6 1. Simple Examples of Propagation
Its values along the characteristic (t, x(t)) are determined by integrating the
nonhomogeneous linear ordinary differential equation
(1.1.8)
d
dt
u(t, x(t)) + d(t, x(t)) u(t, x(t)) = f(t, x(t)) .
There are finite regularity results too. If f, g are k times differentiable with
k ≥ 1, then so is u. Though the equation is first order, u is in general not
smoother than f. This is in contrast to the elliptic case.
The method of characteristics also applies to systems of hyperbolic equa-
tions. Consider vector valued unknowns u(t, x) ∈
CN
. The simplest gener-
alization is diagonal real systems
ut + diag(c1(t, x),...,cN (t, x)) u = 0.
Here uj is constant on characteristics with speed cj(t, x). This idea extends
to some systems
L := ∂t + A(t, x) ∂x + B(t, x),
where A and B are smooth matrix valued functions so that
∀T, ∀α, ∂t,x{A,
α
B} ∈
L∞([−T,
T] × R) .
The method of characteristics applies when the following hypothesis is sat-
isfied. It says that the matrix A has real eigenvalues and is smoothly di-
agonalizable. The real spectrum as well as the diagonalizability are related
to a part of the general theory of constant coefficient hyperbolic systems
sketched in the Appendix 2.I to Chapter 2.
Hypothesis 1.1.2. There is a smooth matrix valued function, M(t, x), so
that
∀T, ∀α, ∂t,xM
α
and ∂t,x(M
α −1)
belong to
L∞([0,T]
× R)
and
(1.1.9) M
−1
A M = diagonal and real.
Examples 1.1.4. 1. The hypothesis is satisfied if for each t, x the matrix
A has N distinct real eigenvalues c1(t, x) c2(t, x) · · · cN (t, x). Such
systems are called strictly hyperbolic. To guarantee that the estimates on
M, M
−1
are uniform as |x| → ∞, it suffices to make the additional assump-
tion that
inf
(t,x)∈[0,T ]×R
min
2≤j≤N
cj(t, x) − cj−1(t, x) 0 .
2. More generally the hypothesis is satisfied if for each (t, x), A has
uniformly distinct real eigenvalues and is diagonalizable. It follows that the
multiplicity of the eigenvalues is independent of t, x.
