Reference request for the focussing example

Question

I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. 48, No 3.

There is a mention on energy estimates being fixed time estimates at a specified time, but requiring a lot of regularity in $L^p.$ Then there is a mention of the focussing example.

The article talks of how it is an example with initial data dispersed near the unit sphere and the solution $u$ focuses at the origin at time $t=1$ with high $L^{\infty}$ norm.

I was wondering if anyone can provide any references for the Focussing example, or if they know anywhere I can find out more about it?

Thank you in advance for you comments.

Answer 1

The example is almost trivial. For argument sake fix the number of spatial dimensions to be 3. If you consider radially symmetric solutions to the wave equation, one observes that if

$$ u_{tt} = u_{xx} + u_{yy} + u_{zz} $$

is radially symmetric (meaning that the dependence on $x,y,z$ is only through the dependence on $r := \sqrt{x^2 + y^2 + z^2}$) we have that

$$ (ru)_{tt} = (ru)_{rr} $$

or that the function $v:= ru$ solves a one-dimensional wave equation on $\mathbb{R}_+$. The boundary condition that $v = 0$ when $r = 0$ means that we can extend $v$ to be an odd solution of the one-dimensional wave equation on $\mathbb{R}$. L'Hopital's rule show that $u(t, r=0) = v'(t, r=0)$, and so we see explicitly the regularity loss. By plugging in for $v$ a pair of traveling waves, you can construct solutions $u$ to the original wave equation such that at the initial time $\|u\|_{L^p}$ is arbitrarily small, but at time 1 $\|u\|_{L^\infty}$ is arbitrarily large.