Definition 3. The Cauchy problem

is well posed if any gives rise to a unique -solution , and if there are constants and such that for all and all .Before we proceed and analyze under which conditions on the operator the Cauchy problem (3.73, 3.74) is well posed, let us make the following observations:

- In the constant coefficient case, inequality (3.75) is equivalent to inequality (3.11), and in this sense Definition 3 is a generalization of Definition 1.
- If and are the solutions corresponding to the initial data , then the difference satisfies the Cauchy problem (3.73, 3.74) with and the estimate (3.75) implies that In particular, this implies that converges to if converges to in the -sense. In this sense, the solution depends continuously on the initial data. This property is important for the convergence of a numerical approximation, as discussed in Section 7.
- Estimate (3.75) also implies uniqueness of the solution, because for two solutions and with the same initial data the inequality (3.76) implies .
- As in the constant coefficient case, it is possible to extend the solution concept to weak ones by taking sequences of -elements. This defines a propagator , which maps the solution at time to the solution at time and satisfies similar properties to the ones described in Section 3.1.2: (i) for all , (ii) for all , (iii) for , is the unique solution of the Cauchy problem (3.73, 3.74), (iv) for all and all . Furthermore, the Duhamel formula (3.23) holds with the replacement .

Like in the constant coefficient case, we would like to have a criterion for well-posedness that is based on the coefficients of the differential operator alone. As we have seen in the constant coefficient case, well-posedness is essentially a statement about high frequencies. Therefore, we are led to consider solutions with very high frequency or, equivalently, with very short wavelength. In this regime we can consider small neighborhoods and since the coefficients are smooth, they are approximately constant in such neighborhoods. Therefore, intuitively, the question of well-posedness for the variable coefficient problem can be reduced to a frozen coefficient problem, where the values of the matrix coefficients are frozen to their values at a given point.

In order to analyze this more carefully, and for the sake of illustration, let us consider a first-order linear system with variable coefficients

where are complex matrices, whose coefficients belong to the class of bounded, -functions with bounded derivatives. As mentioned above, the Fourier transform of this operator does not yield a simple, algebraic symbol like in the constant coefficient case.This leads us to the following statement: a necessary condition for the linear, variable coefficient Cauchy problem for the equation to be well posed is that all the corresponding problems for the frozen coefficient equations are well posed. For a rigorous proof of this statement for the case in which is time-independent; see [397]. We stress that it is important to replace by its principal part when freezing the coefficients. The statement is false if lower-order terms are retained; see [259, 397] for counterexamples.

Now it is natural to ask whether or not the converse statement is true: suppose that the Cauchy problems for all frozen coefficient equations are well posed; is the original, variable coefficient problem also well posed? It turns out this localization principle is valid in many cases under additional smoothness requirements. In order to formulate the latter, let us go back to the first-order equation (3.77). We define its principal symbol as

In analogy to the constant coefficient case we define:Definition 4. The first-order system (3.77) is called

- weakly hyperbolic if all the eigenvalues of its principal symbol are purely imaginary.
- strongly hyperbolic if there exist and a family of positive definite, Hermitian matrices , , whose coefficients belong to the class , such that for all , where .
- symmetric hyperbolic if it is strongly hyperbolic and the symmetrizer can be chosen independent of .

We see that these definitions are straight extrapolations of the corresponding definitions (see
Definition 2) in the constant coefficient case, except for the smoothness requirements for the symmetrizer
.^{8}
There are examples of ill-posed Cauchy problems for which a Hermitian, positive-definite symmetrizer
exists but is not smooth [397] showing that these requirements are necessary in
general.

The smooth symmetrizer is used in order to construct a pseudo-differential operator

from which one defines a scalar product , which, for each , is equivalent to the product. This scalar product has the property that a solution to the equation (3.77) satisfies an inequality of the form see, for instance, [411]. Upon integration this yields an estimate of the form of Eq. (3.75). In the symmetric hyperbolic case, we have simply and the scalar product is given by We will return to the application of this scalar product for deriving energy estimates below. Let us state the important result:Theorem 3. If the first-order system (3.77) is strongly or symmetric hyperbolic in the sense of Definition 4, then the Cauchy problem (3.73, 3.74) is well posed in the sense of Definition 3.

For a proof of this theorem, see, for instance, Proposition 7.1 and the comments following its formulation in Chapter 7 of [411]. Let us look at some examples:

Example 18. For a given, stationary fluid field, the non-relativistic, ideal magnetohydrodynamic equations reduce to the simple system [120]

for the magnetic field , where is the fluid velocity. The principal symbol for this equation is given by In order to analyze it, it is convenient to introduce an orthonormal frame such that is parallel to . With respect to this, the matrix corresponding to is with purely imaginary eigenvalues , . However, the symbol is not diagonalizable when is orthogonal to the fluid velocity, , and so the system is only weakly hyperbolic.One can still show that the system is well posed, if one takes into account the constraint , which is preserved by the evolution equation (3.85). In Fourier space, this constraint forces , which eliminates the first row and column in the principal symbol, and yields a strongly hyperbolic symbol. However, at the numerical level, this means that special care needs to be taken when discretizing the system (3.85) since any discretization, which does not preserve , will push the solution away from the constraint manifold, in which case the system is weakly hyperbolic. For numerical schemes, which explicitly preserve (divergence-transport) or enforce (divergence-cleaning) the constraints, see [159] and [136], respectively. For alternative formulations, which are strongly hyperbolic without imposing the constraint; see [120].

Example 19. The localization principle can be generalized to a certain class of second-order systems [261][308]: For example, we may consider a second-order linear equation of the form

, , where now the matrices , , , and belong to the class of bounded, -functions with bounded derivatives. Zooming into a very small neighborhood of a given point by applying the expansion in Eq. (3.78) to , one obtains, in the limit , the constant coefficient equation with where we have used the fact that . Eq. (3.89) can be rewritten as a first-order system in Fourier space for the variable see Section 3.1.5. Now Theorem 2 implies that the problem is well posed, if there exist constants and and a family of positive definite Hermitian matrices , , which is -smooth in all its arguments, such that and for all , where .In particular, it follows that the Cauchy problem for the Klein–Gordon equation on a globally-hyperbolic spacetime with , is well posed provided that is uniformly positive definite; see Example 17.

Consider a first-order linear system of the form (3.77), which is strongly hyperbolic. Then, for each , and the principal symbol is diagonalizable and has purely complex eigenvalues. In the constant coefficient case with no lower-order terms () an eigenvalue of with corresponding eigenvector gives rise to the plane-wave solution

If lower-order terms are present and the matrix coefficients are not constant, one can look for approximate plane-wave solutions, which have the form where is a small parameter, a smooth-phase function and a slowly varying amplitude. Introducing the ansatz (3.93) into Eq. (3.77) and taking the limit yields the problem Setting and , a nontrivial solution exists if and only if the eikonal equation is satisfied. Its solutions provide the phase function whose level sets have co-normal . The phase function and determine approximate plane-wave solutions of the form (3.93). For this reason we call the characteristic speed in the direction , and a corresponding characteristic mode. For a strongly hyperbolic system, the solution at each point can be expanded in terms of the characteristic modes with respect to a given direction , The corresponding coefficients are called the characteristic fields.Example 20. Consider the Klein–Gordon equation on a hyperbolic spacetime, as in Example 17. In this case the eikonal equation is

which yields . The corresponding co-normals is null; hence the surfaces of constant phase are null surfaces. The characteristic modes and fields are where and is the Klein–Gordon field.Example 21. In the formulation of Maxwell’s equations discussed in Example 15, the characteristic speeds are , and , and the corresponding characteristic fields are the components of the vector on the right-hand side of Eq. (3.54).

Here we focus our attention on first-order linear systems, which are symmetric hyperbolic. In this case it is not difficult to derive a priori energy estimates based on integration by parts. Such estimates assume the existence of a sufficiently smooth solution and bound an appropriate norm of the solution at some time in terms of the same norm of the solution at the initial time . As we will illustrate here, such estimates already yield quite a lot of information on the qualitative behavior of the solutions. In particular, they give uniqueness, continuous dependence on the initial data and finite speed of propagation.

The word “energy” stems from the fact that for many problems the squared norm satisfying the estimate is directly or indirectly related to the physical energy of the system, although for many other problems the squared norm does not have a physical interpretation of any kind.

For first-order symmetric hyperbolic linear systems, an a priori energy estimate can be constructed from the symmetrizer in the following way. For a given smooth solution of Eq. (3.77), define the vector field on by its components

where . By virtue of the evolution equation, satisfies where the Hermitian matrix is defined as If , Eq. (3.100) formally looks like a conservation law for the current density . If , we obtain, instead of a conserved quantity, an energy-like expression whose growth can be controlled by its initial value. For this, we first notice that our assumptions on the matrices , and imply that is bounded on . In particular, since is uniformly positive, there is a constant such that Let be a tubular region obtained by piling up open subsets of hypersurfaces. This region is enclosed by the initial surface , the final surface and the boundary surface , which is assumed to be smooth. Integrating Eq. (3.100) over and using Gauss’ theorem, one obtains where is the unit outward normal covector to and the volume element on that surface. Defining the “energy” contained in the surface by and assuming for the moment that the “flux” integral over is positive or zero, one obtains the estimate where we have used the inequality (3.102) and the definition of in the last step. Defining the function this inequality can be rewritten as which yields upon integration. This together with (3.105) gives which bounds the energy at any time in terms of the initial energy.In order to analyze the conditions under which the flux integral is positive or zero, we examine the sign of the integrand . Decomposing where is a unit vector and a positive normalization constant, we have

where is the principal symbol in the direction of the unit vector . This is guaranteed to be positive if the boundary surface is such that is greater than or equal to all the eigenvalues of the boundary matrix , for each . This is equivalent to the condition Since is symmetric, the supremum is equal to the maximum eigenvalue of . Therefore, condition (3.109) is equivalent to the requirement that be greater than or equal to the maximum characteristic speed in the direction of the unit outward normal .With these arguments, we arrive at the following conclusions and remarks:

- Finite speed of propagation. Let be a given event, and set
Define the past cone at as
^{9}The unit outward normal to its boundary is , which satisfies the condition (3.109). It follows from the estimate (3.107) applied to the domain that the solution is zero on if the initial data is zero on the intersection of the cone with the initial surface . In other words, a perturbation in the initial data outside the ball does not alter the solution inside the cone . Using this argument, it also follows that if has compact support, the corresponding solution also has compact support for all . - Continuous dependence on the initial data. Let be smooth initial data with compact support. As we have seen above, the corresponding smooth solution also has compact support for each . Therefore, applying the estimate (3.107) to the case , the boundary integral vanishes and we obtain In view of the definition of , see Eq. (3.104), and the properties (3.81) of the symmetrizer, it follows that which is of the required form; see Definition 3. In particular, we have uniqueness and continuous dependence on the initial data.
- The statements about finite speed of propagation and continuous dependence on the data can easily be generalized to the case of a first-order symmetric hyperbolic inhomogeneous equation , with a bounded, -function with bounded derivatives. In this case, the inequality (3.113) is replaced by
- If the boundary surface does not satisfy the condition (3.109) for the boundary integral to be positive, then suitable boundary conditions need to be specified in order to control the sign of this term. This will be discussed in Section 5.2.
- Although different techniques have to be used to prove them, very similar results hold for strongly hyperbolic systems [353].
- For definitions of hyperbolicity of a geometric PDE on a manifold, which do not require a decomposition of spacetime, see, for instance, [205, 353], for first-order systems and [47] for second-order ones.

Example 22. We have seen that for the Klein–Gordon equation propagating on a globally-hyperbolic spacetime, the characteristic speeds are the speed of light. Therefore, in the case of a constant metric (i.e., Minkowksi space), the past cone defined in Eq. (3.111) coincides with the past light cone at the event . A slight refinement of the above argument shows that the statement remains true for a Klein–Gordon field propagating on any hyperbolic spacetime.

Example 23. In Example 21 we have seen that the characteristic speeds of the system given in Example 15 are , and , where is assumed for strong hyperbolicity. Therefore, the past cone corresponds to the past light cone provided that . For , the formulation has superluminal constraint-violating modes, and an initial perturbation emanating from a region outside the past light cone at could affect the solution at . In this case, the past light cone at is a proper subset of .

Living Rev. Relativity 15, (2012), 9
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