### Decay of waves IIIa: nonlinear tails in Minkowski space redux

Before we move on to the geometric case, I want to flesh out the nonlinear case mentioned in the end of the last post a bit more. Recall that it was shown for generic nonlinear (actually semilinear; for quasilinear and worse equations we cannot use Duhamel’s principle) wave equations, if we put in compact support for the initial data, we expect the first iterate to exhibit a tail. One may ask whether it is possible that, in fact, this is an artifact of the successive approximation scheme; that in fact somehow it always transpires that a conspiracy happens, and all the higher order iterates cancel out the tail coming from the first iterate. This is rather unlikely, owing to the fact that the convergence to $\phi_\infty$ is dominated by a geometric series. But to just make double sure, here we give a nonlinear system of wave equations such that the successive approximation scheme converges after finitely many steps (in fact, after the first iterate), and so we can also explicitly compute the rate of decay for the nonlinear tail. While the decay rate is not claimed to be generic (though it is), the existence of one such example with a fixed decay rate shows that for a statement quantifying over all nonlinear wave equations, it would be impossible to demonstrate better decay rate than the one exhibited.

In this particular case we will consider the following system of wave equations on 1+3 dimensional Minkowski space, with the assumption of spherical symmetry in force.

Equation 21
$-\partial_t^2 \phi + \triangle \phi = 0$
$-\partial_t^2 \psi + \triangle \psi = 0$
$-\partial_t^2 \zeta + \triangle \zeta = (\partial_t\phi + \partial_r\phi)\cdot(\partial_t\psi - \partial_r\psi)$

The system as a whole, as an equation on the vector $(\phi,\psi,\zeta)$ is nonlinear, since the right hand side depends nonlinearly on the vector itself. However, we recognize that taken individually the components of the vector decouple, and each component solves a (possibly inhomogeneous) wave equation with no nonlinearities. (So we cheat a little bit. But this is how we can make the successive approximation scheme converge after finitely many steps.)

Recall that under the spherical symmetry change of coordinates, writing again $u = \frac12(t-r)$ and $v=\frac12(t+r)$, we can rewrite the equations as

Equation 21′
$\partial_u\partial_v(r\phi) = 0$
$\partial_u\partial_v(r\psi) = 0$
$\partial_u\partial_v(r\zeta) = -(v-u) \partial_v\phi \cdot \partial_u \psi$

Why this particular form on nonlinearities? It is known that generic nonlinearities, quadratic in $\partial\phi$, lead to formation of singularities in finite time, with the number of spatial dimensions is less than or equal to three. In dimension 1, we can explicitly illustrate this: consider the wave equation $-\partial_u\partial_v \varphi = (\partial_v\varphi)^2$. Then along the curves of constant $v$, the quantity $\partial_v \varphi$ verifies an equation of the form $y' = -y^2$, whose solution is $y = \frac{1}{x+c}$. So if $\partial_v\varphi$ is negative at any point when $t = 0$, the solution will develop a singularity in finite time.

In low spatial dimensions, the rates of dispersion are slower (the energy density is expected to drop as the inverse of the area of the wave front, which in $d$ dimensions grows like $t^{d-1}$), and compared to the positive feedback coming from a generic quadratic derivative nonlinearity, may not be enough to radiate away the self-interaction sufficiently fast. This motivated my thesis advisor, Sergiu Klainerman, to study, when he was doing his own PhD, the conditions sufficient to guarantee the quadratic nonlinearities are not too strong. This lead to the recognition that the so-called null structures play an important role in the study of wave equations in low spatial dimensions. To cut the story short, the form chosen for the nonlinearity above renders it a null form, so it should be better behaved (decay faster) than the most general quadratic type nonlinearities. By choosing this example, my goal is to illustrate the fact that even in the good scenarios, one expect to have some fairly strong nonlinear tail effects. (A detailed discussion of this topic would take us too far from the focus of this post. Perhaps I’ll write about it in a part IV to this series.) (That we need a $\phi$ and a $\psi$ is also related. If we only have the function $\phi$ and choose the nonlinearity for $\zeta$ to be $\partial_v\phi\partial_u\phi$, we see that the null form cancellation is too strong, and we won’t be able to see the effect at the level of the first iterate.)

Anyway, back to the example.

Figure 4 (again)

The key observation is captured already in Figure 4 (which we recall again here above): when the red dot is “sufficiently far in the future”, the crosshatched region only intersects the “outgoing legs” of the cyan region (using the fact the incoming wave, after bouncing off the center axis, becomes outgoing also). In other words, where the crosshatched region and the cyan region intersects, the wave is purely outgoing.

Now suppose the cyan region is a representation of the support of $\phi$ and $\psi$, and we are trying to evaluate $\zeta$ at the red dot (the part coming from initial data can be neglected due to linearity). To do so we integrate $-(v-u)\partial_v\phi\partial_u\psi$ in the intersection of the crosshatched and cyan regions. Now in that region, Equation 21′ implies that $r\phi$ and $r\psi$ are outgoing waves, that is there $\partial_v$ derivatives vanish. Furthermore the wave equation also means their $\partial_u$ derivatives are constant along lines of constant $u$.

Equation 22
$0 = \partial_v (r\phi) = \phi + r\partial_v\phi$
$c(u) = \partial_u(r\psi) = -\psi + r\partial_u\psi$

Using the constancy of $r\phi$ along constant $u$ lines, we have that $\phi = c_\phi(u)/r$, and similarly $\psi = c_\psi(u)/r$. So we get that

Equation 23
$\displaystyle r\partial_v\phi = - \frac{c_\phi(u)}{r}$
$\displaystyle \partial_u\psi = \frac{c(u)}{r} + \frac{c_\psi(u)}{r^2}$

Now, the area integrated over is determined both by the cyan and crosshatched regions. The cyan region coming from $\phi,\psi$ determins the $u$ limits of the integration, while the crosshatched region determines the $v$ limits. We will just assume that the $u$ limits are some fixed values $u_l$ and $u_r$, which is completely determined by the compact support of the initial data. The $v$ limits, however, depends on where we will evaluate $\zeta$. If we choose to evaluate $\zeta$ at point $(u_0,v_0)$, we see that by the requirement $v_0 - u_0 = 0$ on the central axis, the lower bound of the region is precisely $v = u_0$, and the upperbound $v = v_0$. Therefore we have that, for the red dot sufficiently far in the future where the picture in Figure 4 is a good representation,

Equation 24
$\displaystyle (v_0 - u_0)\zeta(u_0,v_0) = \int_{u_0}^{v_0}\int_{u_r}^{u_l} \frac{c_\phi(u)}{r}\left( \frac{c(u)}{r} + \frac{c_\psi(u)}{r^2}\right)~du~dv$

Now, sufficiently far into the future, $u_0$ is sufficiently large, and hence $r = v - u \geq u_0 - u_l$ is very large. This means that we can throw away the second term in the parentheses as negligible. For generic $\phi$ and $\psi$ (note that if $\phi = \psi$ there is actually a cancellation so that the contribution from the nonlinearity vanishes), we can find a lowerbound once $u_0, v_0$ become large enough (the upper bound is trivial)

Equation 25
$\displaystyle C_{\phi,\psi} \int_{u_0}^{v_0}v^{-2} dv \geq \left| \int_{u_0}^{v_0}\int_{u_r}^{u_l} \frac{c_\phi(u)c(u)}{r^2}~du~dv \right| \geq C_{\phi,\psi}^{-1} \int_{u_0}^{v_0} v^{-2} dv$

And in Equation 25 we capture the behaviour of the nonlinear tail. For any fixed $r_0 = v_0 - u_0$, the integral $\int_{u_0}^{v_0} v^{-2} dv \sim (v_0 - u_0)(v_0 + u_0)^{-2} = r_0 t_0^{-2}$. So we have that

Equation 26a
$\displaystyle \zeta(t,r_0) \sim \frac{1}{t^2}$ for fixed $r_0$

giving us the “interior decay” rate. On the otherhand, if we were to take the limit toward $v_0\nearrow +\infty$ with $u_0$ fixed, we have that the total integral is controlled by $u_0^{-1}$, and so we have the “radiative decay” rate coming purely from the $r$ weight:

Equation 26b
$\displaystyle \zeta(v,u_0) \sim \frac{1}{v}$ for any sufficiently large fixed $u_0$.

In both Equations 26a and 26b, the “eventually vanishing” phenomena observed in the linear case for data with compact support are destroyed.

Lastly, just note that Equations 26a,b can be glued together with boundedness on bounded regions to get the global estimate

$\displaystyle \zeta(v,u) \sim \frac{1}{(1+|u|)(1+|v|)}$

which captures a more generic (in the sense of stability under nonlinear perturbations) behaviour of spherically symmetric waves on 1+3 dimensional Minkowski space.