Advertisement

Acta Mathematica Sinica, English Series

, Volume 34, Issue 4, pp 655–661 | Cite as

Extension Problems Related to the Higher Order Fractional Laplacian

  • Yu Kang Chen
  • Zhen Lei
  • Chang Hua Wei
Article
  • 105 Downloads

Abstract

Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)] characterized the fractional Laplacian (−Δ) s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 < s < 1. In this paper, we extend this result to all s > 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of L p norm using the Caffarelli–Silvestre’s extension technique.

Keywords

Fractional Laplacian quasi-geostrophic equations energy equality 

MR(2010) Subject Classification

42B37 35P30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Brezis, H.: Analyse Fonctionnelle. Thorie et applications. Collect. Math. Appl. Maitrise, Masson, Paris, 1993Google Scholar
  2. [2]
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math., 35, 771–831 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. of Math., 171, 1903–1930 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Capella, A., Dávila, J., Dupaigne, L., et al.: Regularity of radial extremal solutions for some non local semilinear equations. Comm. Part. Diff. Eqs., 36, 1353–1384 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chang, S. Y. A., González, M. d. M.: Fractional Laplacian in conformai geometry. Adv. Math., 226, 1410–1432 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen, Y., Wei, C.: Partial regularity of solutions to the fractional Navier–Stokes equations. Discrete Contin. Dyn. Syst., 36(10), 5309–5322 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Chen, Q., Miao, C., Zhang, Z.: New Bernstein’s inequality and the 2D dissipative quasi-geostrophic equa-tion. Comm. Math. Phys., 271, 821–838 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Córdoba, A., Córdoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys., 249, 511–528 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys., 255, 161–181 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Katz, N. H., Pavlovic, N.: A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation. Geom. Funct. Anal., 12, 355–379 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Li, D.: On a frequency localized Bernstein inequality and some generalized Poincáre-type inequalities. Math. Res. Lett., 20, 933–945 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Ren, W., Wang, Y., Wu, G.: Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations. Commun. Contemp. Math., 18(6), 1650018 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago, Chicago, 1995Google Scholar
  15. [15]
    Tang, L., Yu, Y.: Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations. Comm. Math. Phys., 334, 1455–1482 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Yang, R.: On higher order extensions for the fractional Laplacian. arXiv: 1302.4413 (2013)Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiP. R. China
  2. 2.School of Mathematical Sciences and Shanghai Center for Mathematical SciencesFudan UniversityShanghaiP. R. China
  3. 3.Department of MathematicsZhejiang Sci-Tech UniversityHangzhouP. R. China

Personalised recommendations