J. Colliander (UBC)¶
Remarks:
The $+$ case is defocusing $-$ is focusing.
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.
We consider an even more general NLS equation.
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 $\partial_{x_j} \phi = \partial_j \phi = \phi_j$.
$T_{0j} = T_{j0} = 2 \Im (\overline{\phi} \phi_j)$
$L_{jk} = L_{kj} = - \partial_j \partial_k |\phi|^2 + 4 \Re (\overline{\phi_j} \phi_k)$
$\{ f, g\}_m = \Im (f \overline{g})$
$\{ f, g\}^j_{p} = \Re (f \partial_j \overline{g} - g \partial_j \overline{f} )$
Local mass (non)conservation identity:
$$
\partial_t T_{00} + \partial_j T_{0j} = 2 \{ \mathcal{N} , \phi \}_m
$$
Local momentum (non)conservation identity:
$$\partial_t T_{0j} + \partial_k L_{kj} = 2 \{ \mathcal{N} , \phi
\}_p^j
$$
Consider $\mathcal{N} = F' (|\phi|^2) \phi$ for polynomial $F:{\mathbb{R}}^+ \rightarrow \mathbb{R}$.
Thus mass is conserved for these nonlinearities.
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.
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 $$ \partial_t V_a = M_a + \int_{{\mathbb{R}}^d} a (x) \{ \mathcal{N}, \phi\}_m (t,x) dx, $$ $$ \partial_t M_a = \int_{{\mathbb{R}}^d} (-\Delta \Delta a) |\phi|^2 + 4 a_{jk} \Re (\overline{\phi_j} \phi_k) + 2 a_j \{ \mathcal{N}, \phi\}_p^j dx. $$
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 $a (x) = |x|$.
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.
$ 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.
It is bounded: $$\Big|M[u](t)\Big| \lesssim \| u(t) \|_{L^2_x}^3 \| \nabla u(t) \|_{L^2_x}.$$
We calculate $$ \partial_t M[u] = \int_{{\mathbb{R}}^d_y} |u(t,y)|^2 \{\partial_t M_y [u] \} + \{\partial_t |u(y)|^2\} M_y [u] dy. $$
$\exists ~I, II, III, IV$ such that $I, III \geq 0$ and $II + IV \geq 0$ and $ \partial_t M[u] = I + II + III + IV. $ Integrating in time gives $$\int_0^T \int_{{\mathbb{R}}^3} |u(t,x)|^4 dx dt \lesssim \| u(t) \|_{L^\infty_T L^2_x}^3 \| \nabla u(t) \|_{L^\infty_T L^2_x}~ . $$
(Hassell 04)
satisfies an NLS-type equation on ${\mathbb{R}}^{1+6}$ $$ (i \partial_t + \Delta_1 + \Delta_2) \Psi = [F' (|\phi_1 |^2) + F' (|\phi_2 |^2)] \Psi. $$
$y=0$ 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...
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 ${\mathbb{R}}^3$.
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 $$ \| D^{\frac{3-d}{2}} |u|^2 \|^2_{L^2_{t,x}} \lesssim \| u_0 \|_{L^2_x}^3 \| \nabla u \|_{L^\infty_t L^2_x}. $$
(Hassel-Tao) [C-Holmer-Visan-Zhang]
$\overline{x}=$ 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$
satisfies a defocusing NLS equation on ${\mathbb{R}}^{1+4}$.