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

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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Correspondence to J. Xiao.

Additional information

YW was supported by NSFC No. 11201143 and AARMS Postdoctoral Fellowship; JX was supported by NSERC of Canada (FOAPAL # 202979463102000).

Communicated by D. Chae

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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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Keywords

  • Nonnegative weak solution
  • fractional Laplacian
  • uniqueness

Mathematics Subject Classification

  • 35Q30
  • 76D05