Sharp well-posedness and ill-posedness of the Navier-Stokes initial value problem in Besov-type spaces

Abstract

We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than this critical value. The well-posedness result is proved by using some sharp bilinear estimates obtained from some Hardy-Littlewood type inequalities. The ill-posedness assertion is proved by refining the arguments of Wang [18] and Yoneda [20].

This is the second version of the paper with the same title publicized in ArXiv under the number 1505.00865. It remedies the incorrect proof of Lemma 4.2 of the previous version and some other small mistakes. The main result of this paper has been written in the book of Lemarié-Rieusset "The Navier-Stokes Problem in the 21st Century" (CRC press, 2016) as Theorem 9.6 (without proof)