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. A survey Ta2006a of recent advances in nonlinear wave equations based on ther criticality is available.

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

A nonlinear equation can often be written as a linear equation with a time-dependent potential $V$, which in turn depends nonlinearly on the solution. For instance, the NLS can be written as $i\partial _{t}u+\Delta u=Vu$, where $V=\pm |u|^{p-1}$.

In scattering theory one is often interested in the asymptotic behavior of a solution starting from small localized data. Simple heuristics stemming from Gronwall's inequality suggest that the time-dependent potential $V$ will have an asymptotically negligible impact on the evolution if its supremum norm $\|V(t)\|_{L_{x}^{\infty }(\mathbb {R} ^{d})}$ is absolutely integrable in time. On the other hand, the dispersive inequality suggests that the linear solution decays in supremum norm at a certain rate, which in the case of Schrodinger-type equations is $\langle t\rangle ^{-d/2}$. Assuming that the nonlinear solution also decays at this rate, one arrives at a heuristic as to when the linear behavior is expected to dominate (the short-range or scattering-subcritical regime), when the nonlinear behavior is expected to dominate (the long-range or scattering-supercritical regime), or when the two are finely balanced (the critical-range or scattering-critical regime). In the case of the NLS, these three regimes occur when $p>1+{\frac {2}{d}}$, $p=1+{\frac {2}{d}}$, and $p<1+{\frac {2}{d}}$ respectively.

Note that in contrast to the scaling-critical concept, it is generally the high powers which are scattering-subcritical rather than the low powers. Informally, this is because high power nonlinearities are weaker than low power nonlinearities when the solution is small and decaying.

Critical values of mass and energy

For equations are mass-critical (resp. energy-critical), such as the mass critical NLS, perturbation theory alone can often give global existence, regularity, and linear-type control when the mass (resp. energy) is sufficiently small. On the other hand, the existence of soliton solutions or blowup solutions at high mass (resp. energy) in focusing settings indicates that nonlinear behavior can occur when the mass (resp. energy) is sufficiently large. This indicates that there must be a critical value of mass (or energy), below which linear behavior always occurs, and above which nonlinear behavior can sometimes occur. Determining the exact critical value is often an important challenge; in many cases it is suspected that this value is equal to the mass (or energy) of the ground state.

There has been substantial progress recently on defocusing energy critical equations. An induction on energy strategy for attacking these problems appeared in Bo1999b in $\mathbb {R} ^{3}$. The radial symmetry assumption used in Bo1998b was removed using a new Morawetz inequality in CoKeStTkTa-p. Extensions to higher dimensions and to cases involving lower order subcritical terms have been carried out (and need to be properly surveyed here).

An optimal size condition for the focusing quintic energy critical nonlinear Schrodinger equation in $\mathbb {R} ^{3}$ with radial data has been identified in KnMe2006a. Beneath the threshold, scattering holds. Above the threshold, there exist finite time blowup solutions.

For energy supercritical problems, very little is known. A first positive result in Ta2006b establishes global regularity for a logarithmically supercritical nonlinear wave equation.