Navier-Stokes equation in super-critical spaces Ep,qs

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 E-p(,q)s with exponentially decaying weights (s < 0, 1 < p, q < infinity) for which the norms are defined by parallel to f parallel to E-p,q(s )= (Sigma(k is an element of Zd)2(s vertical bar k vertical bar q)parallel to F-1 chi(k+[0,1])dFf parallel to(q)(p))(1/q) The space E-p,(q)s is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding H-sigma subset of E-2,(1)s for any sigma < 0 and s < 0. It is known that H-sigma (sigma < d/2 - 1) is a super-critical space of NS, it follows that E-2,(1)s (s < 0) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to E-2,(1)s (s < 0) and their Fourier transforms are supported in R-I(d) := {xi is an element of R-d : xi(i) >= 0, i = 1, ..., d}. Similar results hold for the initial data in E-r,(1)s with 2 < r <= d. Our results imply that NS has a unique global solution if the initial value up is in L-2 with supp (u) over cap0 subset of R-I(d) (C) 2020 L'Association Publications de l'Institut Henri Poincare. Published by Elsevier B.V. All rights reserved.