Dispersive effect of the Coriolis force and the local well-posedness for the fractional Navier–Stokes–Coriolis system

Abstract

This paper discusses the Cauchy problem for the fractional Navier–Stokes–Coriolis equation (FNSC). The FNSC equation refers to that obtained by replacing the Laplacian in the Navier–Stokes–Coriolis equation by the more general operator \((-\Delta )^{\alpha }\) with \(\alpha >0\). We prove the time-local existence and uniqueness of the mild solution for every \(\varOmega \in \mathbb {R}\backslash \{0\}\) and \(u_0\in \dot{H}^s(\mathbb {R}^3)^3\) with \(1/4<\alpha \leqslant 3/2\), \(3/2-\alpha<\,s\, <5/4\). Furthermore, we give a lower bound for the time interval of its local existence in terms of \(|\varOmega |\) and \(\Vert u_0\Vert _{\dot{H}^s}\). It follows from our characterization that the existence time T of the solution can be arbitrarily large provided the speed of rotation \(|\varOmega |\) is sufficiently fast.

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Acknowledgements

The authors would like to express their deep gratitude to the referee for giving many valuable suggestions. X. Sun is supported by NSF of China (Grant: 11461059, 11561062, 11601434), SRPNWNU (Grant: NWNU-LKQW-14-2). Y. Ding is supported by NSF of China (Grant: 11371057) and SRFDP of China (Grant: 20130003110003).

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Sun, X., Ding, Y. Dispersive effect of the Coriolis force and the local well-posedness for the fractional Navier–Stokes–Coriolis system. J. Evol. Equ. 20, 335–354 (2020). https://doi.org/10.1007/s00028-019-00531-7

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Keywords

  • Cauchy problem
  • The generalized Navier–Stokes–Coriolis equation
  • Local well-posedness
  • Dispersive effect

Mathematics Subject Classification

  • 35R11
  • 35Q30
  • 35Q35
  • 76B03
  • 76U05
  • 42B37