3.3 Quasilinear equations
Next, we generalize the theory one more step and consider evolution systems, which are described by
quasilinear partial differential equations, that is, by nonlinear partial differential equations, which are linear
in their highest-order derivatives. This already covers most of the interesting physical systems, including the
Yang–Mills and the Einstein equations. Restricting ourselves to the first-order case, such equations have the
form
where all the coefficients of the complex
matrices
, …,
and the nonlinear
source term
belong to the class
of bounded,
-functions
with bounded derivatives. Compared to the linear case, there are two new features the solutions may
exhibit:
- The nonlinear term
may induce blowup of the solutions in finite time. This
is already the case for the simple example where
, all the matrices
vanish
identically and
, in which case Eq. (3.115) reduces to
. In the context
of Einstein’s equations such a blowup is expected when a curvature singularity forms, or it
could also occur in the presence of a coordinate singularity due to a “bad” gauge condition.
- In contrast to the linear case, the matrix functions
in front of the derivative operator
now depend pointwise on the state vector
itself, which implies, in particular, that the
characteristic speeds and fields depend on
. This can lead to the formation of shocks where
characteristics cross each other, like in the simple example of Burger’s equation
corresponding to the case
,
and
. In general,
shocks may form when the system is not linearly degenerated or genuinely nonlinear [250]. The
Einstein vacuum equations, on the other hand, can be written in linearly degenerate form (see,
for example, [6, 7, 348, 8]) and are therefore expected to be free of physical shocks.
For these reasons, one cannot expect global existence of smooth solutions from smooth initial data with
compact support in general, and the best one can hope for is existence of a smooth solution on some finite
time interval
, where
might depend on the initial data.
Under such restrictions, it is possible to prove well-posedness of the Cauchy problem. The
idea is to linearize the problem and to apply Banach’s fixed-point theorem. This is discussed
next.
3.3.1 The principle of linearization
Suppose
is a
(reference) solution of Eq. (3.115), corresponding to initial data
. Assuming this solution to be uniquely determined by the initial data
, we may ask if
a unique solution
also exists for the perturbed problem
where the perturbations
and
belong to the class of bounded,
-functions with
bounded derivatives. This leads to the following definition:
Here, the norms
and
appearing on both sides of Eq. (3.119) are different from each other
because
controls the function
over the spacetime region
while
is a norm controlling the function
on
.
If the problem is well posed at
, we may consider a one-parameter curve
of initial data lying
in
that goes through
and assume that there is a corresponding solution
for each
small enough
, which lies close to
in the sense of inequality (3.119). Expanding
and plugging into the Eq. (3.115) we find, to first order in
,
with
Eq. (3.121) is a first-order linear equation with variable coefficients for the first variation,
, for which
we can apply the theory described in Section 3.2. Therefore, it is reasonable to assume that the linearized
problem is strongly hyperbolic for any smooth function
. In particular, if we generalize the
definitions of strongly and symmetric hyperbolicity given in Definition 4 to the quasilinear case by requiring
that the symmetrizer
has coefficients in
, it follows that the
linearized problem is well posed provided that the quasilinear problem is strongly or symmetric
hyperbolic.
The linearization principle states that the converse is also true: the nonlinear problem is well posed
at
if all the linear problems, which are obtained by linearizing Eq. (3.115) at functions in a suitable
neighborhood of
are well posed. To prove that this principle holds, one sets up the following
iteration. We define the sequence
of functions by iteratively solving the linear problems
for
starting with the reference solution
. If the linearized problems are well posed in
the sense of Definition 3 for functions lying in a neighborhood of
, one can solve each Cauchy
problem (3.123, 3.124), at least for small enough time
. The key point then, is to prove that
does
not shrink to zero when
and to show that the sequence
of functions converges to a solution
of the perturbed problem (3.116, 3.117). This is, of course, a nontrivial task, which requires controlling
and its derivatives in an appropriate way. For particular examples where this program is
carried through; see [259
]. For general results on quasilinear symmetric hyperbolic systems;
see [251, 164
, 412, 51
].