I was referred to the paper C. Kenig and F. Merle, "Global well-posedness, scattering, and blow-up for the energy critical, focusing non-linear Schrodinger equation in the radial case", in which one of the propositions (Proposition 4.2) concerns compactness modulo symmetries of the flow of a certain critical element to the energy-critical NLS

$$\begin{cases}iu_{t}+\Delta u \pm |u|^{\frac{4}{N-2}}u=0 & {(x,t)\in\mathbb{R}^{N}\times\mathbb{R}}\\ u|_{t=0}=u_{0}\in\dot{H}^{1}(\mathbb{R}^{N}) & {}\end{cases} \tag{CP}$$

Proposition 4.1, which I have omitted because I do not think it essential for my question, gives the existence of a critical initial datum $u_{0,C}\in\dot{H}^{1}$ with corresponding solution $u_{C}$ to the Cauchy problem, such that $u_{C}$ has maximal time interval of existence $I$ containing the origin, and $\|u_{C}\|_{S(I)}=+\infty$, where $S(I)$ is a certain space-time norm.

Proposition 4.2Assume $u_{C}$ is as in Proposition 4.1 and that $\|u_{C}\|_{S(I_{+})}=+\infty$, where $I_{+}=(0,+\infty)\cap I$. Then there exists $x(t)\in \mathbb{R}^{N}$ and $\lambda(t)\in\mathbb{R}^{+}$, for $t\in I_{+}$, such that $$K = \{v(x,t) : v(x,t) = \frac{1}{\lambda(t)^{(N-2)/2}}u_{C}(\frac{x-x(t)}{\lambda(t)},t)\}$$ has the property that $\overline{K}$ is compact in $\dot{H}^{1}$.

I took a look at the proof on pg. 13, and I am having trouble arriving at the first claim the authors present, assuming that Proposition 4.2 is false. The aforementioned claim is reproduced below.

Using the notation $u(x,t)=u_{C}(x,t)$, the authors claim that if Proposition 4.2 is false, then there exists an $\eta_{0}>0$ and a sequence $\{t_{n}\}_{n=1}^{\infty}$ of times $t_{n}\geq 0$, such that, for all $\lambda_{0}\in\mathbb{R}^{+}$, $x_{0}\in\mathbb{R}^{N}$, we have $$\|\frac{1}{\lambda_{0}^{(N-2)/2}}u(\frac{x-x_{0}}{\lambda_{0}}, t_{n})-u(x,t_{n}')\|_{\dot{H}_{x}^{1}}\geq \eta_{0}, \quad n\neq n' \tag{*}$$

If Proposition 4.2 is false, then we know that for all functions $x:I_{+}\rightarrow\mathbb{R}^{N}$ and $\lambda:I_{+}\rightarrow\mathbb{R}^{+}$, there exist a sequence of times $\{t_{n}\}$ such that $$\{\frac{1}{\lambda(t_{n})^{(N-2)/2}}u(\frac{x-x(t_{n})}{\lambda(t_{n})},t_{n})\}_{n=1}^{\infty}$$ has no convergent subsequence. However, it's mot clear to me how to obtain what Kenig-Merle write. Rather, it seems to me that (*) follows from assuming that that flow $\{u(t) : t\in I\}$ is not relatively compact in $G \backslash\dot{H}^{1}$, where $G$ is the group of translations and dilations associated to the equation and we equip this space with the quotient metric.

Perhaps something related to what I am asking is the following from pg. 6 of the paper T. Tao, M. Visan, and X. Zhang, "Minimal-mass blowup solutions of the mass-critical NLS".

A function $u\in C_{t,loc}^{0}L_{x}^{2}(I\times\mathbb{R}^{N})$ is

almost periodic modulo $G$if the quotiented orbit $\{Gu(t) : t\in I\}$ is a precompact subset of $G\backslash L_{x}^{2}(\mathbb{R}^{N})$. Equivalently, $u$ is almost periodic modulo $G$ if there exists a compact subset $K\subset L_{x}^{2}(\mathbb{R}^{N})$ such that $u(t)\in GK$, $t\in I$.

Above, $G$ is the group of symmetries now associated to the mass-critical NLS. Proving the equivalence assertion seems relevant to what I am asking in regards to the Kenig-Merle paper. I feel like I'm missing some standard functional analysis/topology argument implicit in the cited works, but it's not clear to me what such an argument is. Any help would be greatly appreciated.