24 1. Differential Equations and Their Solutions Let V : Rn × Rk Rn be a smooth function. Then to each α in Rk we can associate a vector field V (·,α) on Rk, defined by x V (x, α). For this reason it is customary to consider V as a “vector field on Rn depending on a parameter α in Rk”. It is often important to know how solutions of dx dt = V (x, α) depend on the parameter α, and this is answered by the following theorem. 1.4.5. Theorem on Smoothness w.r.t. Parameters. Let V : Rn × Rk Rn be a Cr map, r 1, and let σp α denote the maximum solution curve of dx dt = V (x, α) with initial condition p. Then the map (p, α, t) σα(t) p is Cr. Exercise 1–19. Deduce this from the Theorem on Smoothness w.r.t. Initial Conditions. Hint: This is another one of those cute reduction arguments that this subject is full of. The idea is to consider the vector field ˜ on Rn × Rk defined by ˜ (x, α) = (V (x, α), 0) and to note that its maximal solution with initial condition (p, α) is t (σp α (t),α). You may have noticed an ambiguity inherent in our use of σp to denote the maximal solution curve with initial condition p of a vector field V . After all, this maximal solution clearly depends on V as well as on p, so let us now be more careful and denote it by σp V . Of course, this immediately raises the question of just how σp V depends on V . If V changes just a little, does it follow that σp V also does not change by much? If we return to our smoke particle in the wind metaphor, then this seems reasonable if we make a tiny perturbation of the direction and speed of the wind at every point, it seems that the path of a smoke particle should not be grossly different. This intuition is correct, and all that is required to prove it is another tricky application of Gronwall’s Inequality. 1.4.6. Theorem on the Continuity of σp V w.r.t. V. Let V be a C1 time-dependent vector field on Rn and let K be a Lipschitz con- stant for V , in the sense that V (x, t) V (y, t)≤ K x y for all x, y, and t. If W is another C1 time-dependent vector field on Rn such that V (x, t) W (x, t)≤ for all x and t, then σV p (t) σW p (t) K ( eKt 1 ) .
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