#!/usr/bin/env python
# coding: utf-8

# # Nonlinear Schrödinger as a Dynamical System
# 
# > #### [J. Colliander](http://colliand.com) ([UBC](http://www.math.ubc.ca))
# 
# #### [Ascona Winter School 2016](http://www.math.uzh.ch/pde16/index-Ascona2016.html), [(alternate link)](http://www.monteverita.org/en/90/default.aspx?idEvent=295&archive=)

# ## Lectures
# 
# 1. [Introduction](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture1.ipynb)
# 2. [Conservation](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture2.ipynb)
# 3. [Monotonicity](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture3.ipynb)
# 4. **[Research Frontier](http://nbviewer.jupyter.org/github/colliand/ascona2016/blob/master/colliander-lecture4.ipynb)**
# 
# 
# 
# ###  https://github.com/colliand/ascona2016
# 
# 
# ***

# ## 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 Raphaël 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](http://arxiv.org/pdf/1506.06239.pdf) by Ben Dodson on NLW. Recently, Dodson announced corresponding results for $NLS_3^+ (\mathbb{R}^3)$.

# # Partial Regularity? 

# ![ckn](https://wwejubwfy.s3.amazonaws.com/1982_Caffarelli_Partial_regularity_of_suitable_weak_Comm._Pure_Appl._Math.pdf-2016-01-14-21-18-08.jpg)
# 

# ## 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](https://wwejubwfy.s3.amazonaws.com/friday-morning-ascona.jpg)
#