# Nonlinear Schrödinger as a Dynamical System¶

## Overview of Lecture 1¶

• Initial Value Problem for NLS
• Conserved Quantities
• Well-posedness Theory

# Initial Value Problem for NLS¶

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

## Studies explore interplay between:¶

• Dispersion
• Nonlinearity

Dilation invariance quantifies the balance between these effects.

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

Norms which are invariant under $u \longmapsto u_\lambda$ are critical.

## Critical Regularity¶

In the $L^2$-based Sobolev scale, $$\| D^s u^\lambda (t) \|_{L^2} = \lambda^{-\frac{2}{p-1} - s + \frac{d}{2}} \| D^s u (t)\|_{L^2}.$$

The critical Sobolev index for $NLS_p^{\pm}(\mathbb{R}^d)$ is $$s_c := \frac{d}{2} - \frac{2}{p-1}.$$

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

### The spatial domain $\Omega$?¶

• Infinite measure space like $\mathbb{R}^d$
• Finite measure space like $\mathbb{T}^d$

### The Choice of Sign?¶

• $+$ defocusing
• $-$ focusing

Large data dynamics are completely different.

# Conserved Quantities¶

\begin{align*} {\mbox{Mass}}& = \| u \|_{L^2_x}^2 = \int_{\mathbb{R}^d} |u(t,x)|^2 dx. \\ {\mbox{Momentum}}& = {\textbf{p}}(u) = 2 \Im \int_{\mathbb{R}^2} {\overline{u}(t)} \nabla u (t) dx. \\ {\mbox{Energy}} & = H[u(t)] = \frac{1}{2} \int_{\mathbb{R}^2} |\nabla u(t) |^2 dx {\pm} \frac{2}{p+1} |u(t)|^{p+1} dx . \end{align*}

## Remarks¶

• Conserved quantities constrain the dynamics.
• NLS defines a flow on a sphere in $L^2$.
• Energy vividly reveals focusing vs. defocusing difference.
• Mass is $L^2$; Momentum scales like $H^{1/2}$; Energy involves $H^1$.
• Local conservation laws express how a quantity is conserved: $\partial_t |u|^2= \nabla \cdot 2 \Im (\overline{u} \nabla u)$.

# Variations on Conserved Quantities?¶

## Conserved¶

$$\partial_t Q[u] = 0.$$
• Energy/Compactness methods for building solutions.
• Globalizing control to extend local-in-time solutions.

## Almost Conserved¶

$$\big| \partial_t Q[u] \big| ~\mbox{is small}.$$
• Bourgain's High/Low Frequency Decomposition
• $I$-Method
• Multilinear Correction Terms
• Applications

## Monotone¶

$$\partial_t Q[u] > 0.$$
• Virital identity $\implies$ blow-up.
• Morawetz-type inequalities $\implies$ decay.

# Local Well-posedness Theory¶

• Fixed point argument based on Contraction Mapping
• Show the "Picard iterates" converge
• Key innovation for the analysis: identify the right space!

# Free Schrödinger Evolution¶

$$\begin{equation*} \tag{LS(\mathbb{R}^d)} \left\{ \begin{matrix} (i \partial_t + \Delta) u = 0 \\ u(0,x) = u_0 (x). \end{matrix} \right. \end{equation*}$$

## Explicit Solution Formula¶

$$u_0 \longmapsto u(t,x) = e^{it \Delta} u_0$$

## Explicit Solution Formula¶

• Fourier Multiplier Representation: $$e^{it \Delta} u_0 (x) = c_\pi\int_{\mathbb{R}^d} e^{i x \cdot \xi} e^{-i t |\xi|^2} \widehat{u_0} (\xi) d\xi.$$
• Convolution Representation: $$e^{it \Delta} u_0 (x) = k_\pi \frac{1}{(it)^{d/2}} \int_{\mathbb{R}^d} e^{i \frac{|x-y|^2}{4t}} u_0 (y) dy.$$

## Estimates for Schrödinger Propagator $e^{it \Delta} u_0$¶

• Fourier Multiplier Representation $\implies$ Unitary in $H^s$: $$\| D_x^s e^{it \Delta} u_0 \|_{L^2_x} = \| D_x^s u_0 \|_{L^2_x}.$$
• Convolution Representation $\implies$ Dispersive estimate: $$\| e^{it \Delta} u_0 \|_{L^\infty_x} \leq \frac{C}{t^{d/2}} \| u_0 \|_{L^1_x}.$$
• Spacetime estimates? Strichartz estimates hold, for example, $$\| e^{it \Delta} u_0 \|_{L^4 ( \mathbb{R}_t \times \mathbb{R}^2_x)} \leq C \| u_0 \|_{L^2 (\mathbb{R}^2_x)}.$$

# Schrödinger Evolution with Forcing¶

$$\begin{equation*} \left\{ \begin{matrix} (i \partial_t + \Delta) u = F \\ u(0,x) = u_0 (x). \end{matrix} \right. \end{equation*}$$

### Strichartz Estimates¶

$$\| u \|_{L^q_t L^r_x (\mathbb{R}_t \times \mathbb{R}^d_x )} \leq C \| u_0 \|_{L^2_x} + \| F \|_{L^{Q'}_t L^{R'}_x(\mathbb{R}_t \times \mathbb{R}^d_x )}.$$

Admissibility for independent pairs $(q,r), (Q,R)$ $$\frac{2}{q} + \frac{d}{r} = \frac{d}{2}, ~ q > 2.$$ The $'$ denotes HÃ¶lder dual exponent.

# Local-in-time theory for $NLS^{\pm}_3 (\mathbb{R}^2)$¶

• $\forall ~u_0 \in L^2 (\mathbb{R}^2)~\exists ~T_{lwp} ( u_0 )$ determined by $$\| e^{it \Delta} u_0 \|_{L^4_{tx} ([0,T_{lwp} ] \times \mathbb{R}^2)} < \frac{1}{100} ~{\mbox{such that}}~$$

$\exists$ unique $u \in C([0, T_{lwp} ]; L^2 ) \cap L^4_{tx} ([0,T_{lwp}] \times \mathbb{R}^2)$ solving $NLS_3^{+} (\mathbb{R}^2)$.

• $\forall ~ u_0 \in H^s (\mathbb{R}^2), s>0$, $T_{lwp} \thicksim \| u_0 \|_{H^s}^{-\frac{2}{s}}$ and regularity persists: $u \in C([0,T_{lwp}]; H^s (\mathbb{R}^2))$.
• Define the maximal forward existence time $T^* (u_0)$ by $$\| u \|_{L^4_{tx} ([0,T^* -\delta] \times \mathbb{R}^2) }< \infty$$ for all $\delta > 0$ but diverges to $\infty$ as $\delta \searrow 0$.
• $\exists ~$ small data scattering threshold $\mu_0 > 0$ $$\| u_0 \|_{L^2} < \mu_0 \implies \|u \|_{L^4_{tx} (\mathbb{R} \times \mathbb{R}^2)} < 2 \mu_0.$$

# $H^1$ Global-in-time theory for $NLS^{+}_3 (\mathbb{R}^2)$¶

• $\forall ~ u_0 \in H^1 (\mathbb{R}^2), s>0$, $T_{lwp} \thicksim \| u_0 \|_{H^1}^{-2}$ and regularity persists: $u \in C([0,T_{lwp}]; H^1 (\mathbb{R}^2))$.
• Conserved Quantities $\implies ~T_{lwp} > C(u_0)$ $${\mbox{Energy}} = H[u(t)] = \frac{1}{2} \int_{\mathbb{R}^2} |\nabla u(t) |^2 dx + \frac{2}{p+1} |u(t)|^{p+1}$$ $${\mbox{Mass}} = \| u \|_{L^2_x}^2 = \int_{\mathbb{R}^d} |u(t,x)|^2 dx$$
• Globalize by iteration using $~T_{lwp} > C(u_0)$.