By G. F. D. Duff, D. Naylor
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4 Notes The material in this chapter can be found in many places, for instance Courant-Hilbert [8], Evans [9], John [12], Taylor [23], and Vainberg [24]. We concentrated on smooth solutions of the Cauchy problem, with no attempt to discuss nonsmooth solutions or other boundary value problems for Hamiton–Jacobi equations. Some hints about blowup of smooth solutions are given in the Exercises. 1. Conservation laws in the plane R2x,t are special nonlinear systems in divergence form ∂t u + ∂x (F (u)) = 0.
Prove that if there is x0 where u0 (x0 ) = max u0 > 0, u0 (x0 ) < 0, then max(u0 − u0 ) > max u0 . (b) Use the method of characteristics to solve the equation. Prove that a C 2 solution u exists for 0 ≤ t < T¯ ≡ (max(u0 − u0 ))−1 . (c) Let u ∈ C 2 (R × [0, T [) be a solution of the Cauchy problem. Denote by (x(t), t) the characteristic starting from (x0 , 0) and q(t) = (∂x u)(x(t), t). Establish the ODE satisfied by q, and compute q explicitly. Deduce from this that T ≤ T¯ (hence, as in Exercise 5, T¯ is the lifespan).
Though S is a union of horizontal lines, it is not necessarily everywhere the graph of a smooth function! To see this, observe that at a point m = (y + su0 (y), s, u0 (y)) ∈ S, Tm S is spanned by ∂y F = (1 + su0 (y), 0, u0 (y)), ∂s F = (u0 (y), 1, 0). There exists a nontrivial vertical linear combination of these two vectors if and only if 1 + su0 (y) = 0. If u0 is not everywhere nonnegative, such points always exist (see Exercises 1–7 for a more precise discussion). 30 Chapter 3 Nonlinear First Order Equations The method of characteristics is an effective way of finding solutions.