1.3. Group velocity and the method of nonstationary phase 19 one has the estimate eiφ/ f(x) dx C2 m |α|≤m ∂αf L1 . Exercise 1.3.2. Prove the lemma. Hint. Use (1.3.2). Example 1.3.2. Applied to the phases φ = := j xjξj with ξ belonging to a compact subset of Rd \ 0, the lemma implies the rapid decay of the Fourier transform of smooth compactly supported functions. Conversely, the lemma can be reduced to the special case of the Fourier transform. Since the gradient of φ in the lemma does not vanish, for each x supp f, there is a neighborhood and a nonlinear change of coordinates so that in the new coordinates φ is equal to x1. Using a partition of unity, one can suppose that f is the sum of a finite number of functions each supported in one of the neighborhoods. For each such function, a change of coordinates yields an integral of the form eix1/ g(x) dx = c ˆ(1/, 0,..., 0) , which is rapidly decaying since it is the transform of an element of C0 (Rd). Care must be taken to obtain estimates uniform in the family of phases of the lemma. Exercise 1.3.3. Suppose that f H1(R) and g L2(R) and that u is the unique solution of the Klein–Gordon equation with initial data (1.3.3) u(0,x) = f(x) , ut(0,x) = g(x) . Prove that for any 0 and R 0, there is a δ 0 such that (1.3.4) lim sup t→∞ |x| (1−δ)t−R u2 t + u2 x + u2 dx . Hint. Replace ˆ ˆ by compactly supported smooth functions making an error at most /2 in energy. Then use Lemma 1.3.2 noting that the group velocities are uniformly smaller than those for ξ belonging to the supports of a±. Discussion. Note that as ξ ∞, the group velocities approach ±1. High frequencies will propagate at speeds nearly equal to one. In par- ticular they travel at the same speed. High frequency signals stay together better than low frequency signals. Since singularities of solutions are made of only the high frequencies (modifying the Fourier transform of the data on a compact set modifies the solution by a smooth term), one expects sin- gularities to propagate at speeds ±1. That is proved for the fundamental solution in Exercise 1.2.5. Once this is known for the fundamental solution, it follows for all. The simple proof is an exercise in my book [Rauch, 1991, pp. 164–165].
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