Nonlinear Schrödinger as a Dynamical System

J. Colliander (UBC)

Ascona Winter School 2016, (alternate link)

Overview of Lecture 4

  • Review of the main issue: understand the dynamics.
  • Critical Scattering Conjecture?
    • Robust Methods for Global Well-posedness?
    • Partial Regularity?
    • High Regularity Globalizing Estimates?

Initial Value Problem for NLS

$$ \begin{equation*} \tag{$NLS^{\pm}_p (\Omega)$} \left\{ \begin{matrix} (i \partial_t + \Delta) u = \pm |u|^{p-1} u \\ u(0,x) = u_0 (x), ~ x \in \Omega. \end{matrix} \right. \end{equation*} $$

What happens?

Dilation Invariance

One solution $u$ generates parametrized family $\{u^\lambda\}_{\lambda > 0}$ of solutions:

$$u:[0,T) \times \mathbb{R}^d_x \rightarrow \mathbb{C} ~{\mbox{solves}}~ NLS_p^{\pm}(\mathbb{R}^d)$$

$${\iff}$$$$u^\lambda: [0,\lambda^2 T )\times \mathbb{R}^d_x \rightarrow \mathbb{C} ~{\mbox{solves}}~ NLS_p^{\pm}(\mathbb{R}^d)$$

where $$ u^\lambda (\tau, y) = \lambda^{-2/(p-1)} u( \lambda^{-2} \tau, \lambda^{-1} y). $$

Critical Regularity Regimes

critical Sobolev index Regime
$ s_c < 0$ mass subcritical
$0 < s_c < 1$ mass super/energy subcritical
$s_c = 1$ energy critical
$1 < s_c < \frac{d}{2}$ energy supercritical

Critical Scattering Conjecture?

$ H^{s_c} ({\mathbb{R}}^d) \ni u_0 \longmapsto u$ solves defocusing $NLS_p^{\pm}(\mathbb{R}^d)$ globally in time with globally bounded spacetime Strichartz size.

$ \implies $ the behavior of $u(t)$ is described by associated linear evolutions as $ t \rightarrow \pm \infty$.

Status of Critical Scattering?

  • Energy Critical ($s_c = 1$): $\checkmark$
  • Mass Critical ($s_c = 0$): $\checkmark$
  • $0 < s_c < 1$: OPEN
  • Energy Supercritical ($s_c > 1$): OPEN

Remarks

  • Conditional results under critical norm bounds $$\sup_t \| u(t) \|_{H^{s_c}} < \infty.$$
  • Interesting analogies with global issue for Navier-Stokes.
  • Corresponding problems for NLW might be resolved first.
  • $H^{1/2}$-Critical seems most tractable, e.g. $NLS_3^+ ({\mathbb{R}}^3)$.

Robust GWP Methods down to $s_c + \epsilon$?

Local to Global via Almost Conservation

The almost conservation property $$ \sup_{t \in [0,T_{lwp}]} \widetilde{E} [I u(t)] \leq \widetilde{E}[Iu_0] + N^{-\alpha}$$ led to GWP for $$ s > s_\alpha = \frac{2}{2+\alpha}. $$

Better Bootstraps?

  • Bootstrap arguments using Morawetz control generated improvements.
  • Precise bootstraps with RaphaĆ«l led to results for $s > s_c$.
  • Q: Could improved bootstraps lead to robust method to prove global regularity for $s>s_c$?

Addendum from discussion after the talk! See recent work by Ben Dodson on NLW. Recently, Dodson announced corresponding results for $NLS_3^+ (\mathbb{R}^3)$.

Partial Regularity?

ckn

Notations for $NLS_p^+ (\mathbb{R}^d)$.

$$ u_\lambda (t,x) = [\frac{1}{\lambda}]^{\frac{2}{p-1}} u(\frac{t}{\lambda^2}, \frac{x}{\lambda}). $$

Exponents appearing in dilation invariant spaces used in the study of $NLS_p$: $$ s_c = \frac{d}{2} - \frac{2}{p-1}, ~\mbox{(Scaling invariant Sobolev index)} $$ $$ \frac{2}{q} + \frac{d}{r} = \frac{2}{p-1}, ~(H^{s_c} ~\mbox{admissible Strichartz pairs} ~(q,r))$$ $$ \frac{d}{p_c} = \frac{2}{p-1}, ~\mbox{(Scaling invariant spatial Lebesgue space exponent)}$$ $$ \frac{2+d}{q_c} = \frac{2}{p-1}, ~\mbox{(Diagonal scaling invariant Strichartz exponent)}.$$

The Singular Set

  • Spacetime point $z_0 = (t_0, x_0) \in \mathbb{R} \times {\mathbb{R}}^d.$
  • $Q_\lambda$ denote the parabolic box of scale $\lambda$ behind $z_0$ $$ \{(t,x) \in \mathbb{R}^{1+d}: 0< t_0 - t < \lambda^2, |x - x_0| < \lambda\}. $$

Definition: A spacetime point $z_0$ is called a singular point for a weak solution $u$ of $NLS_p$ if $$ \lim_{\lambda \searrow 0} \int_{Q_\lambda} |u|^{q_c} dx dt = + \infty. $$ Points which are not singular are called regular points. The set of all singular points is denoted $\Sigma$. The diagonal Strichartz norm diverges on all parabolic boxes behind a singular point. Points which are not singular are called regular points for $u$.

Concentration Statement?

Singular Point $\implies$ Consistent $L^{q_c}_{t,x}$ Concentration: If $z_0$ is a singular point for the weak solution $u$ then $z_0$ is also a point of (scaling consistent) concentration. That is, there exists $\epsilon_0 > 0$ (a constant independent of $z_0$) and a sequence of scales $\lambda_j \searrow 0$ satisfying $$ \int_{Q_{\lambda_j}} |P_{< \frac{1}{\lambda_j}} u|^{q_c} dx dt > \epsilon_0. $$

The converse would be an "$\epsilon$-regularity" statement.

Parabolic Hausdorff Dimension of $\Sigma$?

Absorption of Interaction Morawetz $\implies {\mbox{dim}}_{\mathscr{P}} \Sigma \leq 4 (s_c - \frac{1}{4})$

High Regularity Globalizing Estimates?

Consider the $H^2$-critical problem $$ \begin{equation*} \tag{$NLS^{\pm}_5 (\mathbb{R}^5)$} \left\{ \begin{matrix} (i \partial_t + \Delta) u = \pm |u|^{4} u \\ u(0,x) = u_0 (x), ~ x \in \mathbb{R}^5. \end{matrix} \right. \end{equation*} $$

Assume $u_0$ is nice (smooth, compactly supported). Can one prove that $$ \sup_t \| u(t) \|_{H^k (\mathbb{R}^5)} < \infty? $$

Possible ingredients: Almost conservation techniques, a priori spacetime estimates, Gronwall estimate, contradiction arguments, ...?

Thank you!

alpenglow