Low regularity ray tracing for wave equations with Gaussian beams. (English) Zbl 1418.35349

Summary: 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 A. Waters [Commun. Math. Sci. 9, No. 1, 225–254 (2011; Zbl 1295.35011)].


35R01 PDEs on manifolds
35R30 Inverse problems for PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
58J45 Hyperbolic equations on manifolds
35A22 Transform methods (e.g., integral transforms) applied to PDEs


Zbl 1295.35011
Full Text: DOI arXiv Euclid


[1] G. Alessandrini and J. Sylvester, Stability for a multidimensional inverse spectral theorem, Comm. Partial Differential Equations 15 (1990), no. 5, 711-736. · Zbl 0715.35080
[2] 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. · Zbl 1325.35274
[3] 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. · Zbl 0786.93009
[4] C. Bender and S. Orszag, Advanced mathematical methods for scientists and engineers, I: Asymptotics and perturbation theory, Springer, 1999. · Zbl 0938.34001
[5] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal. 14 (1997), no. 2, 157-191. · Zbl 0892.93009
[6] 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. · Zbl 1016.35003
[7] A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Diff. Equations 3 (1978), no. 11, 979-1005. · Zbl 0389.35046
[8] G. Eskin, Lectures on linear partial differential equations, Graduate Studies in Mathematics 123, American Mathematical Society, Providence, RI, 2011. · Zbl 1228.35001
[9] 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. · Zbl 1320.93027
[10] L. Hörmander, Fourier integral operators I, Acta Math. 127 (1971), no. 1-2, 79-183.
[11] 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.
[12] 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. · Zbl 0155.16401
[13] V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Partial Differential Equations 16, (1991), no. 6-7, 1183-1195. · Zbl 0739.35105
[14] V. Isakov and Z. Q. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems 8 (1992), no. 2, 193-206. · Zbl 0754.35184
[15] J.B. Keller, Corrected Bohr-Sommerfield quantum conditions for nonseparable systems, Ann. Physics 4 (1958), 180-188. · Zbl 0085.43103
[16] 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. · Zbl 1037.35098
[17] H. Liu, O. Runborg and N. M. Tanushev, Error estimates for Gaussian beam superpositions, Math. Comp. 82 (2013), no. 282, 919-952. · Zbl 1277.35236
[18] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, vol. III, Grundlehren der mathematischen Wissenschaft 183, Springer, 1972. · Zbl 0227.35001
[19] V. P. Maslov and M. V. Fedoriuk, Semiclassical approximation in quantum mechanics, Mathematical Physics and Applied Mathematics 7, D. Riedel Publishing Co., Dordrecht, 1981.
[20] 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. · Zbl 1223.65076
[21] Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems 6 (1990), no. 1, 91-98. · Zbl 0712.35104
[22] Rakesh and W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations 13 (1988), no. 1, 87-96. · Zbl 0667.35071
[23] 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.
[24] P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z. 201 (1989), no. 4, 541-559. · Zbl 0653.35049
[25] 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. · Zbl 1088.53027
[26] 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. · Zbl 1058.44003
[27] T. Tao Nonlinear dispersive equations: Local and global analysis, CBMS Regional Conference Series in Mathematics 106, American Mathematical Society, Providence, RI, 2006. · Zbl 1106.35001
[28] 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. · Zbl 1295.35011
[29] 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. · Zbl 1304.35773
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.