Abstract
We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a \(1/2\) power of \(t\) decay relative to what would be dictated by energy estimates, in a small part of spacetime.
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Notes
We remark that in the body of the paper, \(X\) will refer to the manifold with boundary given by compactifying such a space.
We may a priori assume that \(h\) is in fact a smooth tensor including \(dr\) components; that we may then change variables to remove these components is a result of Joshi and Sá Barreto [9], Section 2.
If preferred, we could of course replace the norm on the inhomogeneity by the weighted \(L^2\) spacetime norm
$$\begin{aligned}\int _0^T{\left||{(2+t^2)^{1/4} \log (2+t^2) (\Box +V) u}\right||}^2 \,dt.\end{aligned}$$The notation is short for “main face”, as for \(X\) Euclidean (diffeomorphically, not metrically), this can be identified with an open dense subset of the boundary of the radial compactification of Minkowski space. Note that the light cone, \(r/t=1\), hits the boundary in the interior of \(\mathrm mf \). If \(r/t=1\) in \(\mathrm mf \) were blown up inside \(\tilde{M}\), the resulting front face is where Friedlander’s radiation field [4] can be defined by rescaling a solution to the wave equation. Indeed, in the interior of this front face (in \(t>0\)) \(r-t=(r/t-1)/t^{-1}\) becomes a smooth variable along the fibers of the front face.
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The authors gratefully acknowledge partial support from the NSF under grant numbers DMS-0801226 (A.V.) and DMS-0700318, DMS-1001463 (J.W.). A.V. is also grateful for support from a Chambers Fellowship at Stanford University. The authors are grateful to Daniel Tataru, and to an anonymous referee, for many helpful comments, and in particular for suggesting the refined \(\ell ^\infty \)–\(\ell ^1\) version of the estimates in Theorem 1.1.
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Vasy, A., Wunsch, J. Morawetz estimates for the wave equation at low frequency. Math. Ann. 355, 1221–1254 (2013). https://doi.org/10.1007/s00208-012-0817-x
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DOI: https://doi.org/10.1007/s00208-012-0817-x