The Theory of Linear Constant Coefficients Evolution Equations and Generalizations to Quasi-linear Systems

2 The Theory of Linear Constant Coefficients Evolution Equations and Generalizations to Quasi-linear Systems

In this section, I summarize the main results of the theory of first order evolutionary partial differential equation systems. I do this by first developing the theory of linear constant coefficients evolution equation systems in $ℜn$ , that is, equations of the type:

ut = P (D )u,

where $u = u(x,t)$ indicates a “vector” valued function of dimension $s$ in $n+1 ℜ$ , $ut$ its time derivative, $P (D )$ , a $s × s$ matrix whose components smoothly depend on: $|ν| D ν := ∂xν∂1...xνnn- 1$ . For most of the results, no particular form for the dependence of $P$ on $D$ is needed, as long as it is continuous. But for simplicity one can think of $P$ as given by:

∑ P(D ) := AνD ν. |ν|≤m

We shall focus on the Cauchy (or initial value) Problem for the above system, namely under what conditions it is true that given the value of $u$ at $t = 0$ , $f (x)$ , say, there exists a unique solution, $u(t,x)$ , to the above system with $u(0,x ) := f(x )$ . Later we shall mention a related problem which is important on most numerical schemes used in relativity, namely the initial-boundary value problem, where one also prescribes some data on time-like boundaries.

What follows is a short account of Chapter II of [48 ], see also Chapter IV of [36 ]. After this, I indicate what aspects of the theory generalize to quasi-linear systems, and under which further assumptions this is so. I also give some indications of the relation of this theory to the stability issues of numerical simulations. This section can be skipped by those not interested in the mathematical theory itself or those who already know it.

2.1 Existence and uniqueness of smooth solutions
2.2 First order systems
2.3 Generalization to variable coefficient and non-linear systems
2.4 Hyperbolicity and numerical simulations