We know that the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $ \delta(x)=\begin{cases}+\infty, &x=0\\ 0, &x\neq 0 \end{cases} $ and which is also constrained to satisfy the identity $ \int_{-\infty}^{\infty}\delta(x)dx=1 $ However, this is merely a heuristic characterization. The Dirac delta function is rigorously defined either as a distribution or as a measure.
Here comes the problem. I've seen many PDEs that contain delta functions, which are not functions at all, in the traditional sense. For example the following:
Stokes equations with singularly forced: $ \begin{align} -\nabla p+\mu\nabla^2{\bf u}+{\bf g}\delta({\bf x}-{\bf x}_0) &= 0,\\ \nabla\cdot {\bf u} &= 0 \end{align} $ where ${\bf g}=(g_i)_{1\leq i\leq 3}\in{\mathbb R}^3$ is an arbitrary constant vector, ${\bf x}_0$ is an arbitrary point in the domain, and $\delta$ is the three-dimensional Dirac delta function.
Here are my questions:
- For the first equation, the first two terms, $-\nabla p$, $\mu\nabla^2{\bf u}$, are functions in the traditional sense, how should I understand "a traditional sense function plus a distribution/measure"?
- Generally, what does a PDE mean when it contains a measure or distribution?