# American Institute of Mathematical Sciences

September  2011, 4(3): 589-631. doi: 10.3934/krm.2011.4.589

## A Gaussian beam approach for computing Wigner measures in convex domains

 1 Aeroelasticity and Structural Dynamics Department, Onera-The French Aerospace Lab, F-92322 Châtillon, France 2 Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, 200900 Shanghai, China 3 Mechanics, Structures and Materials Laboratory, École Centrale Paris, 92295 Châtenay-Malabry, France

Received  March 2011 Revised  May 2011 Published  August 2011

A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset $\Omega$ of $IR^n$, with initial conditions compactly supported in $\Omega$, and Dirichlet or Neumann type boundary condition. The transport of the microlocal energy density along the broken bicharacteristic flow at the high frequency limit is proved through the use of Wigner measures. Our approach consists first in computing explicitly the Wigner measures under an additional control of the initial data allowing to approach the solution by a superposition of first order Gaussian beams. The results are then generalized to standard initial conditions.
Citation: Jean-Luc Akian, Radjesvarane Alexandre, Salma Bougacha. A Gaussian beam approach for computing Wigner measures in convex domains. Kinetic & Related Models, 2011, 4 (3) : 589-631. doi: 10.3934/krm.2011.4.589
##### References:
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Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French),, in, (1976), 165. Google Scholar [12] M. Combescure, J. Ralston and D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition,, Commun. Math. Phys., 202 (1999), 463. doi: 10.1007/s002200050591. Google Scholar [13] T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, (). Google Scholar [14] S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics,, Multiscale Model. Simul., 1 (2003), 674. doi: 10.1137/S1540345902409797. Google Scholar [15] E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces,, J. Math. Pures Appl. (9), 87 (2007), 144. doi: 10.1016/j.matpur.2006.11.002. Google Scholar [16] I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit,, Asympt. Anal., 14 (1997), 97. Google Scholar [17] P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves],, in, (1991), 1990. Google Scholar [18] P. Gérard, Microlocal defect measures,, Commun. Partial Differential Equations, 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [19] P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem,, Duke Math. J., 71 (1993), 559. Google Scholar [20] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. Google Scholar [21] L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principle of Mathematical Sciences], 256,, Springer-Verlag, (1983). Google Scholar [22] V. 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Phys., 228 (2009), 2951. doi: 10.1016/j.jcp.2009.01.007. Google Scholar [28] S. Leung and J. Qian, The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation,, J. Comput. Phys., 229 (2010), 8888. doi: 10.1016/j.jcp.2010.08.015. Google Scholar [29] P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures],, Rev. Mat. Iberoamericana, 9 (1993), 553. Google Scholar [30] H. Liu and J. Ralston, Recovery of high frequency wave fields for the acoustic wave equation,, Multiscale Model. Simul., 8 (): 428. doi: 10.1137/090761598. Google Scholar [31] H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, (). Google Scholar [32] F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach,, Asymptot. Anal., 32 (2002), 1. Google Scholar [33] P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent Quantum-Vlasov equation in $3$D,, Math. Models Methods Appl. Sci., 3 (1993), 109. doi: 10.1142/S0218202593000072. Google Scholar [34] P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi)classical limits: electrons in a periodic potential,, J. Math. Phys., 35 (1994), 1066. doi: 10.1063/1.530629. Google Scholar [35] P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations,, Pubbl. Ian, 1035 (1997), 1. Google Scholar [36] A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis,", Universitext, (2002). Google Scholar [37] L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary,, J. Math. Pures Appl. (9), 79 (2000), 227. doi: 10.1016/S0021-7824(00)00158-6. Google Scholar [38] M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave Motion, 47 (2010), 421. doi: 10.1016/j.wavemoti.2010.02.001. Google Scholar [39] A. N. Norris, Elastic Gaussian wave packets in isotropic media,, Acta Mech., 71 (1988), 95. doi: 10.1007/BF01173940. Google Scholar [40] G. Papanicolaou and L. Ryzhik, Waves and Transport,, in, 5 (1999), 305. Google Scholar [41] T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures,, Comm. Math. Phys., 175 (1996), 229. doi: 10.1007/BF02102407. Google Scholar [42] M. Pulvirenti, Semiclassical expansion of Wigner functions,, J. Math. Phys., 47 (2006). Google Scholar [43] J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation,, Multiscale Model. Simul., 8 (2010), 1803. doi: 10.1137/100787313. Google Scholar [44] J. Ralston, Gaussian beams and the propagation of singularities,, in, 23 (1982), 206. Google Scholar [45] S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions,, J. Math. Phys., 34 (1993), 2185. doi: 10.1063/1.530112. Google Scholar [46] N. M. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449. Google Scholar [47] N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comput. Phys., 228 (2009), 8856. doi: 10.1016/j.jcp.2009.08.028. Google Scholar [48] N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams,, Multiscale Model. Simul., 6 (2007), 688. doi: 10.1137/060673667. Google Scholar [49] L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193. Google Scholar [50] E. Wigner, On the quantum correction for thermodynamic equilibrium,, Phys. Rev., 40 (1932), 749. doi: 10.1103/PhysRev.40.749. Google Scholar [51] M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems,, J. Phys. A, 20 (1987), 2415. doi: 10.1088/0305-4470/20/9/028. Google Scholar [52] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces,, Duke Math. J., 55 (1987), 919. doi: 10.1215/S0012-7094-87-05546-3. Google Scholar

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##### References:
 [1] R. Alexandre, Oscillations in PDE with singularities of codimension one. Part I : review of the symbolic calculus and basic definitions,, preprint., (). Google Scholar [2] G. Ariel, B. Engquist, N. M. Tanushev and R. Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization,, J. Comput. Phys., 230 (2011), 2303. doi: 10.1016/j.jcp.2010.12.018. Google Scholar [3] V. M. Babič, Eigenfunctions concentrated in a neighborhood of a closed geodesic, (Russian),, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 9 (1968), 15. Google Scholar [4] S. Bougacha, J.-L. Akian and R. Alexandre, Gaussian beams summation for the wave equation in a convex domain,, Commun. Math. Sci., 7 (2009), 973. Google Scholar [5] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, (French) [Exact controllability of waves in nonsmooth domains],, Asympt. Anal., 14 (1997), 157. Google Scholar [6] N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures],, in, 1996/97 (1997), 167. Google Scholar [7] N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach,, Canad. Math. Bull., 48 (2005), 3. doi: 10.4153/CMB-2005-001-3. Google Scholar [8] N. Burq and G. Lebeau, Mesures de défaut de compacité, application au systéme de Lamé, (French) [Microlocal defect measures and application to the Lame system],, Ann. Sci. École Norm. Sup. (4), 34 (2001), 817. Google Scholar [9] F. Castella, The radiation condition at infinity for the high frequency Helmholtz equation with source term: a wave packet approach,, J. Funct. Anal., 223 (2005), 204. doi: 10.1016/j.jfa.2004.08.008. Google Scholar [10] V. Červený, M. M. Popov and I. Pšenčík, Computation of wave fields in inhomogeneous media-Gaussian beam approach,, Geophys. J. R. Astr. Soc., 70 (1982), 109. Google Scholar [11] J. Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French),, in, (1976), 165. Google Scholar [12] M. Combescure, J. Ralston and D. Robert, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition,, Commun. Math. Phys., 202 (1999), 463. doi: 10.1007/s002200050591. Google Scholar [13] T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, (). Google Scholar [14] S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics,, Multiscale Model. Simul., 1 (2003), 674. doi: 10.1137/S1540345902409797. Google Scholar [15] E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces,, J. Math. Pures Appl. (9), 87 (2007), 144. doi: 10.1016/j.matpur.2006.11.002. Google Scholar [16] I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit,, Asympt. Anal., 14 (1997), 97. Google Scholar [17] P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves],, in, (1991), 1990. Google Scholar [18] P. Gérard, Microlocal defect measures,, Commun. Partial Differential Equations, 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [19] P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem,, Duke Math. J., 71 (1993), 559. Google Scholar [20] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C. Google Scholar [21] L. Hörmander, "The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis," Grundlehren der Mathematischen Wissenschaften [Fundamental Principle of Mathematical Sciences], 256,, Springer-Verlag, (1983). Google Scholar [22] V. Ivrii, "Microlocal Analysis and Precise Spectral Asymptotics,", Springer Monographs in Mathematics, (1998). Google Scholar [23] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems",, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123 (2001). Google Scholar [24] A. P. Katchalov and M. M. Popov, The application of the Gaussian beam summation method to the computation of high-frequency wave fields,, Dokl. Akad. Nauk, 258 (1981), 1097. Google Scholar [25] L. Klimeš, Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams,, Geophys. J. R. astr. Soc., 79 (1984), 105. Google Scholar [26] A. Laptev and I. M. Sigal, Global Fourier integral operators and semiclassical asymptotics,, Rev. Math. Phys., 12 (2000), 749. Google Scholar [27] S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime,, J. Comput. Phys., 228 (2009), 2951. doi: 10.1016/j.jcp.2009.01.007. Google Scholar [28] S. Leung and J. Qian, The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation,, J. Comput. Phys., 229 (2010), 8888. doi: 10.1016/j.jcp.2010.08.015. Google Scholar [29] P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures],, Rev. Mat. Iberoamericana, 9 (1993), 553. Google Scholar [30] H. Liu and J. Ralston, Recovery of high frequency wave fields for the acoustic wave equation,, Multiscale Model. Simul., 8 (): 428. doi: 10.1137/090761598. Google Scholar [31] H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, (). Google Scholar [32] F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach,, Asymptot. Anal., 32 (2002), 1. Google Scholar [33] P. A. Markowich and N. J. Mauser, The classical limit of a self-consistent Quantum-Vlasov equation in $3$D,, Math. Models Methods Appl. Sci., 3 (1993), 109. doi: 10.1142/S0218202593000072. Google Scholar [34] P. A. Markowich, N. J. Mauser and F. Poupaud, A Wigner-function approach to (semi)classical limits: electrons in a periodic potential,, J. Math. Phys., 35 (1994), 1066. doi: 10.1063/1.530629. Google Scholar [35] P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations,, Pubbl. Ian, 1035 (1997), 1. Google Scholar [36] A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis,", Universitext, (2002). Google Scholar [37] L. Miller, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary,, J. Math. Pures Appl. (9), 79 (2000), 227. doi: 10.1016/S0021-7824(00)00158-6. Google Scholar [38] M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition,, Wave Motion, 47 (2010), 421. doi: 10.1016/j.wavemoti.2010.02.001. Google Scholar [39] A. N. Norris, Elastic Gaussian wave packets in isotropic media,, Acta Mech., 71 (1988), 95. doi: 10.1007/BF01173940. Google Scholar [40] G. Papanicolaou and L. Ryzhik, Waves and Transport,, in, 5 (1999), 305. Google Scholar [41] T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures,, Comm. Math. Phys., 175 (1996), 229. doi: 10.1007/BF02102407. Google Scholar [42] M. Pulvirenti, Semiclassical expansion of Wigner functions,, J. Math. Phys., 47 (2006). Google Scholar [43] J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation,, Multiscale Model. Simul., 8 (2010), 1803. doi: 10.1137/100787313. Google Scholar [44] J. Ralston, Gaussian beams and the propagation of singularities,, in, 23 (1982), 206. Google Scholar [45] S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions,, J. Math. Phys., 34 (1993), 2185. doi: 10.1063/1.530112. Google Scholar [46] N. M. Tanushev, Superpositions and higher order Gaussian beams,, Commun. Math. Sci., 6 (2008), 449. Google Scholar [47] N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields,, J. Comput. Phys., 228 (2009), 8856. doi: 10.1016/j.jcp.2009.08.028. Google Scholar [48] N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams,, Multiscale Model. Simul., 6 (2007), 688. doi: 10.1137/060673667. Google Scholar [49] L. Tartar, $H$-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 193. Google Scholar [50] E. Wigner, On the quantum correction for thermodynamic equilibrium,, Phys. Rev., 40 (1932), 749. doi: 10.1103/PhysRev.40.749. Google Scholar [51] M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems,, J. Phys. A, 20 (1987), 2415. doi: 10.1088/0305-4470/20/9/028. Google Scholar [52] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces,, Duke Math. J., 55 (1987), 919. doi: 10.1215/S0012-7094-87-05546-3. Google Scholar
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