On stability of weak Navier–Stokes solutions with large L3, ∞ initial data. Issue 4 (3rd April 2018)

• Record Type: Journal Article
• Authors: Barker, T.; Seregin, G.; Šverák, V.
• Is Part Of: Communications in partial differential equations. Volume 43:Issue 4(2018)
• Page Start: 628
• Page End: 651
• Publication Date: 2018-04-03
• Subjects: Large initial data in critical spaces -- Navier–Stokes solutions -- stability of weak solutions; 35K55 -- 35Q30 -- 76D05; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2018.1449219
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We consider the Cauchy problem for the Navier–Stokes equation in ℝ 3 ×]0, ∞[ with the initial datumu 0 ∈ L w e a k 3, a critical space containing nontrivial (−1)−homogeneous fields. For small| | u 0 | | L w e a k 3 one can get global well-posedness by perturbation theory. When| | u 0 | | L w e a k 3 is not small, the perturbation theory no longer applies and, very likely, the local-in-time well-posedness and uniqueness fails. One can still develop a good theory of weak solutions with the following stability property: If u ( n ) are weak solutions corresponding the the initial datumu 0 ( n ), andu 0 ( n ) converge weakly* inL w e a k 3 to u 0, then a suitable subsequence of u ( n ) converges to a weak solution u corresponding to the initial condition u 0 . This is of interest even in the special case u 0 ≡0.