energy trapping for NLS

Question

I am trying to understand the proof of Theorem 3.9, p.9.

---

We consider NLS $$i\partial_t u + \Delta + |u|^{4/(d-2)}u=0, u(x,0)=u_0 \in H^1(\mathbb R^d)$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\mathbb R^d \to \mathbb C$

Assume that $\int_{\mathbb R^d} |\nabla u_0(x)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$ and $\int_{\mathbb R^d} |\nabla u_0(x)|^2 - |u_0(x)|^{2d/d-2} \geq \delta \int_{\mathbb R^d} |\nabla u_0(x)|^2$ for some fix $\delta >0$.

Now, let $I$ be the maximal interval of existence of the above NLS.

My Question: How to show $\int_{\mathbb R^d} |\nabla u(x, t)|^2 dx \leq (1-\delta) \|\nabla W \|^2_{L^2}$ and $\int_{\mathbb R^d} |\nabla u(x, t)|^2 - |u(x, t)|^{2d/d-2} dx \geq \delta \int_{\mathbb R^d} |\nabla u(x, t)|^2 dx$ for all $t\in I.$ In other words, if these inequalities are true at time $t=0$, then why they are true for all time $t\in I$?

(Why this is call energy trapping? Because $E(u(t))\geq 0$?)

[My confusion: The author gives the hint that use continuity argument, what is the meaning of continuity arguement and how to apply here.]

Answer 1

Let me quote Lemma 3.4 from the paper, with some paraphrasing to make extremely clear certain points. The function $W$ here is some function $\mathbb{R}^N\to \mathbb{R}$ solving a certain nonlinear elliptic equation (see start of Section 3 of the paper). Its precise form does not matter for our discussion, other than that it is a fixed function.

Definition If $u:\mathbb{R}^N \to \mathbb{C}$, we say that the energy of $u$ is $$ E(u) := \frac12 \int |\nabla u|^2 \mathrm{d}x - \frac{N-2}{2N} \int |u|^{2N/(N-2)} \mathrm{d}x$$

Lemma 3.4 Let $u:\mathbb{R}^N\to \mathbb{C}$ be some function. Assume that there exists some $\delta_0> 0$ such that $u$ satisfies

• $E(u) \leq (1-\delta_0) E(W)$
• $\|\nabla u\|_{L^2} < \|\nabla W\|_{L^2}$

then the following hold:

• $E(u) \geq 0$
• there exists $\bar{\delta} = \bar{\delta}(\delta_0,N) > 0$ such that $$ \|\nabla u\|_{L^2}^2 \leq (1 - \bar{\delta}) \|\nabla W\|_{L^2}^2 $$ and $$ \int |u|^{2N / (N-2)} \mathrm{d}x \leq (1 - \bar{\delta}) \|\nabla u\|^2_{L^2} $$

Now, in order to prove Theorem 3.9, we look at Lemma 3.4, and see that by energy conservation the first condition that $E(u) \leq (1 - \delta_0) E(W)$ is automatically satisfied if the initial data satisfies it. So it remains to show that $\|\nabla u\| < \|\nabla W\|$ with strict inequality pointwise in time. And here we use the continuity argument:

For contradiction, assume there is some $t > 0$ denoting the first time at which $$ \|\nabla u(t) \| = \|\nabla W\| $$ (there is a first positive time since $\|\nabla u(0)\| < \|\nabla W\|$ and $t \mapsto \|\nabla u(t)\|$ is continuous). Then for every $\tau \in [0,t)$ we have that $\|\nabla u(\tau)\| < \|\nabla W\|$, and hence we can apply Lemma 3.4 to $u(\tau)$ to conclude that $\|\nabla u(\tau)\|^2 \leq (1 - \bar{\delta})\|\nabla W\|^2$ uniformly in $\tau$. But this means $\limsup_{\tau \to t} \|\nabla u(\tau)\| \leq \sqrt{1 - \bar{\delta}} \|\nabla W\| < \|\nabla W\|$ contradicting the continuity of $t \mapsto \|\nabla u(t)\|$. Q.E.D.

---

This result is called energy trapping because it indicates a trapped set in state space, which is bounded by some "energy-like" conditions.

The NLS has a negative nonlinear potential, so in principle conservation of energy does not give a priori bounds on $\|\nabla u\|_{L^2}$ (it is not coercive). In principle it may be the case that you start with a solution with small energy that by losing potential energy through the negatively-signed self-interaction you gain a lot of kinetic energy. This theorem says that such a scenario cannot happen, provided initially your kinetic energy is not too big. In the small data case this is not very hard. In this case, however, the argument is not via "smallness", since "not too big" means here bounded by the first soliton energy which is "medium sized". The argument is that the first soliton actually provides a barrier to the runaway scenario described above. So that the set of initial data where both the conserved and the kinetic energies are less than that of the soliton is a "trapped set" for the flow associated to the NLS: solutions starting in this set can never escape it.