Abstract
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).
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References
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)
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)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
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)
Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. 171, 1903–1930 (2010)
Chae, D.: Remarks on the Liouville type results for the compressible Navier–Stokes equations in \({\mathbb{R}}^N\). Nonlinearity 25, 1345–1349 (2012)
Chae, D.: Liouville-type theorems for the forced Euler equations and the Navier–Stokes equations. Commun. Math. Phys. 326, 37–48 (2014)
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)
Chae, D., Wolf, J.: On Liouville type theorems for steady Navier–Stokes equations in \(\mathbb{R}^3\). J. Differ. Equ. 261, 5541–5560 (2016)
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)
Engelberg, S., Liu, H.L., Tadmor, E.: Critical thresholds in Euler–Poisson equations. Indiana Univ. Math. J. 50, 109–157 (2001)
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)
Grafakos, L.: Classical Fourier Analysis. Grad. Texts in Math, 2nd edn. Springer, New York (2008)
Holm, D., Johnson, S.F., Lonngern, K.E.: Expansion of a cold ion cloud. Appl. Phys. Lett. 38, 519–521 (1981)
Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, New York (1975)
Perthame, B.: Nonexistence of global solutions to the Euler–Poisson equations for repulsive forces. Jpn J. Appl. Math. 7, 363–367 (1990)
Struwe, M.: On partial regularity results for the Navier–Stokes equations. Commun. Pure Appl. Math. 41, 437–458 (1988)
Tang, L., Yu, Y.: Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations. Commun. Math. Phys. 334, 1455–1482 (2015)
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)
Xiao, J.: A sharp Sobolev trace inequality for the fractional-order derivatives. Bull. Sci. Math. 130, 87–96 (2006)
Zhang, X.: Stochastic Lagrangian particle approach to fractal Navier–Stokes equations. Commun. Math. Phys. 311, 133–155 (2012)
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Communicated by D. Chae
YW was supported by NSFC No. 11201143 and AARMS Postdoctoral Fellowship; JX was supported by NSERC of Canada (FOAPAL # 202979463102000).
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Wang, Y., Xiao, J. A Liouville Problem for the Stationary Fractional Navier–Stokes–Poisson System. J. Math. Fluid Mech. 20, 485–498 (2018). https://doi.org/10.1007/s00021-017-0330-9
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DOI: https://doi.org/10.1007/s00021-017-0330-9