of order , where . Here are a few representative examples:
Example 1. The advection equation with speed in the negative direction.
Example 2. The heat equation , where
denotes the Laplace operator.
Example 3. The Schrödinger equation .
Example 4. The wave equation , which can be cast into the form of Eq. (3.1),
We can find solutions of Eq. (3.1) by Fourier transformation in space,symbol of the differential operator , is defined as
At this point we have to ask ourselves if expression (3.9) makes sense. In fact, we do not expect the integral to converge in general. Even if is smooth and decays rapidly to zero as we could still have problems if diverges as . One simple, but very restrictive, possibility to control this problem is to limit ourselves to initial data in the class of functions, which are the Fourier transform of a -function with compact support, i.e., , where1 If the integral in Eq. (3.9) is well-defined and we obtain a solution of the Cauchy problem (3.7, 3.8), which, for each lies in this space. However, this possibility suffers from several unwanted features:
For these reasons, it is desirable to consider initial data of a more general class than . For this, we need to control the growth of . This is captured in the following
The importance of this definition relies on the property that for each fixed time the norm of the propagator is bounded by the constant , which is independent of the wave vector . The definition does not state anything about the growth of the solution with time other that this growth is bounded by an exponential. In this sense, unless one can choose or arbitrarily small, well-posedness is not a statement about the stability in time, but rather about stability with respect to mode fluctuations.
Let us illustrate the meaning of Definition 1 with a few examples:
Example 5. The heat equation .
Fourier transformation converts this equation into . Hence, the symbol is and . The problem is well posed.
Example 6. The backwards heat equation .
In this case the symbol is , and . In contrast to the previous case, exhibits exponential frequency-dependent growth for each fixed and the problem is not well posed. Notice that small initial perturbations with large are amplified by a factor that becomes larger and larger as increases. Therefore, after an arbitrarily small time, the solution is contaminated by high-frequency modes.
Example 7. The Schrödinger equation .
In this case we have and . The problem is well posed. Furthermore, the evolution is unitary, and we can evolve forward and backwards in time. When compared to the previous example, it is the factor in front of the Laplace operator that saves the situation and allows the evolution backwards in time.
Example 8. The one-dimensional wave equation written in first-order form,
Example 9. Perturb the previous problem by a lower-order term,
More generally one can show (see Theorem 2.1.2 in ):
By considering the eigenvalues of the symbol we obtain the following simple necessary condition for well-posedness:
Proof. Suppose is an eigenvalue of with corresponding eigenvector , . Then, if the problem is well posed,
Although the Petrovskii condition is a very simple necessary condition, we stress that it is not sufficient in general. Counterexamples are first-order systems, which are weakly, but not strongly, hyperbolic; see Example 10 below.
Now that we have defined and illustrated the notion of well-posedness, let us see how it can be used to solve the Cauchy problem (3.7, 3.8) for initial data more general than in . Suppose first that , as before. Then, if the problem is well posed, Parseval’s identities imply that the solution (3.9) must satisfy
The family is called a semi-group on . In general, cannot be extended to negative as the example of the backwards heat equation, Example 6, shows.
For the function is called a weak solution of the Cauchy problem (3.7, 3.8). It can also be constructed in an abstract way by using the Fourier–Plancharel operator . If the problem is well posed, then for each and the map defines an -function, and, hence, we can define
According to Duhamel’s principle, the semi-group can also be used to construct weak solutions of the inhomogeneous problem,
In order to extend the solution concept to initial data more general than analytic, we have introduced the concept of well-posedness in Definition 1. However, given a symbol , it is not always a simple task to determine whether or not constants and exist such that for all and . Fortunately, the matrix theorem by Kreiss  provides necessary and sufficient conditions on the symbol for well-posedness.
Theorem 1. Let , , be the symbol of a constant coefficient linear problem, see Eq. (3.5), and let . Then, the following conditions are equivalent:
A generalization and complete proof of this theorem can be found in . However, let us show here the implication (ii) (i) since it illustrates the concept of energy estimates, which will be used quite often throughout this review (see Section 3.2.3 below for a more general discussion of these estimates). Hence, let be a family of Hermitian matrices satisfying the condition (3.25). Let and be fixed, and define for . Then we have the following estimate for the “energy” density ,
Many systems in physics, like Maxwell’s equations, the Dirac equation, and certain formulation of Einstein’s equations are described by first-order partial-differential equations (PDEs). In fact, even systems, which are given by a higher-order PDE, can be reduced to first order at the cost of introducing new variables, and possibly also new constraints. Therefore, let us specialize the above results to a first-order linear problem of the formprincipal symbol, , and the lower-order term . The principal part is the one that dominates for large and hence the one that turns out to be important for well-posedness. Notice that depends linearly on . With these observations in mind we note:
These observations motivate the following three notions of hyperbolicity, each of them being a stronger condition than the previous one:
Definition 2. The first-order system (3.28) is called
The matrix theorem implies the following statements:
Example 10. Consider the weakly hyperbolic system 
However, when , the eigenvalues of are
Example 11. For the system ,
Example 12. Next, we present a system for which the eigenvectors of the principal symbol cannot be chosen to be continuous functions of :
Of course, is symmetric and so can be chosen to be unitary, which yields the trivial symmetrizer . Therefore, the system is symmetric hyperbolic and yields a well-posed Cauchy problem; however, this example shows that it is not always possible to choose as a continuous function of .
Example 13. Consider the Klein–Gordon equation4 The corresponding Cauchy problem is well posed. However, a problem with this first-order system is that it is only equivalent to the original, second-order equation (3.43) if the constraints and are satisfied.
An alternative symmetric hyperbolic first-order reduction of the Klein–Gordon equation, which does not require the introduction of constraints, is the Dirac equation in two spatial dimensions,
Yet another way of reducing second-order equations to first-order ones without introducing constraints will be discussed in Section 3.1.5.
Example 14. In terms of the electric and magnetic fields , Maxwell’s evolution equations,
Example 15. There are many alternative ways to write Maxwell’s equations. The following system [353, 287] was originally motivated by an analogy with certain parametrized first-order hyperbolic formulations of the Einstein equations, and provides an example of a system that can be symmetric, strongly, weakly or not hyperbolic at all, depending on the parameter values. Using the Einstein summation convention, the evolution system in vacuum has the form
In order to analyze under which conditions on and the system (3.50, 3.51) is strongly hyperbolic we consider the corresponding symbol,5 we can write the eigenvalue problem as
In order to analyze under which conditions the system is symmetric hyperbolic we notice that because of rotational and parity invariance the most general -independent symmetrizer must have the form
An important class of systems in physics are wave problems. In the linear, constant coefficient case, they are described by an equation of the form6 With this redefinition one obtains a system of the form (3.1) with and  for the particular case in which the mixed–second-order derivative term in Eq. (3.56) vanishes; that is, when . This result can be generalized in a straightforward manner to the case where the matrices are proportional to the identity:
Theorem 2. Suppose , . (Note that this condition is trivially satisfied if .) Then, the Cauchy problem for Eq. (3.56) is well posed if and only if the symbolhas the following properties: there exist constants and and a family of Hermitian matrices such that for all .
Proof. Since for the advection term commutes with any Hermitian matrix , it is sufficient to prove the theorem for , in which case the principal symbol reduces to3.63). Then, choosing , and we find that . Furthermore, and where
Conversely, suppose that the problem is well posed with symmetrizer . Then, the vanishing of yields the conditions and the conditions (3.63) are satisfied for . □
Remark: The conditions (3.63) imply that is symmetric and positive with respect to the scalar
product defined by . Hence it is diagonalizable, and all its eigenvalues are positive. A practical way of
finding is to construct , which diagonalizes , with
diagonal and positive. Then, is the candidate for satisfying the
Let us give some examples and applications:
Example 16. The Klein–Gordon equation on flat spacetime. In this case, and , and trivially satisfies the conditions of Theorem 2.
Example 17. In anticipation of the following Section 3.2, where linear problems with variable coefficients are treated, let us generalize the previous example on a curved spacetime . We assume that is globally hyperbolic such that it can be foliated by space-like hypersurfaces . In the ADM decomposition, the metric in adapted coordinates assumes the form
Living Rev. Relativity 15, (2012), 9
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