2.3 Generalization to Variable Coefficient 2 The Theory of Linear 2.1 Existence and Uniqueness of

2.2 First Order Systems. 

We shall now restrict consideration to systems which have at most one space derivative, i.e. systems of the form,

  equation137

Using the above theorem, it is easy to see that if a system P (D) is well posed, then so is the system P (D) + B, where B is any constant matrix. For the particular case at hand, this means that we can further restrict attention, without loss of generality, to the the principal part of the operator, namely

displaymath1494

where tex2html_wrap_inline1496 is a tex2html_wrap_inline1378 matrix valued vector in tex2html_wrap_inline1364 . In this case we can improve on the above condition by showing that well posedness implies no growth of the solution, that is that we can choose tex2html_wrap_inline1502 above.

theorem141

If tex2html_wrap_inline1496 satisfies the above condition for some tex2html_wrap_inline1512, then we say that P (D) is strongly hyperbolic, which, as we see, is equivalent for first order equation systems to well posedness. If the operator H does not depend on tex2html_wrap_inline1518, a case that appears in most physical problems, then we say the system is symmetric hyperbolic . Indeed, if H does not depend on tex2html_wrap_inline1518, then there is a base in which it just becomes the identity matrix. (One can diagonalize it and re-scale the base.) Then the above condition in the new base just means that tex2html_wrap_inline1524 -with the upper matrix index lowered- is symmetric for any tex2html_wrap_inline1518, and so each component of tex2html_wrap_inline1496 is symmetric. Even in the general (strongly hyperbolic) case, one can find a base (tex2html_wrap_inline1518 dependent) in which tex2html_wrap_inline1532 can be diagonalized, basically because it is symmetric with respect to the (tex2html_wrap_inline1518 dependent) scalar product induced by tex2html_wrap_inline1512 . In this diagonal version, it is easy to see that the well posedness requires all eigenvalues of tex2html_wrap_inline1538 to be purely imaginary. Thus we see that an equivalent characterization for well posedness of first order systems is that their principal part (i.e. tex2html_wrap_inline1538) has purely imaginary eigenvalues, and that it can be diagonalized by an invertible, tex2html_wrap_inline1518 -dependent, transformation. The classical example of a symmetric hyperbolic system is the wave equation.  

For simplicity we consider the wave equation in 1+1 dimensions. Choosing Cartesian coordinates we have,

displaymath1546

and so defining the ``vector'' tex2html_wrap_inline1548 we have the following first order system:

displaymath1550

There are several other notions of hyperbolicity that appear in the literature:



2.3 Generalization to Variable Coefficient 2 The Theory of Linear 2.1 Existence and Uniqueness of

image Hyperbolic methods for Einstein's Equations
Oscar A. Reula
http://www.livingreviews.org/lrr-1998-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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