a distributional solution

Many natural solutions of PDEs are not smooth enough to plug into the equation in the ordinary way — a shock front, a jump in density, a point source. A distributional solution is a way of declaring something a solution even when its classical derivatives do not exist, by asking it to satisfy the equation in the averaged, integral-against-a-probe sense rather than pointwise. It is what lets the rough but physically real solutions count.

The recipe is to interpret all the derivatives in the equation as distributional derivatives. Take a PDE L u = f. A classical solution makes L u equal f at every point. A distributional solution instead asks that the distribution L u equals the distribution f, which by the definition of distributional derivative means (u, L-adjoint phi) = (f, phi) for every test function phi — every derivative has been integrated by parts onto the smooth probe, where it always exists. So u itself need only be a distribution; the burden of differentiation has been shifted entirely onto phi. If u happens to be smooth, integrating by parts back shows it is also a classical solution, so nothing is lost on smooth solutions.

Distributional solutions matter because they are the natural and often the only sensible notion for the hard cases. Shocks in conservation laws, the fundamental solution itself (which solves L E = delta in the distributional sense), and weak solutions of boundary-value problems all live here. The honest caveat: a distributional solution need not be unique. For nonlinear conservation laws many weak solutions can fit the same data, and an extra entropy condition is needed to single out the physical one.