This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions d ≥ 4. By establishing an ǫ regularity criterion in the spirit of [9], we show that if the mild solution u with initial data inḂThe corresponding result in 3D case has been obtained in [22]. As a by-product, we also prove a regularity criterion for the Leray-Hopf solution in the critical Besov space, which generalizes the results in [15], where blowup criterion in critical Lebesgue space L d (R d ) is obtained. * Corresponding Author

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 < \infty ) for which the norms are defined by \|f\|_{E_{p,q}^{s}} = \left(\:\sum \limits_{k \in \mathbb{Z}^{d}}2^{s|k|q}\|\mathscr{F}^{−1}\chi _{k + [0,1]^{d}}\mathscr{F}f\|_{p}^{q}\right)^{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 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 \mathbb{R}_{I}^{d}: = \{\xi \in \mathbb{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 u_{0} is in L^{2} with \mathrm{\sup p}\:\hat{u}_{0}\: \subset \mathbb{R}_{I}^{d} .

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We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works [6,7,9]. The proof relies on the classical Bona-Smith method as [12], where similar result was obtained in critical Besov spaces B 1 ∞,1 .