On stability of weak Navier–Stokes solutions with large L 3,∞ initial data

Journal article

We consider the Cauchy problem for the Navier–Stokes equation in ℝ3×]0,∞[ with the initial datum (Formula presented.), a critical space containing nontrivial (−1)−homogeneous fields. For small (Formula presented.) one can get global well-posedness by perturbation theory. When (Formula presented.) 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 datum (Formula presented.), and (Formula presented.) converge weakly* in (Formula presented.) to u0, then a suitable subsequence of u(n)converges to a weak solution u corresponding to the initial condition u0. This is of interest even in the special case u0≡0.

Authors: Barker, T; Seregin, G; Šverák, V

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Taylor and Francis
• Journal: Communications in Partial Differential Equations
• Volume: 43
• Issue: 4
• Pages: 628-651
• Publication date: 2018-05-07
• Acceptance date: 2017-10-21
• DOI: 10.1080/03605302.2018.1449219
• EISSN: 1532-4133
• ISSN: 0360-5302
• Keywords: navier–Stokes solutions; stability of weak solutions; large initial data in critical spaces