where the vector valued matrix
, and the matrix
B, are assumed to be smooth
functions of all its arguments.
Systems of this type are called quasi-linear because the derivative appears linearly. This property allows one to use most of the machinery for constant coefficient equations to prove well posedness, thus the local existence is well understood, via linearization techniques. There are few global results, and in general they depend on more refined knowledge of the equation systems for which they apply.
The behavior of solutions to quasi-linear equations is not yet
fully understood. Most of the solutions develop singularities in
a finite time for most initial data, even if they are in
. This is the case for convective systems, or more generally for
genuinely non-linear systems, see [28
,
29
], for the definition and main results, a class which includes
systems like perfect fluids and relativistic dissipative fluids
-for they contain as part of the system the perfect fluid
equations. This is also the case for general relativity, where
singularity theorems (see [5
,
31]) tell us about the development of singularities, although of a
different type. Thus the concept of well posedness has to be
modified to account for the fact that solutions only last for a
finite time and this time depends on the initial data. Basically,
the most we can pretend to show in the above generality is the
same type of well posedness one requires from an ordinary system
of equations. Which is quite a lot! The non-linear aspect of the
equations implies also that it is not possible to generalize
their solutions to be distributions. The minimum
differentiability needed to make sense of an equation depends on
the particular equation. Furthermore, there are cases (e.g.
convection) in which, for some function spaces of low
differentiability, the equation makes sense and some solutions
exist, but they are not unique
.
In order not to worry about the possibility that the
smoothness of the solutions be too stringent a requirement, one
can smooth out the equation using a one parameter family of
mollifiers, and require that the relation
be independent of that parameter family.
To obtain results about well posedness, we just have to
slightly modify the concepts of hyperbolicity already discussed
in the constant coefficient case. Since in the constant
coefficient case the matrices did not depend on the points of
space-time, nor on the solution itself, we had only two cases. In
one case, the norm
H
did not depend on
, and so in some base the matrix
was symmetric. In the other case, the norm
H
did depend on
, and we had a general strongly hyperbolic system. In the latter
case, it can be seen that
is piece-wise continuous and so integrable, which is, in that
case, all that is needed to proceed with the proof. In the
general case with which we are now dealing,
H
would in general depend not only on
, but also on the point of space-time and on the solution,
.
This difference has caused terminology to be not uniform in the literature, so I have taken advantage of this and establish terms in the way I consider best suited for the topic.
Certain authors call some systems symmetric hyperbolic and
others symmetrizable. They call symmetric hyperbolic only those
systems where the symmetrizer does not depend on the unknown
variables nor on the space-time variables (or at most depends
only on the base space variables
H
:=
H
(t,
x)); they call the other systems symmetrizable. This is a rather
arbitrary distinction, since the methods of proof used are valid
for both with no essential difference. Thus, if
H
does not depend on
but depends smoothly on all other variables,
H
:=
H
(t,
x,
u), then we shall still say the system is
symmetric hyperbolic
. In this case the non-singular transformation which symmetrizes
is smooth in all its variables.
The existence and smoothness proof is based, as in the
constant coefficient case, on energy norm estimates, but now
supplemented by Sobolev inequalities. Since the norm is built out
of
H
and it does not depend on
, no passage to Fourier space is needed.
If
H
does also depend on
, and is smooth on all variables,
, we shall say the system is
strongly hyperbolic
. The existence and smoothness proof now requires the
construction of a pseudo-differential norm out of
H, and so pseudo-differential calculus is needed, which implies
that
H
has to be smooth in all its entries, in particular in
.
We shall not discuss weak hyperbolic systems, for they are
generically unstable under perturbations, nor shall we discuss
strictly hyperbolic systems, i.e. systems with strictly different
eigenvalues of
, for they seldom appear in physical processes in more than one
dimension.
With this concept of well posedness we have the following
theorem [See for instance [23] pg. 123]:
Remarks:
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |