Scaling criticality and invariant norms

On the whole space, consider i u_t plus the Laplacian of u equals kappa |u| to the p minus one times u, with p greater than one. Scaling space by lambda forces Schrödinger time to scale by lambda squared. If amplitude scales by lambda to the a, the linear terms carry a plus two powers while the nonlinearity carries ap powers. Equality gives a=2/(p−1). Kappa is fixed; the transformation preserves the equation rather than changing its coefficient.

For a sufficiently regular decaying profile, the homogeneous Sobolev norm of order s then gains the factor lambda to the 2/(p−1)+s−d/2. The invariant order is s_c=d/2−2/(p−1). This calculation uses the whole-space Fourier scaling and homogeneous norms where defined. The inhomogeneous H to the s norm includes a low-frequency term with a different scaling, and a fixed torus is not invariant under arbitrary spatial dilation. Those distinctions matter when transferring the diagnostic to a particular theorem.

Mass is the squared L² norm, so mass-critical means s_c=0, or p=1+4/d. The gradient part of energy lives at homogeneous H¹, so the energy-critical power is p=1+4/(d−2) when d is at least three. In dimensions one and two there is no finite power with that H¹ scaling equation. Cubic NLS in one dimension is mass-subcritical, cubic NLS in three dimensions is energy-subcritical, and quintic NLS in three dimensions is energy-critical.

The names are relative to a specified norm. If the available regularity s exceeds s_c, high-frequency concentration increases that homogeneous norm, giving a subcritical setting. If s is below s_c, concentration can shrink the controlled norm while sharpening the profile, giving a supercritical setting relative to that control. At equality, there is no spare scaling factor to supply a uniform small time just from the norm's size. Critical proofs often need refined spacetime control, small data, concentration analysis, or a structure not visible in dimensional bookkeeping.

For unforced incompressible Navier–Stokes on three-dimensional whole space, velocity rescales as lambda times u at time lambda squared t and position lambda x; pressure rescales by lambda squared. The viscosity stays fixed. Velocity L to the q norms scale by lambda to the one minus three over q, so L³ is invariant. Homogeneous H to the one half is also invariant. In contrast, velocity L² scales by lambda to the minus one half, meaning a concentrated rescaled profile can have smaller energy norm.

Thus the global energy inequality controls a supercritical quantity for three-dimensional velocity. It remains enormously useful for weak compactness, but does not automatically exclude increasingly fine scales in a strong solution. A formal rescaling of one profile neither constructs a singular solution nor proves that singularity is possible for the specified data. Boundary conditions, forcing, and the domain can also obstruct exact scaling. Use the diagnostic to locate what an estimate fails to control, then state the separate theorem that supplies any positive continuation conclusion.