instantaneous smoothing
Hand a diffusion equation the ugliest initial data you can imagine — a jagged, discontinuous, barely-integrable profile — and ask what it looks like an instant later. The astonishing answer is: perfectly smooth. Not smoother, not eventually smooth, but infinitely differentiable for every time t > 0, no matter how small. The roughness is erased the moment the clock starts. This is instantaneous smoothing, the most distinctive and surprising feature of parabolic equations, and the abstract heart of the whole field.
Concretely: if u solves u_t = Laplacian u (or any parabolic equation) on a domain, then for every t > 0 the profile u(., t) is a C^infinity function of x — in fact real-analytic in x, and analytic in t too — even when the initial data u_0 was merely an L^1 or L^2 function with no derivatives at all. The mechanism is visible in the heat kernel: u(x, t) = integral of K_t(x - y) u_0(y) dy is a convolution against an infinitely smooth Gaussian K_t, and convolving anything against a smooth kernel produces a smooth result; the kernel's smoothness is inherited by the answer. The semigroup version is the estimate norm of A e^{-tA} u_0 <= (C/t) norm of u_0 from analyticity: applying a derivative costs only 1/t, so you can apply any number of derivatives at the price of a (still finite) power of 1/t. Iterating across all orders gives C^infinity.
Why it matters and where the honesty lies. Smoothing is the reason parabolic regularity theory exists: rough data is allowed, yet solutions are as nice as you like for positive time, which is exactly what lets you bootstrap weak solutions to classical ones and run nonlinear fixed-point arguments. But smoothing has a price that runs the other way: it is irreversible. Because the forward flow destroys information about fine structure, you cannot run it backward — the backward heat equation is ill-posed (Hadamard). Smoothing forward and ill-posedness backward are two faces of the same arrow of time. Contrast the wave equation, which has NO smoothing: a kink in the initial data stays a kink, propagating along characteristics forever.
Start the heat equation on the line from a step function u_0 = 1 for x < 0 and 0 for x > 0. For any t > 0, u(x, t) = (1/2) erfc(x / sqrt(4t)) — a smooth S-shaped curve, infinitely differentiable, with the sharp jump instantly rounded off. There is no first instant 'just after' the jump survives; smoothness appears at every positive time at once.
A discontinuous step becomes an infinitely smooth profile for every t > 0.
Smoothing is not free reversibility: gaining smoothness forward is exactly why information is lost and the backward problem is ill-posed. And it is specific to parabolic (and elliptic) equations — do not expect it from hyperbolic ones, where singularities persist and travel at finite speed. Smoothing also does not mean instantaneous equilibrium; the profile is smooth at once but still evolves and decays over the long run.