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Ill-posedness in Fourier–Herz spaces of the 3D Navier–Stokes equations with hyper-dissipation

Volume 152 / 2018

Zhijie Nan, Jia Yuan Colloquium Mathematicum 152 (2018), 123-140 MSC: 35Q30, 35B30, 76D03. DOI: 10.4064/cm7030-4-2017 Published online: 8 February 2018

Abstract

We consider the Cauchy problem in $\mathbb {R}^3$ for the 3D Navier–Stokes equations with hyper-dissipation. In [Z. Nan and X. Zheng, J. Differential Equations 6 (2016), 3670–3703], it is shown that there exists a unique global-in-time solution for initial data belonging to the critical Fourier–Herz spaces $F\dot{B}^{{3/2}-{3/r}}_{r,q}$ with $r \gt {12/11}$. In this paper, we study this problem in the border case $r={12/11}$. Using the localization technique in frequency space, we prove that this Cauchy problem is well-posed in $F\dot{B}^{-{5/4}}_{{12/11},2}$ but ill-posed in $F\dot{B}^{-{5/4}}_{{12/11},q}$ with $q \gt 2$.

Authors

  • Zhijie NanSchool of Mathematics and System Science
    Beihang University
    Beijing 100191, P.R. China
    and
    Department of Mathematics
    and Information Engineering
    Nanhu College
    Jiaxing University
    Zhejiang 314001, P.R. China
    e-mail
  • Jia YuanSchool of Mathematics and System Science
    Beihang University
    Beijing 100191, P.R. China
    e-mail

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