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,
ut = AaDau + Bu (1 )

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

P1 (D ) := AaDau,
where a A is a s × s matrix valued vector in n ℜ. 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 α = 0 above.

Theorem 3 A first order system is well posed if and only if there exist, a constant k, and a positive definite Hermitian form H (ω ) such that:

k− 1I ≤ H (ω) ≤ kI and H (ω)Aa ωa − Aa⋆ωaH (ω) = 0 ∀ωa with |ωa | = 1.

If Aa satisfies the above condition for some H (ω ), 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 ωa, a case that appears in most physical problems, then we say the system is symmetric hyperbolic. Indeed, if H does not depend on ωa, 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 Aa ωa – with the upper matrix index lowered – is symmetric for any ωa, and so each component of Aa is symmetric. Even in the general (strongly hyperbolic) case, one can find a base (ωa dependent) in which P (iω ) can be diagonalized, basically because it is symmetric with respect to the (ωa dependent) scalar product induced by H (ω ). In this diagonal version, it is easy to see that the well posedness requires all eigenvalues of a iA ωa 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. iAa ωa) has purely imaginary eigenvalues, and that it can be diagonalized by an invertible, ω a-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,

ϕtt − ϕxx = 0,
and so defining the “vector” u := (ϕ,ϕt,ϕx ) we have the following first order system:
⌊ ⌋ ⌊ ⌋ 0 0 0 0 1 0 ut = |⌈0 0 1|⌉ ux + |⌈ 0 0 0 |⌉u 0 1 0 0 0 0

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

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