The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

Abstract

Abstract. It is shown both locally and globally that L

A necessary condition of potential blowup for the Navier–Stokes system in half-space

“…We refer to X , consisting of measurable functions acting on domains in R 3 , as critical if for u ∈ X such that u λ (x) = λu(λx) we have u λ X = u X . In [15,28], it was shown that when X is the non endpoint Lorentz space with q finite L 3,q (R 3 ) condition (1.6) remains to be true. Recently, Gallagher et al proved in [4] that (1.6) holds for X =Ḃ −1+ holds true?…”

Anisotropic Hardy-Lorentz spaces and their applications

“…For a systematic treatment of Lorentz spaces as well as their dual spaces, we refer the reader to Hunt [37], Cwikel [21] and Cwikel and Fefferman [22,23]; see also [7,8,34,61,66]. It is well known that, due to their fine structures, Lorentz spaces play an irreplaceable role in the study on various critical or endpoint analysis problems from many different research fields and there exists a lot of literatures on this subject, here we only mention several recent papers from harmonic analysis (see, for example, [55,53,63,68]) and partial differential equations (see, for example, [38,51,58]). …”

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

“…The endpoint case r = d, q = ∞ is much more subtle, and it was until 2003 that Escauriaza, Seregin and Sverak [18] solved this endpoint case in 3D, later, Dong and Du [15] extended their result to the case d ≥ 3. On the other hand, there are lots of interests in relaxing the condition (1.2), for instance, Phuc showed the same conclusion for the 3D Leray-Hopf solution u by assuming u ∈ L ∞ (0, T ; L 3,m ) with 3 ≤ m < ∞, see [38]. Besides, according to [22,4], the same result also applies for u in 3D provided u ∈ L ∞ (0, T ;Ḃ s p p,q ) with 3 < p, q < ∞, a natural extension to the higer dimension in such setting is one of aims of our current paper.…”