[Paper Review] Supercritical Nonlinear Schrödinger Equations II: Almost Global Existence

W. M. Wang|arXiv (Cornell University)|Jul 1, 2010

Advanced Mathematical Physics Problems3 references3 citations

TL;DR

This paper establishes almost global existence for supercritical nonlinear Schrödinger equations on the $d$-torus by constructing approximate quasi-periodic solutions via a finitely iterated Newton scheme, leveraging spectral gap estimates and the open mapping theorem. It proves that for generic initial data of size one and small $\delta$, solutions exist for times $|t| \leq \delta^{-A}$ for any $A>1$, with uniform bounds in the analytic norm, resolving the lack of a priori global existence in supercritical regimes.

ABSTRACT

We prove almost global existence for supercritical nonlinear Schrödinger equations on the $d$-torus ($d$ arbitrary) on the good geometry selected in part I. This is seen as the Cauchy consequence of I, since the known invariant measure of smooth solutions are supported on KAM tori. In the high frequency limit, these quantitative solutions could also be relevant to Cauchy problems for compressible Euler equations.

Motivation & Objective

• To address the lack of a priori global existence for supercritical nonlinear Schrödinger equations (NLS) on $\mathbb{T}^d$, where standard conservation laws do not ensure global solutions.
• To establish almost global existence for small-data solutions in the supercritical regime ($p \geq 1$, $d \geq 3$) where local well-posedness fails to extend globally.
• To construct approximate quasi-periodic solutions that satisfy the NLS equation up to $\mathcal{O}(\delta^r)$ error, enabling long-time existence via linearization.
• To prove that the set of initial data leading to such solutions has positive measure, with measure approaching 1 as $\delta \to 0$, ensuring genericity of the result.

Proposed method

• Construct an approximate quasi-periodic solution $v$ with $\mathcal{O}(|\log \delta|)$ basic frequencies using a finitely iterated Newton scheme applied to the initial data $u_0 = u_1 + u_2$, where $\|u_2\| = \mathcal{O}(\delta)$.
• Ensure spectral gap via excision of Fourier amplitudes $\{\hat{u}_1\}$ in an open set $\mathcal{A} \subset (0,1]^b$, relying on the non-degeneracy of the frequency vector and the geometry of resonances.
• Apply the open mapping theorem to extend the construction from a fixed initial approximation to a full neighborhood of initial data $\tilde{u}_0 = u_1 + \tilde{u}_2$ with $\|\tilde{u}_2\| = \mathcal{O}(\delta)$, ensuring robustness.
• Linearize the NLS equation about the approximate solution $v$, proving that the linearized flow is bounded in the analytic norm up to time $\delta^{-r/3}$, with $\|S(t)\| \leq 1 + |t|$.
• Use the Duhamel formula to estimate the remainder $w = u - v$, showing $\|w(t)\| = \mathcal{O}(\delta^{r/2})$ for $|t| < \delta^{-r/10}$, which implies global control of the full solution.
• Establish quantitative bounds on the solution norm and time of existence by scaling the remainder $w = \delta^{r'} w'$ and using $L^2$-type estimates on the linearized flow.

Experimental results

Research questions

• RQ1 Can almost global existence be established for supercritical NLS on $\mathbb{T}^d$ when no conservation law ensures global existence?
• RQ2 What conditions on initial data (in terms of frequency support and amplitudes) allow for the construction of approximate quasi-periodic solutions that control the nonlinear dynamics for times longer than the local existence time?
• RQ3 How can the open mapping theorem be applied to extend the existence of solutions from a single initial data to a full open set of initial data in the analytic topology?
• RQ4 What is the quantitative dependence of the solution lifetime on the small parameter $\delta$, and can it be made arbitrarily long in the sense of $\delta^{-A}$ for any $A>1$?
• RQ5 Can the linearized flow around the approximate solution be controlled uniformly in time up to $\delta^{-A}$, ensuring stability of the construction?

Key findings

• For any $A > 1$, there exists $\delta_0 > 0$ such that for all $\delta \in (-\delta_0, \delta_0)$, the Cauchy problem for the NLS on $\mathbb{T}^d$ has a unique solution $u(t)$ with $|t| \leq \delta^{-A}$, provided the initial data $u_0 = u_1 + u_2$ satisfies $\|u_2\| = \mathcal{O}(\delta)$ and $\{\hat{u}_1\} \in \mathcal{A}$, an open set of positive measure.
• The measure of the set $\mathcal{A}$ of initial frequency amplitudes approaches 1 as $\delta \to 0$, indicating that almost all generic initial data of size one yield long-time solutions.
• The solution satisfies $\|u(t)\| \leq \|u_0\| + \mathcal{O}(\delta)$ in the analytic norm over the time interval $|t| \leq \delta^{-A}$, ensuring uniform control of the solution size.
• The linearized flow $S(t)$ about the approximate solution $v$ satisfies $\|S(t)\| \leq 1 + |t|$ for $|t| < \delta^{-r/3}$, which is crucial for controlling the remainder term in the Duhamel expansion.
• The remainder $w = u - v$ satisfies $\|w(t)\| = \mathcal{O}(\delta^{r/2})$ for $|t| < \delta^{-r/10}$, which ensures that the solution remains close to the approximate quasi-periodic solution over long times.
• The semi-classical counterpart shows that for $\delta = 1$, solutions exist for $|t| \leq K^{-A}$ with $K \to \infty$, and $\|u(t)\| \leq \|u_0\| + \mathcal{O}(1/K^2)$, with the measure of the allowed amplitude set $\mathcal{A}$ approaching 1 as $K \to \infty$.