1.1. The method of characteristics 11 Definition 1.1.4. For Z k 1, the set PCk consists of functions which are piecewise Ck as the set of bounded continuous functions u on [0,T] × R so that for α k and 1 j N, the restriction u|W j belongs to Ck(Wj) and for all α k, ∂α(u|W j ) extends to a bounded continuous function on the closure W j . It is a Banach space with the norm u L∞([0,T ]×R) + |α|≤k 1≤j≤N+1 ∂α t,x ( u W j ) L∞(W j ) . The next result asserts that for piecewise smooth data with singularity at x, the solution is piecewise smooth with its singularities restricted to the characteristics through x. Theorem 1.1.5. Suppose in addition to Hypothesis 1.1.2, that A has N distinct real eigenvalues for all (t, x). If f PCk and g L∞(R) have bounded continuous derivatives up to order k on each side of x, then the solution u belongs to PCk. Sketch of Proof. The construction of u yielded an L∞([0,T]×R) estimate. In addition we need estimates for the derivatives of order k on the wedge Wj. Introduce μk(u, σ) := u(σ) L∞(R) + 2≤|α|≤k 1≤j≤N+1 ∂t,x α ( u Wj ) (σ) L∞(Wj∩{t=σ}) . To estimate un un−1 use the following lemma. Lemma 1.1.6. Assume the hypotheses of the theorem and that cj(t, x) is one of the eigenvalues of A(t, x). Then, there is a constant C(j, T, L) so that if f PCk and ( ∂t + cj(t, x) ∂x ) w = f, w t=0 = 0, then w PCk and μk(w, t) C μk(w, 0) + t 0 μk(f, σ) . Exercise 1.1.3. Prove the lemma. Then finish the proof of the theorem. Exercise 1.1.4. Suppose that u is as in the theorem, f = 0, and that for some 0 and j, the derivatives of u of order k are continuous across γj {0 t }. Prove that they are continuous across γj {0 t T}. Hints. Show that the set of times t for which the solution is Ck on γj {0 t t} is both open and closed. Use finite speed.
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