Navier-Stokes equation in super-critical spaces [formula omitted

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 Ep,qs with exponentially decaying weights (s<0,1<p,q<∞) for which the norms are defined by‖f‖Ep,qs=(∑k∈Zd2s|k|q‖F−1χk+[0,1]dFf‖pq)1/q. The space Ep,qs is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding Hσ⊂E2,1s for any σ<0 and s<0. It is known that Hσ (σ<d/2−1) is a super-critical space of NS, it follows that E2,1s (s<0) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to E2,1s (s<0) and their Fourier transforms are supported in RId:={ξ∈Rd:ξi⩾0,i=1,...,d}. Similar results hold for the initial data in Er,1s with 2<r⩽d. Our results imply that NS has a unique global solution if the initial value u0 is in L2 with suppuˆ0⊂RId.