Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4

Abstract

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

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

“…The proof of this is along the lines of Proposition 2.3 of [36]. We will also make use of a decomposition result for Homogeneous Besov spaces.…”

“…See [44] and [13] for local extensions, as well as [16] and [36] for global extensions. Later in [47], Seregin improved (9):…”

Alternative Theorem of Navier-Stokes Equations in $\mathbb{R}^3$

“…For the initial data in critical spaces, the posedness of global solution of the equations (1.7) is obtained [5,10,26,35] with small initial data. The regularity criterion was established [12,13,24,31,42]. On the other hand, the ill-posedness was showed [2,14,50,53].…”

Exponential Decay and Regularity of Global Solutions for the 3D Navier–Stokes Equations Posed on Lipschitz and Smooth Domains

“…Recent works approach the criticality of the dimension 4 and establish criteria over spaces or domains. In [11] it is shown that for dimensions bigger than three, blow up for some finite time. In [9], the existence, uniqueness and exponential decay of global strong solution to (1.1)-(1.3) posed on Lipschitz bounded and unbounded domains have been established assuming some connections between the geometric sizes of domains and initial data.…”