1.1. First-Order ODE: Existence and Uniqueness 11 we will not give further details here. Instead we refer to Appendix B where you will find a complete proof. 1.1.5. Remark. We give a minor technical point. The argument in Appendix B only gives a local uniqueness theorem. That is, it shows that if x1 : (a, b) → Rn and x2 : (a, b) → Rn are two solutions of the same ODE, then if x1 and x2 agree at a point, then they also agree in a neighborhood of that point, so that the set of points in (a, b) where they agree is open. But since solutions are by definition continuous, the set of points where x1 and x2 agree is also a closed subset of (a, b), and since intervals are connected, it then follows that x1 and x2 agree on all of (a, b). 1.1.6. Remark. The existence and uniqueness theorems tell us that for a given initial condition x0 we can solve our initial value problem (uniquely) for a short time interval. The next question we will take up is for just how long we can “follow a smoke particle”. One important thing to notice is the uniformity of the in the existence theorem—not only do we have a solution for each initial condition, but moreover given any p0 in Rn, we can find a fixed interval I = (t0 − , t0 + ) such that a solution with initial condition p exists on the whole interval I for all initial conditions suﬃciently close to p0. Still, this may be less than what you had hoped and expected. You may have thought that for each initial condition p in Rn we should have a solution xp : R → Rn of the differential equation with xp(t0) = p. But such a global existence theorem is too much to expect. For example, taking n = 1 again, consider the differential equation dx dt = x2 with the initial condition x(0) = x0. An easy calculation shows that the unique solution is x(t) = x0 1−x0t . Note that, for each initial condition x0, this solution “blows up” at time T = 1 x0 , and by the Uniqueness Theorem, no solution can exist for a time greater than T . But, you say, a particle of smoke will never go off to infinity in a finite amount of time! Perhaps the smoke metaphor isn’t so good after all. The answer is that a real, physical wind field has bounded velocity, and it isn’t hard to show that in this case we do indeed have

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