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. 139–173DOI 10.1016/J.ANIHPC.2020.06.002