Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Singular behavior [in PDEs]

An example of an equation that yields inconsistent behavior is the diffusion equation with a negative diffusion constant:

tu[t, x] -xxu[t, x]

This equation makes any variation in u as a function of x eventually become infinitely rapid.

Many equations used in physics can lead to singularities: the Navier–Stokes equations for fluid flow yield shock waves, while the Einstein equations yield black holes. At a physical level, such singularities usually indicate that processes not captured by the equations have become important. But at a mathematical level one can simply ask whether a particular equation always has solutions which are at least as regular as its initial conditions. Despite much work, however, only a few results along these lines are known.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]