The bellow theorem and its proof are presented exactly as in James C. Robinson's book, Infinite-Dimensional Dynamical Systems. My doubts are right after it.
Theorem 14.5. If $ f \in H $ then the squeezing property holds for the 2D Navier-Stokes equations.
Note that the constant $ C $ may change from line to line in the proof.
Proof. The equation for the difference $ w(t) = u(t) - v(t) $ is $$\frac{dw}{dt} + \nu Aw + B(u, w) + B(w, v) = 0,$$ and we will write $$p = P_n w, \quad q = Q_n w, \quad w = p + q.$$ First we take the inner product of (14.26) with $ p $, using $ b(u, w, p) = b(u, p + q, p) = b(u, q, p) $, $$\tfrac{1}{2} \frac{d}{dt} |p|^2 + \nu\|p\|^2 = -b(u, q, p) - b(w, v, p).$$ Using the bounds on $ b $ in (9.27) and the existence of an absorbing set in $ H, V $, and $ D(A) $ (see Chapter 12), we can obtain $$\tfrac{1}{2} \frac{d}{dt} |p|^2 + \nu\|p\|^2 \geq -k \left( |u|^{1/2} |Au|^{1/2} |q||p|| - |w||p||v|^{1/2}|Av|^{1/2} \right)$$ $$\geq -C_1 \left( |q|\lambda^{1/2}|p| + |w|\lambda^{1/2}|p| \right)$$ $$\geq -C\lambda^{1/2}|p|(|p| + |q|),$$ where $ \lambda = \lambda_u $, so that $$|p| \frac{d|p|}{dt} \geq -|p| \left( \nu\lambda|p| + C\lambda^{1/2}|p| + C\lambda^{1/2}|q| \right)$$ $$\geq -\lambda^{1/2}|p| \left( \nu\lambda^{1/2} + C)(|p| + |q|) \right).$$ Now take the inner product with $ q $, $$\tfrac{1}{2} \frac{d}{dt} |q|^2 + \nu\|q\|^2 = -b(u, p, q) - b(w, v, q)$$ $$\leq k|p||q|| + k|w||q||$$ $$\leq C(|p| + |q|)||q||,$$ Provided that the expression in the parentheses is negative, $$(\nu\lambda^{1/2} - C)|q| > C|p|,$$ then we have $$|q|\frac{d|q|}{dt} \leq \lambda^{1/2}|q|(-\nu\lambda^{1/2}|q| + C|p| + C|q|).$$ We now choose $ n $ large enough that $$\nu\lambda_n^{1/2} - C > 2C.$$ If (14.1) does not hold then we follow the previous approach, seeking the first time at which the inequality $$(\nu\lambda^{1/2} - C)|Qw(t)| > 2C|Pw(t)|$$ fails to hold. In this case we take $$\Phi(t) = \Phi(p(t), q(t)) = (|p| + |q|)\exp\left(\frac{\nu\lambda^{1/2}|q|}{C(|p| + |q|)}\right).$$ The argument is essentially unchanged from here.
I am studying the proof of the squeezing property for the 2D Navier-Stokes equations, specifically Theorem 14.5 from the book Nonlinear Dynamical System by James C. Robinson. The goal is to show that for two solutions $ u(t) $ and $ v(t) $ on the global attractor, the difference $ w = u - v $ satisfies that at time $ t=1 $, either the high modes are dominated by the low modes ($ |Qw(1)| \leq |Pw(1)| $), or a strict contraction occurs ($ |w(1)| \leq \delta |w(0)| $ for some $ \delta < 1 $).
I understand the overall structure and the need to handle the case where the "nice" inequality $$ (\nu \lambda_n^{1/2} - C)|Qw(t)| > 2C|Pw(t)| $$ fails for some time, leading to the definition of a critical time $ t_0 $. My struggle, which I believe is shared by many, is with the seemingly deus ex machina introduction of the auxiliary function: $$ \Phi(t) = \big( |Pw(t)| + |Qw(t)| \big) \exp\left( \frac{\nu \lambda_n^{1/2} |Qw(t)|}{C (|Pw(t)| + |Qw(t)|)} \right). $$
My specific questions are:
- What is the conceptual motivation for this specific form of $\Phi$? Is it derived from a more general principle (e.g., an energy method, or a known trick from ODEs)? The exponential term, in particular, seems crucial but non-obvious.
- Are there alternative paths to prove this contraction in the interval $ [t_0, 1] $ that avoid this specific construction? Perhaps a more direct Gronwall-type argument or a different functional inequality?
My Attempts and Context:
- I have worked through the subsequent "elementary but tedious calculations" (as the author calls them) and verified that $ d\Phi/dt \leq 0 $ under the given inequalities. My issue is not with the verification, but with the genesis of $\Phi$.
- I've also attempted a more direct approach. Starting from the differential inequalities for $ a = |Pw| $ and $ b = |Qw| $: $$ \begin{aligned} \frac{da}{dt} &\geq -\lambda a - C(a + b), \\ \frac{db}{dt} &\leq -\nu \lambda b + C(a + b), \end{aligned} $$ (where $ \lambda = \lambda_n^{1/2} $), I tried to find a constant $ \theta $ such that $ L = a + \theta b $ decays. This leads to a condition on $ \theta $, but it only gives a contraction rate $ \delta $ that depends on the length of the interval $ (1 - t_0) $. As $ t_0 \to 1^- $, this rate $ \delta \to 1^- $, failing to guarantee uniform contraction. This numerical observation highlights why a more sophisticated approach like the $\Phi$-function is necessary—it must provide a contraction estimate that is uniform as the time interval shrinks.
This is where I'm stuck. The naive approach fails for short time intervals, and the $\Phi$-function works elegantly but opaquely. I suspect there is a deeper insight I'm missing, perhaps related to the structure of the nonlinearity or the geometry of the attractor.
Any insight into the thought process behind constructing such auxiliary functions, or references to texts that discuss this methodology in a more pedagogical way, would be immensely appreciated.