16 1. Simple Examples of Propagation along codimension one characteristic hypersurfaces in space-time. The sin- gularities propagate satisfying transport equations along rays generating the hypersurface. The general class goes under the name conormal solutions. M. Beals’ book [Beals, 1989] is a good reference. They describe propagating wavefronts. Luneberg’s book [Luneberg, 1944] recounts his discovery that the propagation laws for fronts of singularities coincide with the physical laws of geometric optics. 1.3. Group velocity and the method of nonstationary phase The Klein–Gordon equation has constant coefficients, and so it can be solved explicitly using the Fourier transform. The computation of the singularities of the fundamental solution of the Klein–Gordon equation in Exercise 1.2.5 suggests that the main part of solutions travel with speed equal to 1. One might expect that the energy in a disk growing linearly in time at a speed 1 would be small for t 1. For compactly supported data, such a disk would contain no singularities for large time. Thus it is not unreasonable to guess that for each σ 1 and R 0, (1.3.1) lim sup t→∞ |x|R+σt u2 t + u2 x + u2 dx = 0. Either the method of characteristics or the energy method shows that speeds are no larger than one. The idea about the Klein–Gordon energy expressed in (1.3.1) is dead wrong. The main part of the energy travels strictly slower than speed 1, even though singularities travel with speed exactly equal to 1. The solution of the Cauchy problem for the Klein–Gordon equation in dimension d, utt Δ u + u = 0 , (t, x) R1+d , is given by u = ± (2π)−d/2 a±(ξ) ei(±ξ t+xξ) , ξ := ( 1 + |ξ|2 )1/2 , ˆ(0,ξ) = a+(ξ) + a−(ξ) , ˆt(0,ξ) = i ξ ( a+(ξ) a−(ξ) ) . The conserved energy is equal to 1 2 ut 2 + |∇xu|2 + u2 dx = ξ 2 ( |a+(ξ)|2 + |a−(ξ)|2 ) . Exercise 1.3.1. Verify these formulas. Verify conservation of energy by an integration by parts argument as in Exercise 1.2.3. Hint. Follow the computation that starts §1.4.
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