Two fates for a corner
The first three guides of this rung were all about size: the maximum principle caps where a solution can peak, comparison ranks two solutions, and Harnack's inequality forbids a positive solution from being huge in one spot and tiny right next door. This guide asks a different question, about texture. Take a solution that starts out rough — a temperature with a sharp corner, a plucked string with a kink, a charge sitting at a single point — and watch what the equation does to that roughness as the solution evolves. The astonishing fact is that the answer is dictated by the type of the equation, and it comes in exactly two flavours that could not be more opposed.
On one side sit the elliptic and parabolic equations — Laplace's equation and the heat equation are the headline examples. They are smoothers. The instant you are inside the region (for elliptic) or at any positive time (for parabolic), the solution is infinitely differentiable, no matter how jagged the data was. Corners vanish. On the other side sit the hyperbolic equations — the wave equation is the model. They are propagators. A kink in the data does not soften at all; it survives in full, riding along a characteristic curve at a finite speed, and you can point to exactly where it has travelled to.
Why diffusion sands off every corner
Start with the smoothing side, because it has a satisfying mechanism. Recall the heat kernel picture from the heat rung: solving u_t = k u_xx is convolving your data against a Gaussian, and a Gaussian is infinitely smooth. Convolution always inherits the smoothness of the smoother factor, so even data with a jump becomes C-infinity the instant t > 0. The Fourier view says the same thing more sharply: roughness lives in the high-frequency modes, and the factor e^(-k n^2 t) crushes the mode of wavenumber n. For any t > 0, the n^2 in that exponent annihilates the high harmonics almost completely, and what is left has no fast wiggles left to be rough with.
Made into a theorem, this is the instantaneous smoothing property of parabolic equations: for any t > 0 the solution is C-infinity in space, even real-analytic, regardless of how bad the initial data was. Elliptic equations carry the matching statement at a single frozen time — there is no time at all, only the interior of a region — under the name interior regularity. Here is its punchline, the elliptic regularity theorem in one breath: if Laplacian u = f and f is smooth on some open set, then u is smooth there too, however rough u or its boundary data may be elsewhere. A harmonic function (f = 0) is automatically real-analytic in the interior. The equation forces smoothness from the inside out.
There is a lovely way to feel why an elliptic equation must do this. A harmonic function obeys the mean-value property: its value at a point equals its average over any small sphere around that point — exactly the averaging that powered the maximum principle in guide 1. Averaging is a smoothing operation. A function that is forced to equal its own local averages everywhere can have no isolated spikes, no creases, no corners; the averaging would instantly betray them. Smoothness is not an extra gift here — it is baked into what 'being equal to your own averages' even means.
Why waves carry a kink forever
Now the opposite fate. The cleanest possible window is d'Alembert's formula for the wave equation u_tt = c^2 u_xx on the line: the general solution is u(x,t) = F(x - c t) + G(x + c t), a right-mover and a left-mover gliding at speed c without changing shape. Nothing in that formula smooths anything. If your initial profile F has a corner at x = 0, then at time t that corner sits at x = c t, every bit as sharp as it began. The wave equation transports the data; it never averages it. A kink is a passenger, not a problem to be dissolved.
This is the heart of propagation of singularities: in a hyperbolic equation, a singularity in the data does not vanish and does not appear from nowhere — it travels, and it travels along the characteristic curves, the same curves you learned to find back in the first-order and classification rungs. Where the data is smooth, the solution is smooth; where it has a kink, that kink is conserved and carried at finite speed. The technical name for the set where a solution fails to be smooth is its singular support, and the theorem is that for a wave equation the singular support simply moves along the characteristics. You can predict exactly where the roughness will be at any later time.
Finite speed vs infinite speed: where the line is drawn
Smoothing and speed are two faces of one coin. The wave equation has finite speed of propagation: a disturbance confined to one spot at t = 0 cannot be felt outside the cone |x| <= c t. There is a sharp wavefront, an exact region the signal has not yet reached, fenced off by the characteristics — the domain of dependence is a genuine, finite-sized triangle, not the whole line. The heat equation, by contrast, has infinite speed of propagation: the Gaussian heat kernel is strictly positive for every x at every t > 0, so a pulse of heat at the origin is felt everywhere instantly — undetectably small far away, but never exactly zero.
Notice the deep coupling. Infinite speed is what lets the heat equation smooth: to feel a faraway source instantly is to be a global average of the data, and global averages are smooth. Finite speed is what forces the wave equation not to smooth: if a signal can only reach you by travelling along a characteristic, then whatever rode in on that characteristic — including its corner — arrives intact. So the smoother is the gossip who hears everything at once and blends it into a rumour; the propagator is the runner who carries one sealed letter at fixed speed and delivers it unopened. You do not get to be both.
SMOOTHERS PROPAGATORS
elliptic / parabolic hyperbolic
Laplacian u = 0, u_t = k u_xx u_tt = c^2 u_xx
-------------------------------- --------------------------------
texture | corners vanish; C-inf | corners kept; ride forever
speed | infinite (positive K) | finite, equals c
carrier | global average | F(x - c t) + G(x + c t)
fate of | erased, instantly | moves along the characteristic
a kink | smoothed away | x = c t (singular support)
reverse? | no (irreversible) | yes (time-reversible)Honest edges: what smoothing does and does not promise
Three cautions, so you do not over-read the rule. First, smoothing is interior. Elliptic regularity smooths the inside of a region; it says nothing about the boundary, where the solution can stay exactly as rough as the boundary data you imposed. A harmonic function with a jagged Dirichlet condition is analytic in the interior yet pinned to its corners on the edge — smoothness blooms only as you step away from the boundary. The parabolic version is the same: smooth for t > 0, but the data at t = 0 is allowed to be as ugly as you like.
Second, the price of smoothing is paid backward in time. Diffusion smooths because it destroys fine detail, and a process that destroys information cannot be run in reverse. That is why the backward heat equation is ill-posed — the very thing that makes forward diffusion so well-behaved makes the past unrecoverable. The wave equation pays no such price: because it merely transports, losing nothing, it runs equally cleanly backward in time. Smoothing and irreversibility are bundled together; finite speed and time-reversibility are bundled together. You cannot unbundle them.
Third, watch the word singularity itself. A fundamental solution — the heat kernel, or the wave equation's response to a point source — is a response to a Dirac delta, which is a distribution, not a function, and so it is singular by construction. Smoothing is a statement about how honest data propagates, not about the kernel: the heat kernel smooths anything you convolve it with, yet the kernel itself carries a singularity sitting right at t = 0, x = 0. And for nonlinear hyperbolic equations the rule has a genuine exception — smooth data can create a singularity that was not there before. A smooth solution of a conservation law can steepen until it forms a shock in finite time, and after that a weak solution is not even unique without an entropy condition. So 'singularities only propagate, never appear' is a linear statement; the nonlinear world can manufacture them.
How to read off the behaviour
Put it to work. Faced with a second-order linear equation A u_xx + 2B u_xy + C u_yy + ... = 0, you already know how to classify it: compute the discriminant B^2 - A C. The very same sign that named the type now tells you the texture behaviour for free — that is the payoff of having classified equations in the first place.
- Compute the discriminant B^2 - A C of the principal (highest-order) part — exactly the classification step from the type rung.
- If B^2 - A C < 0 it is elliptic (no real characteristics): expect interior smoothing — the solution is C-infinity inside, however rough the boundary data.
- If B^2 - A C = 0 it is parabolic (one characteristic family): expect instantaneous smoothing for t > 0, infinite speed, and irreversibility.
- If B^2 - A C > 0 it is hyperbolic (two real characteristics): expect NO smoothing — singularities ride along the characteristics at finite speed, and the past is recoverable.
Carry away one sentence. Elliptic and parabolic equations smooth their data instantly and globally; hyperbolic equations propagate every singularity, undimmed, along their characteristics at finite speed — and which fate you get is read straight off the type, the same B^2 - A C you computed long ago. The next and final guide of this rung adds the last chapter to the qualitative story: not just whether a solution stays rough, but how it decays over long times, using energy methods to watch a vibrating or diffusing system settle toward rest.