Parabolic regularity and scaling

For uₜ=Δu, the rescaled function uᵣ(x,t)=u(rx,r²t) satisfies the same equation on the correspondingly rescaled domain. One spatial derivative gains a factor r and one time derivative gains r². Thus an interior estimate naturally uses a backward cylinder with spatial radius r and time depth r². The word backward here describes the neighborhood feeding a forward estimate; it does not mean the ill-posed backward heat evolution is being solved. With a fixed positive diffusivity κ, the corresponding diffusion time scale is proportional to r²/κ.

For a smooth homogeneous heat solution on a cylinder twice as large, an interior gradient estimate bounds the gradient on the smaller cylinder by a constant times r⁻¹ times the solution supremum. Higher spatial and time derivatives receive their matching powers. Constants depend on dimension and the chosen nested-cylinder proportions. The estimate is local and stays away from the initial or lateral boundary. It cannot be extended to t=0 with the same constant for arbitrary rough data, nor can it guarantee smooth attainment of incompatible boundary and initial values.

Parabolic Hölder spaces compare space increments with the square root of time increments. For a scalar nondivergence equation uₜ−aᵢⱼuᵢⱼ=f, uniformly positive definite coefficients that are Hölder continuous with exponents α in space and α/2 in time, together with similarly Hölder forcing, support interior Schauder estimates. In their a priori form these bound the second spatial and first time derivatives, including their Hölder behavior, by the solution size and forcing norm on a larger cylinder. Such an estimate assumes an appropriate solution and does not alone construct one.

Uniform ellipticity by itself does not make discontinuous coefficients smooth. A divergence-form equation with a jump in conductivity can have a continuous temperature but a jump in its gradient while its flux remains continuous. The weak formulation is designed to accommodate that behavior. Differentiating the coefficient as if it were a classical smooth function would introduce unjustified terms. Distinguish divergence from nondivergence form when invoking a regularity theorem, and match the data spaces and coefficient assumptions to that theorem rather than borrowing a conclusion from constant-coefficient heat flow.

A regularity argument usually begins with a solution obtained by a kernel, Galerkin method, or another existence construction. It then proves additional local properties under suitable assumptions. A global classical solution up to a boundary needs more: boundary smoothness, regularity of prescribed values, and compatibility where the initial slice meets the lateral boundary. An interior derivative estimate does not address those corners. Similarly, a nonlinear equation may be uniformly parabolic only while its solution remains in a bounded range separated from degeneracy.

Fourier modes provide a quantitative view of how high derivatives become controlled after positive time. A mode n has gradient size proportional to n exp(−n²t) for unit diffusivity. Initially this grows with n, but at fixed positive time the exponential dominates. Maximizing the analogous continuous-frequency expression gives a scale proportional to $t^{-1/2}$, consistent with the interior estimate. This explains how initial high-frequency roughness can disappear without any uniform derivative bound at the initial instant. The argument describes smoothing, not a stable inversion of the lost frequencies.