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12 1. Simple Examples of Propagation Denote by Φj(t, x) the flow of the ordinary differential equation x = cj(t, x). That is x(t) = Φj(t, x) is the solution with x(0) = x. The solution operator for the pure transport equation (∂t + cj∂x)u = 0 with initial value g is then u(t) = g(Φj(−t, x)) . The values at time t are the rearrangements by the diffeomorphism Φ(−t, ·) of the initial function. Because of the uniform boundedness of the derivatives of cj on slabs [0,T] × R, one has ∂α t,x Φ ∈ L∞([0,T] × R) . The derivative ∂xΦ measures the expansion or contraction by the flow. It is the length of the image of an infinitesimal interval divided by the original length. In particular Φ can at most expand lengths by a bounded quan- tity. The inverse of Φ(t, ·) is the flow by the ordinary differential equation from time t to time 0, so the inverse also cannot expand by much. This is equivalent to a lower bound, ( ∂xΦ ) −1 ∈ L∞([0,T] × R) . The diffeomorphism Φ(t, ·) can neither increase nor decrease length by much. Therefore the maps u(0) → u(t) are uniformly bounded maps from Lp(R) to itself for all p ∈ [1, ∞]. The case p = ∞ is equivalent to the Haar inequalities. There are analogous estimates u(t) Lp(R) ≤ C u(t) Lp(R) + t 0 Lu(σ) Lp(R) dσ , with constant independent of p. This in turn leads to an existence theory like that just recounted but mk(u, t) is replaced by ∑ |α|≤k ∂α t,x u(t) Lp(R) . For these one dimensional hyperbolic Cauchy problems, there is a wide class of spaces for which the evolution is well posed. The case of p = 1 is particularly important for the theory of shock waves in d = 1. Brenner’s Theorem 3.3.5 shows that only the case p = 2 remains valid for typical hyperbolic equations in dimension d 1. 1.2. Examples of propagation of singularities using progressing waves D’Alembert’s solution (see Example 1.1.2) of the one-dimensional wave equa- tion, (1.2.1) utt − uxx = 0 , is the sum of progressing waves (1.2.2) f(x − t) + g(x + t) .