Nonlinear Schrödinger as a Dynamical System¶

Overview of Lecture 3¶

• Generalized Virial Identity
• A Priori Spacetime Estimates
• Lin-Strauss Morawetz Estimate
• Interaction Morawetz Estimates

Variations on Conserved Quantities?¶

Monotone¶

$$\partial_t Q[u] > 0.$$

Generalized Virial Identities¶

Lagrangian NLS¶

$$\begin{equation*} \left\{ \begin{matrix} (i \partial_t + \Delta) u = \pm F'(|u|^2) u \\ u(0,x) = u_0 (x) \\ \end{matrix} \right. \end{equation*}$$

Generalized NLS Equation (GNLS)¶

$$\begin{equation*} \left\{ \begin{matrix} (i \partial_t + \Delta ) \phi = \mathcal{N} \\ \phi(0,x) = \phi_0 (x) \\ \end{matrix} \right. \end{equation*}$$

Remarks:

• Assume $F' \geq 0$. The $+$ case is defocusing $-$ is focusing.
• Generalized NLS with Lagrangian derivation.
• $U(1)$ solution symmetry: $u \rightarrow e^{i\theta} u$.

Time Invariant Quantities¶

The following quantities do not change with time: $${\mbox{Mass}} = \int_{{\mathbb{R}}^d} |u(t,x)|^2 dx.$$ $${\mbox{Momentum}} = 2 \Im \int_{{\mathbb{R}}^2} {\overline{u}(t)} \nabla u (t) dx.$$ $${\mbox{Energy}} = H[u(t)] = \frac{1}{2} \int_{R^2} |\nabla u(t) |^2 dx {\mp} F(|u(t)|^2) dx .$$ $\implies$ a priori conservation controls (defocusing case): $$\| u \|_{L^\infty_t L^2_x} \leq \| u_0 \|_{L^2}$$ $$\| \nabla u \|_{L^\infty_t L^2_x } \leq E[u_0].$$ These are very useful bounds but do not give any decay in time.

Local Conservation Laws¶

We consider an even more general NLS equation.

• Suppose $\phi:[0,T] \times {\mathbb{R}}^d \rightarrow \mathbb{C}$ solves generalized NLS (GNLS)
$$(i \partial_t + \Delta ) \phi = \mathcal{N}$$

for $\mathcal{N} = \mathcal{N} (t, x, \phi ):[0,T] \times {\mathbb{R}}^d \times \mathbb{C} \rightarrow \mathbb{C}$. Assume $\phi$ is nice.

• Not necessarily Lagrangian; No $U(1)$ symmetry.

• Express mass and momentum (non)conservation for $GNLS$.

Write $\partial_{x_j} \phi = \partial_j \phi = \phi_j$.

Local mass/momentum (non)conservation¶

• mass density:
$T_{00} = |\phi|^2$
• momentum density/mass current:
$T_{0j} = T_{j0} = 2 \Im (\overline{\phi} \phi_j)$
• (linear part of the) momentum current:
$L_{jk} = L_{kj} = - \partial_j \partial_k |\phi|^2 + 4 \Re (\overline{\phi_j} \phi_k)$
• mass bracket: $\{ f, g\}_m = \Im (f \overline{g})$

• momentum bracket:
$\{ f, g\}^j_{p} = \Re (f \partial_j \overline{g} - g \partial_j \overline{f} )$

Local mass (non)conservation identity:
$$\partial_t T_{00} + \partial_j T_{0j} = 2 \{ \mathcal{N} , \phi \}_m$$ Local momentum (non)conservation identity: $$\partial_t T_{0j} + \partial_k L_{kj} = 2 \{ \mathcal{N} , \phi \}_p^j$$

Local mass/momentum (non)conservation¶

Consider $\mathcal{N} = F' (|\phi|^2) \phi$ for polynomial $F:{\mathbb{R}}^+ \rightarrow \mathbb{R}$.

• We calculate the mass bracket $$\{ F' (|\phi|^2) \phi , \phi \}_m = \Im ( F' (|\phi|^2) \phi \overline{\phi}) =0.$$

Thus mass is conserved for these nonlinearities.

• We calculate the momentum bracket $$\{ F' (|\phi|^2) \phi , \phi \}_p^j = - \partial_j G (|\phi|^2)$$ where $G(z) = z F'(z) - F(z) \thicksim F(z)$.

Thus the momentum bracket contributes a divergence and momentum is conserved for these nonlinearities.

Generalized Virial Identity¶

Let $a: {\mathbb{R}}^d \rightarrow \mathbb{R}$ be a virial weight function. Form the virial potential $$V_a (t) = \int_{{\mathbb{R}}^d} a(x) |\phi(t,x)|^2 dx.$$

Form the Morawetz action $$M_a (t) = \int_{{\mathbb{R}}^d} \nabla a \cdot 2 \Im (\overline{\phi} \nabla \phi) dx.$$

Conservation identities lead to the generalized virial identities $$\partial_t V_a = M_a + \int_{{\mathbb{R}}^d} a (x) \{ \mathcal{N}, \phi\}_m (t,x) dx,$$ $$\partial_t M_a = \int_{{\mathbb{R}}^d} (-\Delta \Delta a) |\phi|^2 + 4 a_{jk} \Re (\overline{\phi_j} \phi_k) + 2 a_j \{ \mathcal{N}, \phi\}_p^j dx.$$

Remarks on Virial Identities¶

• The virial potential is a weighted average of the mass density against the virial weight $a$.

• The Morawetz action is a contraction of the momentum density against $\nabla a$. Vector fields not arising as gradients could also be considered.

• Useful estimates emerge from monotonicity and boundedness of terms in the virial identities.

• Monotone quantities provide dynamical insights.

• Idea of Morawetz Estimates: Cleverly choose the weight function $a$ so that $\partial_t M_a \geq 0$ but $M_a \leq C (\phi_0)$ to obtain spacetime control on $\phi$. This strategy imposes various constraints on $a$ which suggest choosing $a (x) = |x|$.

Variance Identity¶

• Consider $GNLS$ with $\mathcal{N} = \pm |u|^{4/d} u$. This is the $L^2$ critical focusing equation $NLS_{1+{\frac{4}{d}}}^{\pm} ({\mathbb{R}}^d)$.
• Choose $a(x) = |x|^2$. Calculations reveal that $$\partial_t^2 \int_{{\mathbb{R}}^d} |x|^2 |u(t,x)|^2 dx = 16 H[u(t)].$$
• In the focusing case, we can consider initial data $u_0$ with $H[u_0] <0$ and finite variance. Such data must blow up in finite time.

A Priori Spacetime Estimates¶

[Lin-Strauss] Morawetz identity¶

Consider $(i \partial_t + \Delta ) \phi = F' (|\phi|^2) \phi$ with $F' \geq 0$ and $x \in {\mathbb{R}}^3$. Choose $a(x) = |x|$. Observe that $a$ is weakly convex, $\nabla a = \frac{x}{|x|}$ is bounded, and $-\Delta \Delta a = 4 \pi \delta_0$. From monotonicity $\partial_t M_a \geq 0$ and the bound $|M_a| \leq \sqrt{H[u_0]}$ emerges the Lin-Strauss Morawetz identity $$M_a (T) - M_a (0) = \int\limits_0^T \int\limits_{{\mathbb{R}}^3} 4 \pi \delta_0 (x) |\phi (t,x)|^2 + (\geq 0) + 4 \frac{G(|\phi|^2)}{|x|} dx dt.$$

This implies the spacetime control estimate (centered at $x=0$) $$(H[u_0])^{1/2} \|u_0\|_{L^2} \gtrsim \int\limits_0^T \int\limits_{{\mathbb{R}}^3} \frac{G(|\phi|^2)}{|x|} dx dt.$$

[Morawetz] "Reward and Anchor."

[Ginibre-Velo] $H^1$-Scattering.

[Bourgain] and [Grillakis] truncation¶

• Let $\chi_{B_R}$ denote a smooth cutoff adapted to $B_R = \{ |x| < R \}$.
• Choose cutoff virial weight $a(x) = \chi_{B_R} (x) |x|.$ and calculate $$M_a \Big|_0^T \geq \int\limits_0^T \int\limits_{{\mathbb{R}}^3} 4 \pi \delta_0 (x) |\phi (t,x)|^2 + 4 \int\limits_0^T \int\limits_{|x| < R/2} \frac{G(|\phi|^2)}{|x|} dx dt$$
• $| M_a \Big|_0^T | \leq R^{-1} T H[u_0] + R H[u_0] \implies$ choose $R \thicksim T^{1/2} \implies$ $$\int\limits_0^T \int\limits_{|x| < T^{1/2}} \frac{G(|\phi|^2)}{|x|} dx \lesssim T^{1/2} \|\nabla \phi \|_{L^\infty_{[0,T]} L^2_x}^2.$$

[Bourgain], [Grillakis]: Energy critical bubbles sparse along time axis.

Interaction Morawetz Estimates¶

Averaging over [Lin-Strauss] center?¶

• Translation invariance? Weight $|x|^{-1}$ difficult in proofs.
• Recenter [L-S] at fixed $y \in {\mathbb{R}}^d$. Set $a(x) = |x-y|$.
• Recentered Morawetz action can be expressed $$M_y [u] (t) = \int_{{\mathbb{R}}^d} \frac{(x-y)}{|x-y|} 2 \Im ( u \nabla \overline{u}) (t,x) dx.$$
• Monotonicity $\partial_t M_y [u] \geq 0$: mass repelled from $y \in {\mathbb{R}}^d$.
• Can we average with respect to center $y$ and obtain new translation invariant spacetime control?
• Yes, if we average against the natural density $|u(t,y)|^2$.

Interaction Morawetz via Averaging¶

• Define the Morawetz interaction potential $$M[u](t) = \int_{{\mathbb{R}}^d_y} |u(t,y)|^2 M_y [u] (t) dy.$$

It is bounded: $$\Big|M[u](t)\Big| \lesssim \| u(t) \|_{L^2_x}^3 \| \nabla u(t) \|_{L^2_x}.$$

We calculate $$\partial_t M[u] = \int_{{\mathbb{R}}^d_y} |u(t,y)|^2 \{\partial_t M_y [u] \} + \{\partial_t |u(y)|^2\} M_y [u] dy.$$

• Local conservation and [Lin-Strauss] Morawetz $\implies$ monotonicity:
$\exists ~I, II, III, IV$ such that $I, III \geq 0$ and $II + IV \geq 0$ and $\partial_t M[u] = I + II + III + IV.$ Integrating in time gives $$\int_0^T \int_{{\mathbb{R}}^3} |u(t,x)|^4 dx dt \lesssim \| u(t) \|_{L^\infty_T L^2_x}^3 \| \nabla u(t) \|_{L^\infty_T L^2_x}~ .$$

2-particle interaction Morawetz¶

(Hassell 04)

• Suppose $\phi_1, \phi_2$ are two solutions of $(i \partial_t + \Delta ) \phi = F' (|\phi|^2) \phi$ with $F' \geq 0$ and $x \in {\mathbb{R}}^3$. The 2-particle wave function $$\Psi (t, x_1, x_2) = \phi_1 (t, x_1) \phi_2 (t, x_2)$$

satisfies an NLS-type equation on ${\mathbb{R}}^{1+6}$ $$(i \partial_t + \Delta_1 + \Delta_2) \Psi = [F' (|\phi_1 |^2) + F' (|\phi_2 |^2)] \Psi.$$

• Note that $[F' (|\phi_1 |^2) + F' (|\phi_2 |^2)] \geq 0$ so defocusing.
• Reparametrize ${\mathbb{R}}^6$ using center-of-mass coordinates $(\overline{x}, y)$ with $\overline{x} = \frac{1}{2} (x_1 + x_2) \in {\mathbb{R}}^3$. Note that $y=0$ corresponds to the diagonal $x_1 = x_2 = \overline{x}$. Apply the generalized virial identity with the choice $a(x_1, x_2) = |y|$. Dismissing terms with favorable signs, one obtains...

2-particle interaction Morawetz¶

$$\| \nabla u \|_{L^\infty_{[0,T]} L^2_x} \|u_0\|_{L^2}^3 \geq \int_0^T \int_{{\mathbb{R}}^6} (-\Delta_6 \Delta_6 |y|) |\Psi ( x_1, x_2)|^2 d x_1 dx_2 dt$$$$\geq c\int_0^T \int_{{\mathbb{R}}^6} \delta_{\{y=0\}} (x_1, x_2) |\phi_1(x_1) \phi_2( x_2)|^2 dx_1 dx_2 dt$$$$\geq c\int_0^T \int_{{\mathbb{R}}^3} |\phi_1(t, \overline{x}) \phi_2(t, \overline{x})|^2 d \overline{x} dt.$$

Specializing to $\phi_1 = \phi_2$ gives the 2-particle Morawetz estimate $$\int_0^T \int_{{\mathbb{R}}^3} |\phi(t,x)|^4 dx dt \leq C \| \nabla \phi \|_{L^\infty_{[0,T]} L^2_x} \| \phi_0 \|_{L^2_x}^3$$ valid uniformly for all defocusing NLS equations on ${\mathbb{R}}^3$.

"The" 2-particle Morawetz Estimate¶

Efforts to extend the $L^4(\mathbb{R}_t \times {\mathbb{R}}^3_x)$ interaction Morawetz to the ${\mathbb{R}}^2_x$ setting led to...

Theorem: Finite energy solutions of any defocusing $NLS^+ ({\mathbb{R}}^d)$ satisfy $$\| D^{\frac{3-d}{2}} |u|^2 \|^2_{L^2_{t,x}} \lesssim \| u_0 \|_{L^2_x}^3 \| \nabla u \|_{L^\infty_t L^2_x}.$$

• [C-Grillakis-Tzirakis], [Planchon-Vega]: independently
• Simple proof of $H^1$-scattering in mass supercritical case.[Nakanishi]
• Simplified proof extends to $H^s$ for certain $s<1$.
• Applied by Dodson to resolve the $L^2$ scattering conjecture.

4-particle Morawetz Estimate¶

(Hassel-Tao) [C-Holmer-Visan-Zhang]

• ${\mathbb{R}}^4 = \{ {\bf{x}}=(x_1, x_2, x_3, x_4): x_i \in \mathbb{R}; i=\{ 1,2,3,4 \}$: $\overline{x}=$ center of mass $= \frac{1}{4}(x_1 + x_2 + x_3 + x_4)$. Define $y = (x_1 - \overline{x}, x_2 - \overline{x}, x_3 - \overline{x}, x_4 - \overline{x})$. Here $y \in {\mathbb{R}}^3$. ${\mathbb{R}}^4 \ni {\bf{x}}= (x_1, x_2, x_3, x_4) \iff (\overline{x}, y) \in \mathbb{R} \times {\mathbb{R}}^3$
• The 4-particle wave function $$\Psi (t, {\bf{x}}) = \prod_{i=1}^4 \phi_1 (t, x_i)$$ satisfies a defocusing NLS equation on ${\mathbb{R}}^{1+4}$.
• Choice of virial weight $a({\bf{x}}) = |y|$ spawns $$\int_0^T \int_{\mathbb{R}} |\phi|^8 d{\overline{x}} dt \lesssim \| \phi \|^7_{L^\infty_T L^2_x} \|\nabla \phi \|_{L^\infty_T L^2_x}.$$
• Q: How does this estimate generalize to other dimensions?