A book I'm reading ("An Introduction to PDE" by Pinchover and Rubinstien) compares the propagation of singularities in 1D wave equations and in 3D wave equations.
Authors states:
We proved (in an earlier discussion) that even when (1D wave equation) do have singular points, the solution is singular in exactly the same way, and the singularity propagate along characteristic curves
So far, so good. They proceed to discuss the 3D case:
The situation is different in the three-dimensional case, as smooth initial data might develop singularities over time.
To back those statements up, writers supply an example:
Given a wave equation, with initial condition (in polar coordinates): $$ u_{tt} - \Delta u = 0 \\ u(r, 0) = f(r) \\ u_{t}(r, 0) = 0 \\ $$ Where:
$$ f(r) = \begin{cases} 1 & r \leq 1, \\ 0 & r > 1 \end{cases} $$
Mark $\tilde{f}$ as the even expansions of the function $f$, one obtains a radial solution:
$$ u(r,t) = \frac{1}{2r}[(r+t)\tilde{f}(r+t) + (r-t)\tilde{f}(r-t)] $$
And now to the actual point: $$ u(r,t) \xrightarrow[r \to 0]{} f(t) + tf'(t) $$ (we've used the fact that $\tilde{f}$ is even and thus $\tilde{f}'$ is odd)
This function has a singularity at t=1, as pointed by the authors. They state:
... the above formula implies that the solution blows up at t=1 ... The reason for that spontaneous creation of singularities is geometric ... the singularity that started it's life as a 2D object (a sphere) later turned into a zero-dimensional object (a point)
Stuff I do not understand:
- This is not a case of "smooth initial data developing a singularity over time" or a "spontaneous creation of singularities". The initial data is obviously not smooth. Is this phenomena (creation of singularities over time) possible? May you kindly point me to an example?
- I think I can see where the authors are going - I'd expect waves propagating from the boundary of a circular pool to clash in the middle and make a mess. However, I think this is not completely clear from the example. Is my intuition correct? Can anyone supply me with a short analysis for such case, or a reference to such analysis?
Many thanks!