J. Colliander (UBC)¶
Dilation invariance quantifies the balance between these effects.
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.
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 |
Admissibility for independent pairs $(q,r), (Q,R)$ $$ \frac{2}{q} + \frac{d}{r} = \frac{d}{2}, ~ q > 2. $$ The $'$ denotes Hölder dual exponent.
$\exists$ unique $u \in C([0, T_{lwp} ]; L^2 ) \cap L^4_{tx} ([0,T_{lwp}] \times \mathbb{R}^2)$ solving $NLS_3^{+} (\mathbb{R}^2)$.
for all $\delta > 0$ but diverges to $\infty$ as $\delta \searrow 0$.