3.2 Linear problems with variable coefficients
Next, we generalize the theory to linear evolution problems with variable coefficients. That is, we
consider equations of the following form:
where now the complex
matrices
may depend on
and
. For simplicity, we assume
that each matrix coefficient of
belongs to the class
of bounded,
-functions
with bounded derivatives. Unlike the constant coefficient case, the different
-modes couple when
performing a Fourier transformation, and there is no simple explicit representation of the solutions through
the exponential of the symbol. Therefore, Definition 1 of well-posedness needs to be altered. Instead of
giving an operator-based definition, let us define well-posedness by the basic requirements a Cauchy
problem should satisfy:
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
.
3.2.1 The localization principle
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.
However, given a specific point
, we may zoom into a very small neighborhood
of
. Since the coefficients
and
are smooth, they will be approximately constant in
this neighborhood and we may freeze the coefficients of
and
to their values at the point
. More precisely, let
be a smooth solution of Eq. (3.77). Then, we consider the formal
expansion
As a consequence of Eq. (3.77) one obtains
Taking the pointwise limit
on both sides of this equation we obtain
where
. Therefore, if
is a solution of the variable coefficient
equation
, then,
satisfies the linear constant coefficient problem
obtained by freezing the coefficients in the principal part of
to their values at the point
and by replacing the lower-order term
by the
forcing term
. By adjusting the scaling of
, a similar conclusion can be obtained when
is a higher-derivative operator.
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:
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
.
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.
3.2.2 Characteristic speeds and fields
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).
3.2.3 Energy estimates and finite speed of propagation
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
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
.