ABSTRACT
Critical case p = N = 2: application of the (T − t)-scaling In the critical case p = N , we have that νk = 1 in (135), so that, in the limit k → +∞, we arrive at the same Navier-Stokes equations, but now with uniformly bounded data and solutions in both L3(IRN ) and L∞(IRN ). Moreover, by passing to the limit s0 = s0j → +∞, we actually deal with the following class of solutions:
w¯(s) ∈ L∞ ∩ LN for all s ≥ 0: |w¯(s)| ≤ 1, ‖w¯(s)‖N ≤ C. (140)
In other words, using the scaling in the critical case p = N , we eventually get into the special class of solutions (140), so a key restriction of the NavierStokes equations is achieved. Obviously, no blow-up or other singularities are available in the class (140). Thus, in the critical case, in the class (140), the Navier-Stokes equations induce a smooth gradient dynamical system with the positive Lyapunov function as in (132), which is strictly monotone on such nontrivial solutions. Hence, this admits the unique globally asymptotically stable trivial equilibrium 0. However, for p = N , proving non-blow-up of solutions leads to a hard
problem of nonexistence of suitable ancient solutions. Similar to the previous KS problem, we demonstrate an example of an application of the (T − t)- scaling to get the result in the critical case p = N = 2 of the obvious particular interest. We begin with Leray’s blow-up scaling [273] for (128) by setting
v(x, t) = 1√ T−t w(y, τ), y =
x√ T−t , τ = − ln(T − t), (141)
to get the following rescaled equation:
wτ +P(w · ∇)w = B∗w, B∗ = Δ− 12 y · ∇ − 12 I. (142)
Here, B∗ is the adjoint Hermite operator with the discrete spectrum
σ(B∗) = { λk = − 12 − k2 , k = 0, 1, 2, ...
