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, ∂α t,x {A, 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.
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