ABSTRACT
Namely, as a first simple constraint, which is inherited from our previous study of the KS-type equations, we assume that
‖v(t)‖p ≤ C for t ≥ 0 (p > 2). (133) Then, seeking L∞-bound and, hence, assuming (65), we perform the scaling (66), where we impose the preservation of the Lp-norm of the rescaled function, i.e.,
‖vk‖p = ‖wk‖p =⇒ ak = C− p N
k → 0, and bk = a2k. (134) As usual, we next perform a passage to the limit in the NSEs. This can
be done in the framework of the original model (7), as well as of the nonlocal parabolic representation (128), which we actually do. Note that this passage to the limit in the integral term causes no difficulty for our sequences of uniformly bounded smooth solutions {wk}, where the Ascoli-Arzela´ classic theorem [242, Ch. 2] applies.
