Singular behavior [in PDEs]

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

∂[u[t, x], t] ==-∂[u[t, x], x, 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.