### A cute little estimate for wave-type equations

#### by Willie Wong

Let us consider first the linear wave equation

on (1+d)-dimensional Minkowski space. A well known property of the wave equation is the *conservation of energy*, which can be written as

, where

and proven by simply multiplying the wave equation by and integration-by-parts along a spatial slice.

Now, using the fact that partial derivatives commute, we also have that all derivatives of solve wave equations

, where is a multiindex

and hence has an associated energy conservation . This, in particular, implies that for any , the quantity

evaluated at some time is bounded by the same quantity evaluated at time .

Now let us take a slightly more geometric point of view: the fact that “partial derivatives commute against the d’Alembertian operator” is, in fact, a manifestation of the fact that translations are symmetries of Minkowski space. More precisely, consider an arbitrary pseudo-Riemannian manifold , and let denote its associated Laplace-Beltrami operator, which in coordinates can be written as $\frac{1}{\sqrt{|g|}} \partial_i g^{ij}\sqrt{|g|} \partial_j$, we see that if is a Killing vector field, that is, the Lie derivative , then . In other words, we have that for an arbitrary thrice differentiable function on ,

whenever is Killing.

Going back to Minkowski space, we see that if is a Killing vector field of the Minkowski metric (for example, translation, rotation, Lorentzian boosts), then for a solution to the wave equation, so will be a solution to the wave equation. And in particular, the computation for energies gives that is conserved.

What if we are in a variable coefficient case where the spatial translations are not Killing vector fields? Going back to the general pseudo-Riemannian case, we have that, after a computation

which means that , for a solution to the wave equation, now solves an inhomogeneous wave equation, and we no longer have a conservation law. In fact, the best estimate we can get in this situation is from Gronwall’s inequality applied to the energy identity, that the energy grows possibly exponentially.

Now suppose we are dealing with the equation in divergence form

with independent of . Since are no longer Killing with respect to the implicit metric, at first sight, it seems that we lose control over the higher order energies. (The lowest order energy can still be controlled in the same way: with the assumption that the coefficients is time-independent, the integration by parts carry through and we have that

is conserved in time.) But in this particular case we can actually take advantage of elliptic regularity and get controls for all higher derivatives. This trick is used a few times in the work of Dafermos-Rodnianski on decay of waves on black hole backgrounds.

What we do is first, notice that the equation is invariant under , so we have

which implies that is a conserved energy. The elliptic regularity we will exploit is the assumption that is sufficiently smooth, and uniformly elliptic, that is

Assumption on

for constants with for sufficiently large , and that there exists constants such that for any , we have .

Now consider the following integral:

for some arbitrary function (which doesn’t have to be a solution to the wave equation. Using elliptic regularity, the integrand is controlled by

which we “anti-derive by parts”

We integrate both sides. The first term on the right hand side we perform a double integration by parts, the second and third terms on the right hand side we use the elliptic regularity on , and just case the derivative of the coefficient matrix in sup norm. So we have

and applying Cauchy-Schwarz (Young’s weighted inequality) to the last term, which gives

we can conclude, by absorbing the last term to the left hand side, that

Now examine the first term in the integrand on the right hand side. Suppose for some solution of the wave equation. Commuting all the way in, we have

In the first term on the right hand side, we can apply the equation. So combining everything we have that

Now defining , the above is a recursion relation

By iterating the recursion relation, we have that for some large constant depending on such that

The significance of the above expression is that, by definition,

and so, if we know that our data has initial data with and , the above derivation shows that the quantity

will remain bounded for all times by a constant depending on above and the Sobolev norms of the initial data. And in particular, the “bad scenario” in which the higher order energy grows exponentially cannot happen.