Critical

In the theory of nonlinear PDE, we use the terms sub-critical, critical, and super-critical to denote a significant transition in the behaviour of a particular equation with respect to a specified regularity class (or conserved quantity). Typically, subcritical equations behave in an approximately linear manner, supercritical equation behave in a highly nonlinear manner, and critical equations are very finely balanced between the two.

This distinction into sub-critical, critical, and super-critical behaviour can be made in several contexts:

• Given a fixed equation with a scale invariance (or approximate scale invariance), one can classify various quantities (such as homogeneous Sobolev norms $\|u(0)\|_{{\dot {H}}_{x}^{s}(R^{d})}$, or various conserved quantities) as sub-critical, critical, or super-critical with respect to this scaling invariance. Example: In the cubic NLS on R^3, the energy or ${\dot {H}}_{x}^{1}(R^{3})$ norm is sub-critical, the momentum or ${\dot {H}}^{1/2}(R^{3})_{x}$ norm is critical, and the mass or $L_{x}^{2}(R^{3})$ norm is super-critical.

• Conversely, given a fixed regularity class or conserved quantity, one can classify an equation as sub-critical, critical, or super-critical with respect to that class or quantity. Example: the NLW and NLS on $R^{3}$ is energy-subcritical for $p<5$, energy-critical for $p=5$, and energy-supercritical for $p>5$.

• Occasionally one also discusses the sub-criticality, criticality, or super-criticality of regularities with respect to other symmetries than scaling, such as Galilean invariance or Lorentz invariance.

• When analyzing the long-time asymptotics of nonlinear equations with data which is small and highly

localized, one can distinguish equations into scattering-subcritical (short-range), scattering-critical (critical-range), and scattering-supercritical (long-range) classes; this measures the relative asymptotic strength of the nonlinear and linear components of the equation, and is generally unrelated to the distinction arising from scale invariance or other symmetries. Example: The NLS on $R^{3}$ is scattering-subcritical for $p>5/3$, scattering-critical for $p=5/3$, and scattering-supercritical for $p<5/3$.

• In focusing equations with a critical conserved quantity (e.g. mass), the behavior for small values of this quantity is often quite different from that at large values. In many equations there is a critical value of this quantity, below which linear-type behavior is expected, and above which nonlinear behavior can occur. (When one is exactly at the critical value, soliton-type behavior is very typical.) Example: in a mass critical NLS, masses less than the mass $M(Q)$ of the ground state are considered subcritical, masses greater than $M(Q)$ are supercritical, and the mass $M(Q)$ itself is critical.

Criticality with respect to scaling

A large number of equations studied here enjoy a scale invariance, which typically takes a form such as

$u^{(\lambda )}(t,x):={\frac {1}{\lambda ^{a}}}u({\frac {t}{\lambda ^{b}}},{\frac {x}{\lambda ^{c}}})$

where the scaling parameter $\lambda >0$ is arbitrary, and $a,b,c$ are constants (with $b,c$ positive) which depend on the various parameters of the equation (the linear and nonlinear components of the equation, and the ambient dimension). Thus any given solution $u$ has fine-scale counterparts $u^{(\lambda )}$ with $\lambda \ll 1$, as well as coarse-scale counterparts $u^{(\lambda )}$ with $\lambda \gg 1$. Typically, the fine-scale counterparts are rapidly oscillating, highly concentrated, and can exhibit nonlinear behaviour in short amounts of time, whereas the coarse-scale counterparts are very smooth, spread out over large regions of space and time, and behave linearly for long periods of time.

Many quantities $Q(u)$ associated to a solution, such as a spatial norm (e.g. $Q(u)=\|u(0)\|_{{\dot {H}}_{x}^{s}(\mathbb {R} ^{d})}$), a spacetime norm (e.g. $Q(u)=\|u\|_{L_{t}^{q}L_{x}^{r}(\mathbb {R} \times \mathbb {R} ^{d})}$), or a conserved quantity (e.g. $Q$ could be mass, momentum, or energy) transform in a simple manner under the above scale invariance. Indeed, if $Q$ is homogeneous, then we obtain a relationship of the form

$Q(u^{(\lambda )})=\lambda ^{\alpha }Q(u).$

for some constant $\alpha$ which depends on the scaling symmetry and on the degree of homogeneity of $Q$; from a dimensional analysis perspective, $\alpha$ measures the dimension of $Q$ in terms of the unit of length (which is represented here by $\lambda$). For instance, for the NLS, the mass

$M(u):=\int _{\mathbb {R} ^{d}}|u(0,x)|^{2}\ dx$

interacts with the scaling invariance via the formula

$M(u^{(\lambda )})=\lambda ^{d-{\frac {4}{p-1}}}M(u)$

so in this case $\alpha =d-{\frac {4}{p-1}}$. The sign of $\alpha$ is then used to classify the quantity $Q$:

• If $\alpha <0$, then the fine-scaled copy $u^{(\lambda )}$ has a much larger value of $Q$ than the original solution $u$. Hence, if we are keeping the value of $Q$ fixed, then we do not expect fine-scale behavior to occur, while conversely we expect plenty of coarse-scale behavior. In this case we say that the quantity $Q$ is sub-critical for this equation, or equivalently that the equation is sub-critical with respect to this quantity.

• If $\alpha >0$, then the fine-scaled copy $u^{(\lambda )}$ has a much smaller value of $Q$ than the original solution $u$. Hence, if we are keeping the value of $Q$ fixed, then we expect a substantial amount of fine-scale behavior to occur, though conversely we do not expect much coarse-scale behavior. In this case we say that the quantity $Q$ is super-critical for this equation, or equivalently that the equation is super-critical with respect to this quantity.

• If $\alpha =0$, then $Q$ is scale-invariant, and so for a fixed value of $Q$ we expect the same behavior at all scales. In this case we say that the quantity $Q$ is critical for this equation, or equivalently that the equation is critical with respect to this quantity.

Typically, we expect the sub-critical classification at low dimensions, high regularities, and nonlinearities with low exponents or few derivatives, and conversely expect the super-critical classification at high dimensions, low regularities, and nonlinearities with high exponents or many derivatives. There is typically only one value $s_{c}$ for which the norm $\|u(0)\|_{{\dot {H}}_{x}^{s_{c}}(\mathbb {R} ^{d})}$ is critical; this value is known as the critical regularity.

The behavior of an equation at short and long time scales, and at given levels of regularity, is often controlled by the distinction into sub-critical, critical, and super-critical quantities via the scaling heuristic.

The division of quantities into sub-critical, critical, and super-critical behavior can also be extended somewhat to inhomogeneous quantities (such as inhomogeneous Sobolev norms $H_{x}^{s}(\mathbb {R} ^{d})$), for instance by splitting such quantities into homogeneous components of differing degrees. However, the fine-scale and coarse-scale behavior can then become distinct. For instance, for the cubic NLS on R^3, the $H_{x}^{1}(\mathbb {R} ^{3})$ norm is sub-critical at fine scales and super-critical at coarse scales, thus excluding both fine-scale and coarse-scale behaviour simultaneously. A similar analysis allows one to extend these useful concepts to the case when the equation itself is inhomogeneous (as is for instance the case with the nonlinear Klein-Gordon equation). See also high-frequency limit and low-frequency limit.

Criticality with respect to other symmetries

It is sometimes profitable to generalize the trichotomy of sub-critical, critical, and super-critical quantities for an equation by replacing the scaling symmetry with another non-compact symmetry, such as translation symmetry, Galilean symmetry, or Lorentz symmetry. Whereas the scaling symmetry links fine-scale to coarse-scale behavior, the translation symmetry relates behavior near the origin to behavior away from the origin, while Galilean and Lorentz symmetries relate low-velocity (or low center frequency) behavior to high-velocity (or high center frequency) behavior. The classification becomes less rigorous, because most quantities do not transform cleanly under these other symmetries. Nevertheless, on a heuristic level at least one can classify at least some quantities and norms. For instance:

• With respect to translation symmetry, translation-invariant quantities such as $\|u(0)\|_{H_{x}^{s}(\mathbb {R} ^{d})}$ or $\|u\|_{L_{t}^{q}L_{x}^{r}(\mathbb {R} \times \mathbb {R} ^{d})}$ are critical. On the other hand, weighted norms such as $\|u(0)\|_{L_{x}^{2}(\langle x\rangle ^{s}dx)}$ are sub-critical for $s>0$ and supercritical for $s<0$.

• With respect to Galilean symmetry, Lebesgue norm quantities such as $\|u(0)\|_{L_{x}^{2}(\mathbb {R} ^{d})}$ or $\|u\|_{L_{t}^{q}L_{x}^{r}(\mathbb {R} \times \mathbb {R} ^{d})}$ are critical. On the other hand, Sobolev norm quantities such as $\|u(0)\|_{H_{x}^{s}(\mathbb {R} ^{d})}$ are sub-critical for $s>0$ and supercritical for $s<0$.

• With respect to Lorentz symmetry for the free wave equation, the quantity $\|u(0)\|_{{\dot {H}}_{x}^{1/2}(\mathbb {R} ^{d})}^{2}+\|u_{t}(0)\|_{{\dot {H}}_{x}^{-1/2}(\mathbb {R} ^{d})}^{2}$ is critical. Regularities higher than this are then considered Lorentz-subcritical, while regularities lower than this are Lorentz-supercritical.

These classifications are useful in augmenting the scaling heuristic with considerations arising from other symmetries, and also in suggesting what types of norms and quantities would be relevant in any given problem (for instance, with a problem which is critical with respect to Galilean symmetry, one would expect to rely primarily on norms and quantities which are similarly critical).

Criticality with respect to scattering

To be written...

Critical values of mass and energy

To be written...