I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at $$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$ where $F(u) = u^5$ (for example). The goal is to prove a dispersive estimate $$\sup_t \langle t\rangle^{1/3} \|u(t)\|_\infty <\infty.$$
The step I'm wondering about is this: if $$\|g\| \equiv \|g\|_1 + \|g\|_{H^1}$$ is sufficiently small, then $$\|u(t)\|_2 + \|u_x(t)\|_2 \lesssim \|g\|_{H^1}.$$
Clearly we just need to show the derivative bounded, by the conservation of $L^2$ norm.
Eventually, we get this result: $$\|\partial_x u(t)\|_2^2 \le \epsilon(1 + \|\partial_x u(t)\|_2^4)$$ for some $\epsilon >0$ small and for all $t \in \mathbb{R}$. Now, $\{x \in \mathbb{R} : \ x^2 \le \epsilon (1+ x^4)\} = I_1 \cup I_2 \cup I_3$, where $I_1 = [-r,r], I_2 = [\rho, \infty)$ and $I_3 = (-\infty, - \rho]$ for some $0< r << \rho$, which is easy to see. For $\epsilon>0$ small enough, $\rho > 2r$.
By Kato's theorem $\partial_x u: \mathbb{R} \to L^2(\mathbb{R})$ is weakly continuous, and the norm $\|\cdot \|_2$ is always weakly lower semicontinuous. Also by assumption we know $\|\partial_x u(0) \|_2 \in I_1$ if we take $\|g\|$ small enough.
This is supposed to ensure that, in fact, $\|\partial_x u(t)\|_2 \in I_1$ for all time $t$, since the image is supposed to lie in the connected component containing $\|\partial_x u(0)\|_2$, which is $I_1$. This would give the desired goal, which is that $\|\partial_x u\|_2$ is $O(1)$ (and hence $O(\epsilon)$).
However, I don't know how to get the required connectedness. It is true that $\partial_x u(\mathbb{R})$ is weakly connected in $H^1$. We also know that $\|\partial_x u\|_2(\mathbb{R}) \subset I_1 \cup I_2$. But then lower semi-continuity of the norm doesn't seem to be enough to ensure that $\|\partial_x u\|_2(\mathbb{R})$ is connected. In the paper by Christ and Weinstein (hyperlinked), page 100, they use an identical argument but specifically make note of the fact that the norm is varying continuously.
So, how can we make do with $\partial_x u: \mathbb{R} \to L^2 $ being only weakly continuous? Any bootstrapping/continuity argument I can come up with ultimately needs that the norm can't jump, which I don't know if we have.
Any help would be much appreciated. Thanks!
