar
  15. 15.
    Hmidi, T., Li, D.: Small \(\dot{B}^{-1}_{\infty,\infty }\) implies regularity. Dyn. Partial Differ. Equ. 14, 1–4 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang, R., Xiao, J., Yang, D.: Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants. Anal. Appl. 14, 679–703 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kato, T.: Strong solutions of the Navier-Stokes equations in Morrey spaces. Bol. Soc. Brasil. Math. 22, 127–155 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157, 22–35 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koskela, P., Xiao, J., Zhang, Y., Zhou, Y.: A quasiconformal composition problem for the \(Q\)-spaces. J. Eur. Math. Soc. 19, 1159–1187 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kufner, A., Persson, I.-E.: Weighted Inequalities of Hardy Type. World Scientific Publishing, River Edge (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapman and Hall/CRC, Boca Raton (2002)CrossRefzbMATHGoogle Scholar
  23. 23.
    Lemarié-Rieusset, P.G.: The Navier-Stokes equations in the critical Morrey-Campanato space. Rev. Mat. Iberoam 23, 897–930 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lemarié-Rieusset, P.G.: The Navier-Stokes Problem in the 21st Century. CRC Press, Boca Raton (2016)CrossRefzbMATHGoogle Scholar
  25. 25.
    Mazýa, V.G., Verbitsky, I.E.: Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator. Invent. Math. 162, 81–136 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Seregin, G.: Liouville type theorem for stationary Navier-Stokes equations. Nonlinearity 29, 2191–2195 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Seregin, G.: A Liouville type theorem for steady-state Navier-Stokes equations. arXiv:1611.0156v1 [math.AP] 4 Nov 2016
  28. 28.
    Strichartz, R.S.: Bounded mean oscillation and Sobolev spaces. Indiana Univ. Math. J. 29, 539–558 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, Y., Xiao, J.: Homogeneous Campanato-Sobolev classes. Appl. Comput. Harmon. Anal. 39, 214–247 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, Z., Xiao, J., Zhou, Y.: The \({\cal{Q}}_{\alpha }\)-restriction problem (2016) (preprint)Google Scholar
  31. 31.
    Xiao, J.: Homothetic variant of fractional Sobolev space with application to Navier-Stokes system. Dyn. Partial Differ. Equ. 4, 227–245 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Xiao, J.: Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited. Dyn. Partial Differ. Equ. 11, 167–181 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Xiao, J.: A sharp Sobolev trace inequality for the fractional-order derivatives. Bull. Sci. Math. 130, 87–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Yang, Q., Qian, T., Li, P.: Fefferman-Stein decomposition for \(Q\)-spaces and micro-local quantities. Nonlinear Anal. 145, 24–48 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yue, H., Dafni, G.: A John-Nirenberg type inequality for \(\cal{Q}_\alpha ({\mathbb{R}}^n)\). J. Math. Anal. Appl. 351, 428–439 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada
  2. 2.Department of MathematicsBeijing Jiaotong UniversityBeijingChina

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