# New Trends in the Theory of Hyperbolic Equations by Michael Reissig, Bert-Wolfgang Schulze

By Michael Reissig, Bert-Wolfgang Schulze

This booklet provides numerous fresh advancements within the conception of hyperbolic equations. The rigorously chosen invited and peer-reviewed contributions care for questions of low regularity, serious progress, ill-posedness, decay estimates for strategies of other non-linear hyperbolic versions, and introduce new ways in keeping with microlocal equipment.

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This inequality in turn follows easily from the fact that χ and ψ are rapidly decreasing functions in the Schwartz class S(R). 62) and see that it is suﬃcient to apply the obvious estimate C , N ≥ 2. 63) and of the theorem. 2. 42) and u is a solution to utt − ∆u = F, with initial data u(0, x) = ut (0, x) = 0. 68) for all s > n/2 and 1/2 < θ < 1. Proof. 45). Further, we can localize the space Fourier transform in x so that suppξ F (t, ξ) ⊂ {|ξ| ∼ 2j }, j = 0, 1, 2, . . 70) |ξ| −∞ satisﬁes ϕ1 v ∈ H s,θ .

110) while for |ξ| > τ > 0 we have L− (τ, ξ) ≡ |ξ −ξ1 |−|ξ1 |=τ ξ1 We now introduce polar coordinates ρ = |ξ1 |, ϕ ∈ (0, π), ω ∈ Sn−2 , such that ξ1 = (ρ cos ϕ, ρ sin ϕω). Then the Euclidean metric in Rn can be written (if n ≥ 3) dρ2 + ρ2 (dϕ2 + sin2 ϕdω 2 ), where dω 2 is the standard metric on Sn−2 . A trivial modiﬁcation in the above relations is necessary for the special case n = 2, when ξ1 = (ρ cos ϕ, ρ sin ϕ) and the metric in R simpliﬁes to 2 dρ2 + ρ2 dϕ2 . Wave Maps 35 The ellipsoid S+ (τ, ξ) = {|ξ − ξ1 | + |ξ1 | = τ } in the new coordinates will have the equation (recall that S+ (τ, ξ) is an ellipsoid if and only if τ > |ξ|) ρ= τ 2 − |ξ|2 , 2(τ − |ξ| cos ϕ) ϕ ∈ (0, π).

23) takes the form x b ≤C x a and recalling the assumption a ≥ b we see that also in this case the inequality is true. 24) and this completes the proof of the Lemma for the case 0 ≤ b ≤ a. 25) and this completes the proof of the Lemma. 3. 27) for any T > 0 and any real numbers τ, τ1 , A. Proof. The proof is divided in a few cases. 27). 27) it is suﬃcient to show that (recall that s1 ≥ s) A τ1 s N −N1 ≤ C |ττ1 | − A s . 2 and the assumptions N ≥ N1 , N − N1 ≥ s. 2 and the assumptions N ≥ N1 , N − N1 ≥ s.