Decay of waves IIb: Minkowski space, with right-hand side

by Willie Wong

In the first half of this second part of the series, we considered solutions to the linear, homogeneous wave equation on flat Minkowski space, and showed that for compactly supported initial data, we have strong Huygens’ principle. We further made references to the fact that this behaviour is expected to be unstable. In this post, we will further illustrate this instability by looking at Equation 1 first with a fixed source F = F(t,x), and then with a nonlinearity F = F(t,x, \phi, \partial\phi).

Duhamel’s Principle

To study how one can incorporate inhomogeneous terms into a linear equation, and to get a qualitative grasp of how the source term contributes to the solution, we need to discuss the abstract method known as Duhamel’s Principle. We start by illustrating this for a very simple ordinary differential equation.

Consider the ODE satisfied by a scalar function \alpha:

Equation 13
\displaystyle \frac{d}{ds}\alpha(s) = k(s)\alpha(s) + \beta(s)

when \beta\equiv 0, we can easily solve the equation with integration factors

\displaystyle \alpha(s) = \alpha(0) e^{\int_0^s k(t) dt}

Using this as a sort of an ansatz, we can solve the inhomogeneous equation as follows. For convenience we denote by K(s) = \int_0^s k(t) dt the anti-derivative of k. Then multiplying Equation 13 through by \exp -K(s), we have that

Equation 14
\displaystyle \frac{d}{ds} \left( e^{-K(s)}\alpha(s)\right) = e^{-K(s)}\beta(s)

which we solve by integrating

Equation 15
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + e^{K(s)} \int_0^s e^{-K(t)}\beta(t) dt

If we write K(s;t) = \int_t^s k(u) du, then we can rewrite Equation 15 as given by an integral operator

Equation 15′
\displaystyle \alpha(s) = e^{K(s)}\alpha(0) + \int_0^s e^{K(s;t)}\beta(t) dt

So what does Equations 15 and 15′ mean? The term \exp K(s;t) \beta(t) by itself would indicate a solution to the homogeneous version of ODE 13, evaluated at time s, with initial data prescribed at time t as \beta(t). So in other words, the total solution of the ODE 13 at time s corresponds to the “free evolution” of the initial data prescribed at time t = 0, plus an infinitesimal part for each time instant between 0 < t < s corresponding to the "free evolution" of initial data \beta(t) from time t to s.

Qualitatively, what is going on is that at every time instant, the "source term" gives a small kick to the solution. Because of the linearity of the corresponding homogeneous problem, each small kick evolves independently of the free evolution, and also independently of other small kicks coming from other instants in time. At the end to evaluate the solution at time s, all one needs to do is to add up the contribution from the sum total of these small kicks, and add to it the free evolution of the homogeneous problem given by the initial data.

Now, recall how many statements about scalar ODEs can be transferred to vector valued ODEs: we can treat \alpha, \beta to be vector valued in Equation 13, with k to be matrix valued. The defining \exp K the usual way for matrices, we get that Equations 15 and 15' equally well apply in this context.

Similarly, with a bit more care, we can generalise this to infinite dimensional systems.

For the wave equation, observe that we can write it as a first order ODE on some Hilbert space: let \alpha = (\phi, \partial_t \phi), then the wave equation can be written as

Equation 16
\displaystyle \frac{d}{dt} \begin{pmatrix}\phi(t) \\ \partial_t\phi(t) \end{pmatrix} = \begin{pmatrix} 0 & 1\\ \triangle & 0\end{pmatrix}\begin{pmatrix} \phi(t) \\ \partial_t\phi(t)\end{pmatrix} + \begin{pmatrix} 0 \\ F(t)\end{pmatrix}

so at least on a formal level (which can in fact be made rigorous) we can obtain the solution for the inhomogeneous wave equation by considering the solutions for the homogeneous version.

To recapitulate, and to introduce some terminologies, equations like Equation 13 above, where \alpha, \beta are “time dependent” functions that are scalar-, vector-, or Hilbert-space-valued, and where k is a “time dependent” family of scalar-multipliers, linear transformations, or linear operators (in the three cases mentioned) are first order evolution equations. The corresponding object \exp K(s;t) (which is now is a two-parameter, depending on initial and final times, family of multipliers, linear transformations, or linear operators) goes under the names of free propagator, Green’s function, or fundamental solution, as it denotes the (linear) change for solutions of the homogeneous problem between initial time t and final time s.

As an aside: Interesting to note is that each evolution equation corresponds to a category. The objects are sets representing the time-dependent spaces of data. The morphisms are precisely \exp K that maps between the different times. Well-posedness of the initial value problem relative to one element o in the class of objects is then the statement that o is an initial object (existence) and that for any two objects a,b the class of morphisms hom(a,b) consists of at most one arrow (uniqueness). Time reversibility will then be the statement our category is in fact a groupoid, and hence that o is also a terminal object. Linearity amounts to requiring the objects be vector spaces and the arrows be vector space homomorphisms. Time translation symmetry would be the existence of a functor collapsing our category to a monoid. (At least, if I am not mistaken….)

Example: 1D wave equation

Let us illustrate this by considering the wave equation in 1 spatial dimension. (Which, again, using the reflection trick described in the last post, gives us a solution to the case with Dirichlet boundary conditions, and also then to the spherically symmetric wave equation in 3 spatial dimensions.) We first try to write down the free propagator in terms of the first-order formalism above. Recall Equation 7 for the solution to the homogeneous problem with initial data (f,g)

Equation 17a
\displaystyle \phi(t,x) = \frac12 \left(f(x-t) + f(x+t) + \int_{x-t}^{x+t} g(y) dy \right)

Now observe that \partial_t\phi also solves a homogeneous wave equation by linearity, and using that for a solution \partial_t^2\phi = \triangle \phi, we have that the initial data for \partial_t \phi is (g,f''). Hence we also have the expression

Equation 17b
\displaystyle \partial_t\phi(t,x) = \frac12 \left( g(x-t) + g(x+t) + \int_{x-t}^{x+t} f''(y) dy\right) = \frac12 \left( g(x-t) + g(x+t) + f'(x+t) - f'(x-t)\right)

The time translation symmetry tells us that the propagator \exp K(s;t) = \exp K(s-t). And Equations 17a,b tell us what the map \exp K(s;s'): \left(\phi(s'),\partial_t\phi(s')\right)\mapsto \left(\phi(s),\partial_t\phi(s)\right) looks like. Now, going back up to Equation 16, we see that the vector representation of F only contributes to the second slot. So we have

Equation 18
\displaystyle \left(\exp K(s-t) F(t)\right)(x) = \frac12 \int_{x-s+t}^{x+s-t} F(t,y) dy
\displaystyle \partial_s \left(\exp K(s-t)F(t)\right)(x) = \frac12 \left( F(x-s+t) + F(x+s-t) \right)

and thus, the source part of the full solution to the inhomogeneous problem, would be

Equation 19
\displaystyle \phi_{inhomo}(s,x) = \int_0^s \exp K(s-t) F(t) dt = \frac12 \int_{0}^s \int_{x+t-s}^{x-t+s} F(t,y) dy dt

which perhaps is easier seen with a picture.

Figure 2

In Figure 2, we assume that we want to consider the initial value problem for Equation 16 (of equivalently, the inhomogeneous wave equation) in spatial dimension 1. Let us assume that the initial data is prescribed on the x axis, and the initial data is identically zero. (In other words, we just want to consider purely the contribution coming from the source term, by setting the first term on the right hand side of Equation 15′ to be 0. In the picture, the green region represents the support of the source function F. Suppose we want to evaluate the solution \phi at the red dot, then Equation 19 tells us that \phi is given by the total integral of F in the blue, crosshatched region, which is bounded below by the initial time slice, and above by the null lines emanating from the red dot.

(In general, given a space-time point (t,x) where now space can be arbitrary number of dimensions, we will write the two sets \mathcal{C}^{\pm}(t,x):= \{ (s,y) : s - t = \pm \left| y - x\right|\} which represents the future and past light rays emanating from our point. We see that they form two cones, intersection only at the vertex, in space-time. We will also denote the sets \mathcal{I}^{\pm}(t,x) to be respectively the future of \mathcal{C}^+ and past of \mathcal{C}^-.)

Another thing to read off from the picture, and from Equations 18 and 19, is that the solution \phi will not “settle down to a constant value” until the support set of the inhomogeneity F is completely contained in \mathcal{I}^+(t,x). So in particular, if the support set for F has infinite spatial (or temporal) extent, then the solution will see “tail behaviour”. This supports the interpretation, using Duhamel’s Principle, of the right-hand side of the wave equation as a “source term”.

A quick detour to Penrose diagram

To better illustrate the global behaviour of wave equations, we will use the so-called (Carter-)Penrose diagrams for space-time. To study late time behaviour, we will need to think about points infinitely far into the future. And so we want a convenient way to represent the space-time and its infinite extent on a finite piece of paper. This requires some creative rescaling of space-time coordinates. Now, there are many ways to map the infinite to the finte. For example, consider the tangent function. It is a one-to-one mapping between the open interval (-\pi,\pi) and the entire real line. The Penrose diagram is a particular type of mapping. Now, if you look at any mapping from \mathbb{R}^2, representing the 1+1 dimensional space-time, to a finite region, it would be necessary that some of the straight-lines be mapped to lines that are no longer straight. (If all straight lines are preserved, the mapping must be affine by the fundamental theorem of projective geometry.) So the question is what, if any, curves we want to keep “straight”. As we have already seen, what is most important in the study of wave equations is the null lines x\pm t = const; they, being characteristics of the partial differential equation, encode most of the phenomenon that we want to study. The Penrose diagram is a way of representing the 1+1 dimensional Minkowski space-time as a set with finite extent, such that the null lines (which are also the level sets of the u,v coordinate functions) are preserved.

Figure 3

Figure 3 shows some examples of Penrose diagrams. (a) is the Penrose diagram for the 1+1 dimensional Minkowski space, the boundaries of the diamond are precisely where the coordinate functions u,v attain \pm \infty values; we include the diagonal lines to indicate the level sets of u,v. (b) is the Penrose diagram for the r \geq 0 Minkowski half space; this also corresponds to the Penrose diagram for the 1+3 dimensional Minkowski space after we quotient out by the spherical symmetry. Notice that the central axis r = 0 is drawn slightly curved to illustrate the fact that non-null straight lines don’t necessarily map to straight lines in the compactified diagram. In (a) and (b) we see that the open circles and the dashed lines represent “idealized boundaries”, points that are not in fact in the space-time; in (b) the solid line for the central axis indicates the fact that boundary is in fact a physical region of the space-time. (c) is a representation of Figure 2 in compactified form; observe that the bottom boundary of the blue triangle is drawn slightly curved also. (d) shows the “domain of influence” of a source. In the picture the cyan ball indicates the space-time support set of the inhomogeneity. Under the assumption of either Dirichlet boundary condition (if we consider this as a 1+1 problem on the Minkowski half space) or symmetric reduction from a 1+3 dimensional problem, the pink region is the region in space-time where a priori, the contribution from the inhomogeneity to the solution of the wave equation can be non-zero. The larger the space-time extent of the inhomogeneity, the larger the domain of influence.

Nonlinear tails: an example

Be start by giving the motivation for the construction. Consider the following Penrose diagram, in which we study the future evolution of some wave equation in 1+1 dimensional space with Dirichlet boundary (which could be the symmetry reduction of a spherically symmetric wave equation in 1+3 dimensional Minkowski space).

Figure 4

The initial data is prescribed to be only non-vanishing on the blue line. So if we were to solve the purely linear homogeneous problem, using what we discussed in the previous post in the series, we see that the initial data splits into an incoming part, which approaches the central axis, bounces off, and then radiates to infinity; and an outgoing part which immediately radiates outward. And as we described before, following any future directed time like curve (which can be represented by a curve converging to the top open circle, the “future timelike infinity”) we will eventually escape the region influenced by this data.

Equivalently, we can consider the value of the solution at a point in the space time (here the red dot). Using the representation given by Duhamel’s principle, we see that for the inhomogeneous wave equation with possible source term, the solution can only be non-vanishing at the red dot if the initial data and any inhomogeneities have space-time support that intersects the crosshatched region. So for our linear homogeneous problem, where the initial data (blue curve) does not intersect the crosshatched set determined by strong Huygens’ principle, at the red dot the solution would vanish.

Now suppose that instead of an outside source, we impose a nonlinear self-interaction on the right hand side. That is, the source term is assumed to be generated from non-linear effects of the wave itself. To be slightly more precise, assume our equation takes the form

Equation 20
-\partial_t^2 \phi(t,x) + \triangle\phi(t,x) = F\circ\phi(t,x)

where F is now only a function of the value of the displacement \phi, with F(0) = F'(0) = 0 (these conditions ensure that the interaction is non-linear, and that waves won’t be spontaneously generated from 0 initial data; in other words, 0 is still a solution to the equation).

Now, because the interaction is nonlinear, if \phi = O(\epsilon) is small, we can expect, by Taylor expanding F, that the source term contribution would be F(\phi) = O(\epsilon^2), which is even smaller. Thus intuitively we expect to be able to solve the equation using successive approximation, namely: we start with the given initial data (f,g), we solve the homogeneous wave equation for it, and call the solution \phi_0. Then we run through the following iterative procedure:

  1. We are fed the solution from the previous iterate, \phi_n.
  2. We solve the inhomogeneous wave equation with prescribed source: -\partial_t^2 \phi_{n+1} + \triangle \phi_{n+1} = F\circ \phi_n subject to the initial data (f,g). Notice that on the right hand side, the source is given by \phi_n, with on the left hand side, we are solving for \phi_{n+1}. So we can directly obtain an integral expression using Duhamel’s principle.
  3. Feed \phi_{n+1} into this algorithm and repeat.

If this approximation scheme converges, then we will have that \phi_{\infty}, the limiting function, will solve the nonlinear equation. (See Appendix for a bit more details.)

Let us consider for now the first iterate, which is enough to give a heuristic demonstration of why nonlinear tails are generic, and that the complete vanishing of waves within finite time that is possible in the linear system is very unstable.

Going back to Figure 4, suppose the initial data is precisely supported on the blue line. Then the 0th iterate, \phi_0, just solves the linear homogeneous wave equation, and so its support set is represented by the cyan region, and the red dot lies outside of it. But now let us consider the first iterate. It solves the wave equation with source term F\circ \phi_0! So in particular, while it is true that the crosshatched region does not intersection the initial data (so there is no contribution to \phi_1 at the red dot coming from the prescribed initial value (f,g)), the region does intersect the cyan set, on which our inhomogeneity does not identically vanish. And if the homogeneity is assumed to take only non-negative values (for example F(\phi) = |\phi|^2), then it is necessary that \phi_1 is nonvanishing at the red dot. Since the position of the red dot is more or less arbitrary, we see that at the level of the first iterate, we already expect some nonlinear tails.

Appendix: successive approximation and local existence of solutions for nonlinear problems

Observe that using Duhamel’s principle we have a very precise control of the rate of convergence. Consider the difference \psi_{n+1} = \phi_{n+1} - \phi_n. It solves the wave equation

-\partial_t^2\psi_{n+1} + \triangle \psi_{n+1} = F\circ \phi_n - F\circ \phi_{n-1} = F \circ (\phi_{n-1} + \psi_n) - F\circ \phi_{n-1}

with 0 initial condition. The right hand side we again use Taylor expansion (assuming that \psi_n is small), to replace by

F \circ (\phi_{n-1} + \psi_n) - F\circ \phi_{n-1} = F'\circ \phi_{n-1} \cdot \psi_n

But since \phi_{n-1} is also small, we can approximate F'\circ \phi_{n-1} by F'(0) + O(\phi_{n-1}). So by the assumption that F'(0) = 0, we see that the source term is small. In particular, applying Duhamel’s principle again, using that the integral used to factor in the inhomogeneity is taken over a finite space-time region, we have that

\sup_{t\in [0,T]}\sup_{x}|\psi_{n+1}(t,x)| \leq C_T O(\phi_{n-1}) \sup_{t\in[0,T]}\sup_{x} |\psi_n(t,x)|

where the constant C_T depends only on the maximum time T. Using that \phi_{n-1} = \phi_0 + \sum_{j = 1}^{n-1} \psi_j, we see that \phi_\infty = \phi_0 + \sum \psi_j is in fact controlled by a geometric series, and the series converges if (f,g) is bounded by an \epsilon such that C_T\epsilon is less than, say, 1/100.

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