A Gaussian beam approach for computing Wigner measures in convex domains

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

A Gaussian beam approach for computing Wigner measures in convex domains

Jean-Luc Akian ^1, , Radjesvarane Alexandre ^2, and Salma Bougacha ^3,

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

Abstract
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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.

Keywords: reflection., FBI transform, Wigner measures, Gaussian beam summation, Wave equation.

Mathematics Subject Classification: 35L05, 35L20, 81S3.

Citation: Jean-Luc Akian, Radjesvarane Alexandre, Salma Bougacha. A Gaussian beam approach for computing Wigner measures in convex domains. Kinetic and Related Models, 2011, 4 (3) : 589-631. doi: 10.3934/krm.2011.4.589

References:

[1]	R. Alexandre, Oscillations in PDE with singularities of codimension one. Part I : review of the symbolic calculus and basic definitions,, preprint., ().
[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), 2303-2321. doi: 10.1016/j.jcp.2010.12.018.
[3]	V. M. Babič, Eigenfunctions concentrated in a neighborhood of a closed geodesic, (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 9 (1968), 15-63 .
[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), 973-1008.
[5]	N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, (French) [Exact controllability of waves in nonsmooth domains], Asympt. Anal., 14 (1997), 157-191.
[6]	N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures], in "Séminaire Bourbaki," 1996/97, Astérisque, 245 (1997), 167-195.
[7]	N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, Canad. Math. Bull., 48 (2005), 3-15. doi: 10.4153/CMB-2005-001-3.
[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), 817-870.
[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), 204-257. doi: 10.1016/j.jfa.2004.08.008.
[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), 109-128.
[11]	J. Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French), in "Journées Équations aux Dérivées Partielles de Rennes," Astérisque, Soc. Math. France, (1976), 165-181.
[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), 463-480. doi: 10.1007/s002200050591.
[13]	T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, ().
[14]	S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics, Multiscale Model. Simul., 1 (2003), 674-710. doi: 10.1137/S1540345902409797.
[15]	E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192. doi: 10.1016/j.matpur.2006.11.002.
[16]	I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit, Asympt. Anal., 14 (1997), 97-116.
[17]	P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves], in "Séminaire sur les Équations aux Dérivéees Partielles," 1990-1991, École Polytech., Palaiseau, 1991.
[18]	P. Gérard, Microlocal defect measures, Commun. Partial Differential Equations, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.
[19]	P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71 (1993), 559-607.
[20]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.
[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, Berlin, 1983.
[22]	V. Ivrii, "Microlocal Analysis and Precise Spectral Asymptotics," Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998.
[23]	A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, 2001.
[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), 1097-1100.
[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), 105-118.
[26]	A. Laptev and I. M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., 12 (2000), 749-766.
[27]	S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228 (2009), 2951-2977. doi: 10.1016/j.jcp.2009.01.007.
[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), 8888-8917. doi: 10.1016/j.jcp.2010.08.015.
[29]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures], Rev. Mat. Iberoamericana, 9 (1993), 553-618.
[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.
[31]	H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, ().
[32]	F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32 (2002), 1-26.
[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), 109-124. doi: 10.1142/S0218202593000072.
[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), 1066-1094. doi: 10.1063/1.530629.
[35]	P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations, Pubbl. Ian, 1035 (1997), 1-57.
[36]	A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis," Universitext, Springer-Verlag, New York, 2002.
[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), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[38]	M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439. doi: 10.1016/j.wavemoti.2010.02.001.
[39]	A. N. Norris, Elastic Gaussian wave packets in isotropic media, Acta Mech., 71 (1988), 95-114. doi: 10.1007/BF01173940.
[40]	G. Papanicolaou and L. Ryzhik, Waves and Transport, in "Hyperbolic Equations and Frequency Interactions" (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, (1999), 305-382.
[41]	T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures, Comm. Math. Phys., 175 (1996), 229-258. doi: 10.1007/BF02102407.
[42]	M. Pulvirenti, Semiclassical expansion of Wigner functions, J. Math. Phys., 47 (2006), 12 pp.
[43]	J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul., 8 (2010), 1803-1837. doi: 10.1137/100787313.
[44]	J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in Partial Differential Equations," MAA Stud. Math., 23, Math. Assoc. America, (1982), 206-248.
[45]	S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., 34 (1993), 2185-2205. doi: 10.1063/1.530112.
[46]	N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475.
[47]	N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.
[48]	N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul., 6 (2007), 688-709. doi: 10.1137/060673667.
[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), 193-230.
[50]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. doi: 10.1103/PhysRev.40.749.
[51]	M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems, J. Phys. A, 20 (1987), 2415-2423. doi: 10.1088/0305-4470/20/9/028.
[52]	S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., 55 (1987), 919-941. doi: 10.1215/S0012-7094-87-05546-3.

show all references

References:

[1]	R. Alexandre, Oscillations in PDE with singularities of codimension one. Part I : review of the symbolic calculus and basic definitions,, preprint., ().
[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), 2303-2321. doi: 10.1016/j.jcp.2010.12.018.
[3]	V. M. Babič, Eigenfunctions concentrated in a neighborhood of a closed geodesic, (Russian), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 9 (1968), 15-63 .
[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), 973-1008.
[5]	N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, (French) [Exact controllability of waves in nonsmooth domains], Asympt. Anal., 14 (1997), 157-191.
[6]	N. Burq, Mesures semi-classiques et mesures de défaut, (French) [Semiclassical measures and defect measures], in "Séminaire Bourbaki," 1996/97, Astérisque, 245 (1997), 167-195.
[7]	N. Burq, Quantum ergodicity of boundary values of eigenfunctions: a control theory approach, Canad. Math. Bull., 48 (2005), 3-15. doi: 10.4153/CMB-2005-001-3.
[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), 817-870.
[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), 204-257. doi: 10.1016/j.jfa.2004.08.008.
[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), 109-128.
[11]	J. Chazarain, Paramétrix du problème mixte pour l'équation des ondes à l'intérieur d'un domaine convexe pour les bicaractéristiques, (French), in "Journées Équations aux Dérivées Partielles de Rennes," Astérisque, Soc. Math. France, (1976), 165-181.
[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), 463-480. doi: 10.1007/s002200050591.
[13]	T. Duyckaerts, Stabilization of the linear system of magnetoelasticity,, preprint, ().
[14]	S. Filippas and G. N. Makrakis, Semiclassical Wigner function and geometrical optics, Multiscale Model. Simul., 1 (2003), 674-710. doi: 10.1137/S1540345902409797.
[15]	E. Fouassier, High frequency limit of Helmholtz equations: refraction by sharp interfaces, J. Math. Pures Appl. (9), 87 (2007), 144-192. doi: 10.1016/j.matpur.2006.11.002.
[16]	I. Gasser and P. A. Markowich, Quantum hydrodynamics, Wigner transform and the classical limit, Asympt. Anal., 14 (1997), 97-116.
[17]	P. Gérard, Mesures semi-classiques et ondes de Bloch, (French) [Semiclassical measures and Bloch waves], in "Séminaire sur les Équations aux Dérivéees Partielles," 1990-1991, École Polytech., Palaiseau, 1991.
[18]	P. Gérard, Microlocal defect measures, Commun. Partial Differential Equations, 16 (1991), 1761-1794. doi: 10.1080/03605309108820822.
[19]	P. Gérard and E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J., 71 (1993), 559-607.
[20]	P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.
[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, Berlin, 1983.
[22]	V. Ivrii, "Microlocal Analysis and Precise Spectral Asymptotics," Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998.
[23]	A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems", Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, 2001.
[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), 1097-1100.
[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), 105-118.
[26]	A. Laptev and I. M. Sigal, Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., 12 (2000), 749-766.
[27]	S. Leung and J. Qian, Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime, J. Comput. Phys., 228 (2009), 2951-2977. doi: 10.1016/j.jcp.2009.01.007.
[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), 8888-8917. doi: 10.1016/j.jcp.2010.08.015.
[29]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, (French) [On Wigner measures], Rev. Mat. Iberoamericana, 9 (1993), 553-618.
[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.
[31]	H. Liu, O. Runborg and N. M. Tanushev, Error Estimates for Gaussian Beam Superpositions,, preprint, ().
[32]	F. Macià and E. Zuazua, On the lack of observability for wave equations: a Gaussian beam approach, Asymptot. Anal., 32 (2002), 1-26.
[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), 109-124. doi: 10.1142/S0218202593000072.
[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), 1066-1094. doi: 10.1063/1.530629.
[35]	P. A. Markowich, P. Pietra and C. Pohl., Weak limits of finite difference schemes of Schrödinger-type equations, Pubbl. Ian, 1035 (1997), 1-57.
[36]	A. Martinez, "An Introduction to Semiclassical and Microlocal Analysis," Universitext, Springer-Verlag, New York, 2002.
[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), 227-269. doi: 10.1016/S0021-7824(00)00158-6.
[38]	M. Motamed and O. Runborg, Taylor expansion and discretization errors in Gaussian beam superposition, Wave Motion, 47 (2010), 421-439. doi: 10.1016/j.wavemoti.2010.02.001.
[39]	A. N. Norris, Elastic Gaussian wave packets in isotropic media, Acta Mech., 71 (1988), 95-114. doi: 10.1007/BF01173940.
[40]	G. Papanicolaou and L. Ryzhik, Waves and Transport, in "Hyperbolic Equations and Frequency Interactions" (Park City, UT, 1995), IAS/Park City Math. Ser., 5, Amer. Math. Soc., Providence, RI, (1999), 305-382.
[41]	T. Paul and A. Uribe, On the pointwise behavior of semi-classical measures, Comm. Math. Phys., 175 (1996), 229-258. doi: 10.1007/BF02102407.
[42]	M. Pulvirenti, Semiclassical expansion of Wigner functions, J. Math. Phys., 47 (2006), 12 pp.
[43]	J. Qian and L. Ying, Fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beams for the wave equation, Multiscale Model. Simul., 8 (2010), 1803-1837. doi: 10.1137/100787313.
[44]	J. Ralston, Gaussian beams and the propagation of singularities, in "Studies in Partial Differential Equations," MAA Stud. Math., 23, Math. Assoc. America, (1982), 206-248.
[45]	S. L. Robinson, Semiclassical mechanics for time-dependent Wigner functions, J. Math. Phys., 34 (1993), 2185-2205. doi: 10.1063/1.530112.
[46]	N. M. Tanushev, Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6 (2008), 449-475.
[47]	N. M. Tanushev, B. Engquist and R. Tsai, Gaussian beam decomposition of high frequency wave fields, J. Comput. Phys., 228 (2009), 8856-8871. doi: 10.1016/j.jcp.2009.08.028.
[48]	N. M. Tanushev, J. Qian and J. V. Ralston, Mountain waves and Gaussian beams, Multiscale Model. Simul., 6 (2007), 688-709. doi: 10.1137/060673667.
[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), 193-230.
[50]	E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. doi: 10.1103/PhysRev.40.749.
[51]	M. Wilkinson, A semiclassical sum rule for matrix elements of classically chaotic systems, J. Phys. A, 20 (1987), 2415-2423. doi: 10.1088/0305-4470/20/9/028.
[52]	S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., 55 (1987), 919-941. doi: 10.1215/S0012-7094-87-05546-3.

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