Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 2, pp 485–498 | Cite as

A Liouville Problem for the Stationary Fractional Navier–Stokes–Poisson System

  • Y. Wang
  • J. XiaoEmail author


This paper deals with a Liouville problem for the stationary fractional Navier–Stokes–Poisson system whose special case \(k=0\) covers the compressible and incompressible time-independent fractional Navier–Stokes systems in \(\mathbb {R}^{N\ge 2}\). An essential difficulty raises from the fractional Laplacian, which is a non-local operator and thus makes the local analysis unsuitable. To overcome the difficulty, we utilize a recently-introduced extension-method in Wang and Xiao (Commun Contemp Math 18(6):1650019, 2016) which develops Caffarelli-Silvestre’s technique in Caffarelli and Silvestre (Commun Partial Diff Equ 32:1245–1260, 2007).


Nonnegative weak solution fractional Laplacian uniqueness 

Mathematics Subject Classification

35Q30 76D05 


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  1. 1.
    Brandle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave–convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 143, 39–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brauer, U., Rendal, A., Reula, O.: The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models. Class. Quantum Gravity 11, 2283–2296 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903–1930 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chae, D.: Remarks on the Liouville type results for the compressible Navier–Stokes equations in \({\mathbb{R}}^N\). Nonlinearity 25, 1345–1349 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chae, D.: Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326, 37–48 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chae, D., Constantin, P., Wu, J.: Dissipative models generalizing the 2D Navier–Stokes and surface quasi-geostrophic equations. Indiana Univ. Math. J. 61, 1997–2018 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chae, D., Wolf, J.: On Liouville type theorems for steady Navier–Stokes equations in \(\mathbb{R}^3\). J. Differ. Equ. 261, 5541–5560 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chae, D., Yonedab, T.: On the Liouville theorem for the stationary Navier–Stokes equations in a critical space. J. Math. Anal. Appl. 405, 706–710 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Engelberg, S., Liu, H.L., Tadmor, E.: Critical thresholds in Euler–Poisson equations. Indiana Univ. Math. J. 50, 109–157 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations Vol. II, nonlinear steady problems. In: Springer Tracts in Natural Philosophy 39, Springer, New York (1994)Google Scholar
  13. 13.
    Grafakos, L.: Classical Fourier Analysis. Grad. Texts in Math, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  14. 14.
    Holm, D., Johnson, S.F., Lonngern, K.E.: Expansion of a cold ion cloud. Appl. Phys. Lett. 38, 519–521 (1981)ADSCrossRefGoogle Scholar
  15. 15.
    Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, New York (1975)zbMATHGoogle Scholar
  16. 16.
    Perthame, B.: Nonexistence of global solutions to the Euler–Poisson equations for repulsive forces. Jpn J. Appl. Math. 7, 363–367 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Struwe, M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–458 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tang, L., Yu, Y.: Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations. Commun. Math. Phys. 334, 1455–1482 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, Y., Xiao, J.: A uniqueness principle for \(u^p\le (-\Delta )^\frac{\alpha }{2}u\) in the Euclidean space. Commun. Contemp. Math. 18(6), 1650019 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Xiao, J.: A sharp Sobolev trace inequality for the fractional-order derivatives. Bull. Sci. Math. 130, 87–96 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, X.: Stochastic Lagrangian particle approach to fractal Navier–Stokes equations. Commun. Math. Phys. 311, 133–155 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  2. 2.School of MathematicsThe University of EdinburghEdinburghUnited Kingdom
  3. 3.Department of Mathematics and StatisticsMemorial UniversitySt. John’sCanada

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