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
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,
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.,
, where
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
Definition 1. The Cauchy problem (3.7, 3.8
) is called well posed if there are constants
and
such that
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,
The symbol isExample 9. Perturb the previous problem by a lower-order term,
The symbol is More generally one can show (see Theorem 2.1.2 in [259]):
Lemma 1. The Cauchy problem for the first-order equation with complex
matrices
and
is well posed if and only if
is diagonalizable and has only real eigenvalues.
By considering the eigenvalues of the symbol we obtain the following simple necessary condition
for well-posedness:
Lemma 2 (Petrovskii condition). Suppose the Cauchy problem (3.7, 3.8
) is well posed. Then, there is a
constant
such that
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 [257] 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 [259]. 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 form
whereThese 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 [259]
However, when , the eigenvalues of
are
Example 11. For the system [353],
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 equation
in two spatial dimensions, whereAn 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,
This system implies the Klein–Gordon equation (3.43Yet 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,
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 form
where
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 symbol
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 to
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
conditions (3.63
).
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
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