Huygens' principle or finite speed of propagation?

Question

I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.

For context, $v(t,r)$ is radial and solves a septic nonlinear wave equation $$-\partial_{tt}v + \Delta v = F(v),$$ on $(t,x)\in[T, T_1]\times \mathbb{R}^3$ where $F(v) = G(v) + g(r,t)$, where $G(0)=0$ and $g(r,t)$ is supported on $\{r=|x|<t+C\}$. We know that at time $T$, $v$ is supported in the ball $\{r=|x|\le T+C\}$ with $C>0$. The authors then claim, on page 34, that by Huygens' principle, it follows $v(r,t)$ is supported in the ball $\{r=|x|<t+C\}$ for each fixed $t\ge T$

However, I believe this simply follows from the finite propagation speed property of the wave equation. Is this correct, or is the Huygens' principle genuinely needed here?

Answer 1

As a remark, invoquing the finite propagation speed property for the wave equation is hand waving. How do one make this statement a rigorous mathematical concept ?

For a general hyperbolic PDE (or system of PDEs), you can make a definition, by looking at the velocity of planar waves. But then you have to transform this definition into a theorem and this requires pseudo-differential calculus.

For the wave equation, or more generally for so-called symmetric hyperbolic (systems of) PDEs, a much simpler proof occurs, which consists in integrating the energy equation $$\partial_t\frac12(v_t^2+|\nabla v|^2)-{\rm div}(v_t\nabla v)=-v_tF(v)$$ over the characteristic cone $t\in(0,T)$, $|x|<t+c$.

Mind that you need of course $F(0)=0$.

Answer 2

Two points:

1.

In the wave equations community (in contrast to the community for more general hyperbolic systems) it is rather common to sloppily interchange "Huygen's Principle" with "finite speed of propagation". As a point of evidence: you will often find papers referring to the "sharp Huygen's principle" when the fact that the fundamental solution is a distribution supported on a cone is required. The adjective "sharp" is introduced precisely because the phrase "Huygen's principle" is often taken to refer to something more general.

2.

The (sharp) Huygen's principle is a statement about free propagation of linear waves in $\mathbb{R}^3$. It certainly does not hold for general nonlinear wave equations (the solution will scatter on itself and leave a tail).

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Certainly Krieger and Schlag is simply referring to finite speed or propagation at that spot in their paper.

Answer 3

Q: Huygens' principle or finite speed of propagation?

It's the same thing, Huygens principle is a statement of causality, which means finite speed of wave front propagation.

Without Huygens principle you might imagine that the right-hand-side of the wave equation is a source term that allows a nonzero $v(x,t)$ to appear at $|x|>t+C$, without any relationship to the wavefronts at earlier times. The notion that a nonzero $v$ at some later time is causally related to a wavefront at an earlier time is the content of Huygens principle.

Note added in response to comment: the OP asks about $\mathbb{R}^3$, where all wavelets propagate with a single velocity $v\equiv 1$ and Huygens applies. In even dimensions all velocities $\leq v$ contribute. The support statement in the OP still holds, even though Huygens does not.