Navier–Stokes equation in super-critical spaces

  • Hans G. Feichtinger

    Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
  • Karlheinz Gröchenig

    Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
  • Kuijie Li

    School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
  • Baoxiang Wang

    LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
Navier–Stokes equation in super-critical spaces \( E_{p , q}^{s} \) cover

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Abstract

In this paper we develop a new way to study the global existence and uniqueness for the Navier–Stokes equation (NS) and consider the initial data in a class of modulation spaces with exponentially decaying weights for which the norms are defined by

The space is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding for any and . It is known that () is a super-critical space of NS, it follows that () is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to () and their Fourier transforms are supported in . Similar results hold for the initial data in with . Our results imply that NS has a unique global solution if the initial value is in with .

Cite this article

Hans G. Feichtinger, Karlheinz Gröchenig, Kuijie Li, Baoxiang Wang, Navier–Stokes equation in super-critical spaces . Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 139–173

DOI 10.1016/J.ANIHPC.2020.06.002