Rocky Mountain Journal of Mathematics

Low regularity ray tracing for wave equations with Gaussian beams

Alden Waters

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We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb {R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).

Article information

Rocky Mountain J. Math., Volume 49, Number 3 (2019), 1005-1027.

First available in Project Euclid: 23 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]
Secondary: 35R30: Inverse problems 35L20: Initial-boundary value problems for second-order hyperbolic equations 58J45: Hyperbolic equations [See also 35Lxx] 35A22: Transform methods (e.g. integral transforms)

Control theory inverse problems wave equations observability low regularity coefficients Gaussian beams


Waters, Alden. Low regularity ray tracing for wave equations with Gaussian beams. Rocky Mountain J. Math. 49 (2019), no. 3, 1005--1027. doi:10.1216/RMJ-2019-49-3-1005.

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  • G. Alessandrini and J. Sylvester, Stability for a multidimensional inverse spectral theorem, Comm. Partial Differential Equations 15 (1990), no. 5, 711–736.
  • G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc. 27 (2014), no. 4, 953–981.
  • C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), no. 5, 1024–1065.
  • C. Bender and S. Orszag, Advanced mathematical methods for scientists and engineers, I: Asymptotics and perturbation theory, Springer, 1999.
  • N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal. 14 (1997), no. 2, 157–191.
  • C. Castro and E. Zuazua, Concentration and lack of observability of waves in highly heterogeneous media, Arch. Ration. Mech. Anal. 164 (2002), no. 1, 39–72. With an addendum in Arch. Ration. Mech. Anal. 185 (2007), no. 3, 365–377.
  • A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Diff. Equations 3 (1978), no. 11, 979–1005.
  • G. Eskin, Lectures on linear partial differential equations, Graduate Studies in Mathematics 123, American Mathematical Society, Providence, RI, 2011.
  • F. Fanelli and E. Zuazua, Weak observability estimates for 1-D wave equations with rough coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 2, 245–277.
  • L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), no. 1-2, 79–183.
  • L. Hörmander, The analysis of linear partial differential operators, III: Pseudo-differential operators, Grundlehren der mathematischen Wissenschaften 274, Springer, 1994. Corrected reprint of the 1985 original.
  • A. E. Hurd and D. H. Sattinger, Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Am. Math. Soc. 132 (1968), 159–174.
  • V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations 16, (1991), no. 6-7, 1183–1195.
  • V. Isakov and Z. Q. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems 8 (1992), no. 2, 193–206.
  • J.B. Keller, Corrected Bohr-Sommerfield quantum conditions for nonseparable systems, Ann. Physics 4 (1958), 180–188.
  • A. Katchalov, Y. Kurylev and M. Lassas, Inverse boundary spectral problems, Monographs and Surveys in Pure and Applied Mathematics 123, Chapman and Hall/CRC, Boca Raton, FL, 2001.
  • H. Liu, O. Runborg and N. M. Tanushev, Error estimates for Gaussian beam superpositions, Math. Comp. 82 (2013), no. 282, 919–952.
  • J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, vol. III, Grundlehren der mathematischen Wissenschaft 183, Springer, 1972.
  • V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics 7, D. Riedel Publishing Co., Dordrecht, 1981.
  • J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul. 8 (2010), no. 5, 1803–1837.
  • Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems 6 (1990), no. 1, 91–98.
  • Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations 13 (1988), no. 1, 87–96.
  • J. Ralston, Gaussian beams and the propagation of singularities, pp. 206–248 in Studies in partial differential equations, MAA Stud. Math 23, Mathematics Association of America, Washington, DC, 1982.
  • P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z. 201 (1989), no. 4, 541–559.
  • P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not. (2005), no. 17, 1047–1061.
  • P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J. 123 (2004), no. 3, 445–467.
  • T. Tao Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics 106, American Mathematical Society, Providence, RI, 2006.
  • A. Waters, A parametrix construction for the wave equation with low regularity coefficients using a frame of Gaussians, Commun. Math. Sci. 9 (2011), no. 1, 225–254.
  • A. Waters, Stable determination of X-ray transforms of time-dependent potentials from partial boundary data, Comm. Partial Differential Equations 39 (2014), no. 12, 2169–2197.