A profile decomposition approach to the $$L^\infty _t(L^{3}_x)$$ Navier–Stokes regularity criterion

Abstract

Abstract. In this paper we continue to develop an alternative viewpoint on recent studies of Navier-Stokes regularity in critical spaces, a program which was started in the recent work [14] by C. Kenig and the second author. Specifically, we prove that strong solutions which remain bounded in the space L 3 (R 3 ) do not become singular in finite time, a known result established in [8] by Escauriaza, Seregin andŠverák in the context of suitable weak solutions. Here, we use the method of "critical elements" which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a "profile decomposition" for the Navier-Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier-Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin andŠverák [23]. IntroductionWe consider the incompressible Navier-Stokes equations in R d , for (x, t) ∈ R d × (0, T ), where u = u(x, t) is the velocity vector field and π(x, t) is the associated pressure function. For d ≥ 3, global weak solutions are known to exist, but their uniqueness (as well as the smoothness of the solution for smooth data) has remained an open problem since the pioneering work [21]. There exist several conditionalThe first author was partially supported by the A.N.R grant ANR-08-BLAN-0301-01 "Mathocéan", as well as the Institut Universitaire de France. The second author was supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). The third author was partially supported by A.N.R. grant SWAP. 2 ISABELLE GALLAGHER, GABRIEL S. KOCH, AND FABRICE PLANCHONresults, of which Serrin's criterion is perhaps the most well-known: if a weak solution u is such thatthen u is smooth on (0, T ). On the other hand, there is a long line of work on constructing local in time solutions, from [13] to [20]. In this framework of local in time (strong, e.g. unique) solutions, Serrin's criterion may be understood as a non blow-up criterion at time T : e.g. if u is a strong solution with, and if (0.2) is satisfied, then one may (continuously and uniquely) extend the solution u past time T .In the recent important work [8], Escauriaza-Seregin-Šverák obtained the endpoint version of Serrin's criterion 1 : u ∈ L ∞ ([0, T ]; L 3 (R 3 )) implies no blow-up; they work with the so-called suitable weak solutions introduced in [4]. A similar, but much more precise result in the (smaller) space L ∞ ([0, T ];Ḣ 1 2 (R 3 )) was obtained recently by Seregin in [25]; the main point in that result is that it is proved that theḢ 1 2 norm of the solution u(t) blows up as t goes to blow-up time, and not just for a subsequence as is known in the L 3 (R 3 ) case (see [24] for a partial result in that direction) 2 . Our goal here is to obtain such a result by a somewhat different route, following the concentration-compactness methods developed by KenigMerle in the context ...

An alternative approach to regularity for the Navier–Stokes equations in critical spaces

“…[14]). 5 In Theorems 3.1 and 3.2, we establish the existence of a solution ("critical element") u c with initial datum u 0,c and T * (u 0,c ) < +∞ such that 4 At the time of publication of this article, an L 3 (R 3 ) profile decomposition has been established by the second author in [33], and the program in L 3 (R 3 ) has been completed in the collaboration [22] of the second author with I. Gallagher and F. Planchon. 5 Strictly speaking, this condition applies to "Leray-Hopf" weak solutions, but in fact the local theory gives √ tu ∈ L ∞ which implies smoothness.…”

“…22 As noted previously, for any η ∈ (0, 1) one can, of course, find a similar sequence which converges to U 1 (η · T * (V 0,1 )).…”

“…In fact, it has now been pointed out in[22] that one does not actually need to use a result such as Lemma 3.6 to prove Theorems 3.1 and 3.2.…”

The Navier–Stokes Equations in Nonendpoint Borderline Lorentz Spaces

“…See also the papers [7,13] for another approach to regularity using certain profile decompositions. It should be noticed that the uniqueness of u under condition (1.4) had been known earlier (see [15]).…”

Some remarks about the possible blow-up for the Navier-Stokes equations

“…The following law of product in R 2 is also useful. 13 Lemma A.3. For any functions a ∈ L ∞ ∩Ḣ 1 (R 2 ) and b ∈Ḣ 1 2 (R 2 ), there holds ab Ḣ 1…”