A Phase Space Transform Adapted to the Wave Equation

Abstract

Abstract. Wave packets emerged in recent years as a very useful tool in the study of nonlinear wave equations. In this article we introduce a phase space transform adapted to the geometry of wave packets, and use it to characterize and study the associated classes of pseudodifferential and Fourier integral operators.

Gradient NLW on Curved Background in 4+1 Dimensions

“…To characterize the higher regularity properties of these flows is convenient to introduce (see [4]) a metric g α in the phase space, defined by…”

“…From so that the support of χ ±,α θ (t, x, ξ) is contained in H ± α S α (θ). The regularity of these symbols is easily obtained from the transport equations (see again [4]): We use the above partition of unity in the phase space to produce a corresponding pseudodifferential partition of unity. Given a frequency λ > α −2 we define the symbols…”

“…In all three cases we are interested in the local well-posedness of the Cauchy problem in Sobolev spaces H s (R n ) × H s−1 (R n ) with initial data (4) u(0, x) = u 0 (x), ∂ t u(0, x) = u 1 (x)…”

A Multi-Scale Approach to Hyperbolic Evolution Equations with Limited Smoothness

“…Furthermore, the frame of curvelets and the associated curvelet transform used here can be related Downloaded by [University of Southern Queensland] at 09:19 11 October 2014 to the Fourier-Bros-Iagolnitzer (FBI) transform (Bros andIagolnitzer, 1975-1976) as well as with Gaussian beams and the overcomplete frame they form (for example, Shlivinski et al, 2004). Geba and Tataru (2006) adapted the Bargmann transform to the wave equation in a manner related to curvelets to characterize the associated classes of Fourier integral operators. Curvelets have been used in analyzing wave propagators and associated Fourier integral operators by Candès andDemanet (2003, 2005).…”

Diffraction from conormal singularities

“…Here the main interest is in α < 1, for in the C 1,1 setting one has at least a partial understanding of wave propagation without a geometric structure to the singularities of the metric, such as conormality (though of course one does need some geometric structure to obtain a theorem analogous to ours), as then the Hamilton vector field is Lipschitz, and automatically has unique integral curves; see Smith's paper [16] where a parametrix was constructed, and also the work of Geba and Tataru [2]. We also recall that, in a different direction, for even lower regularity coefficients, Tataru has shown Strichartz estimates [17]; these are not microlocal in the sense of distinguishing reflected vs. transmitted waves as above.…”