- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Numerical analysis of a non equilibrium two-component two-compressible flow in porous media
Asymptotics of wave models for non star-shaped geometries
1. | Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France |
References:
[1] |
F. Abou Shakra, Asymptotics of the critical non-linear wave equation for a class of non star-shaped obstacles, to appear in JHDE, arXiv:1206.0272. |
[2] |
F. Abou Shakra, On 2D NLS on non-trapping exterior domains, preprint, arXiv:1304.7628. |
[3] |
H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[4] |
H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. Henri Poinaré, 15 (1998), 783-789.
doi: 10.1016/S0294-1449(99)80005-5. |
[5] |
M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430.
doi: 10.1007/s00208-011-0772-y. |
[6] |
M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Annales de l'Institut Henri Poincare, 26 (2009), 1817-1829.
doi: 10.1016/j.anihpc.2008.12.004. |
[7] |
C. O. Bloom and N. D. Kazarinoff, Local energy decay for a class of nonstar-shaped bodies, Arch. Rat. Mech. Anal., 55 (1974), 73-85. |
[8] |
C. O. Bloom and N. D. Kazarinoff, "Short wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions," Lecture Notes in Mathematics, 522, Springer-Verlag, Berlin-New York, 1976. |
[9] |
N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical wave in 3-D domains, J. Amer. Math. Soc., 21 (2008), 831-845.
doi: 10.1090/S0894-0347-08-00596-1. |
[10] |
J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 62 (2009), 920-968.
doi: 10.1002/cpa.20278. |
[11] |
J. Colliander, J. Holmer, M. Visan and X. Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$, Commun. Pure Appl. Anal., 7 (2008), 467-489.
doi: 10.3934/cpaa.2008.7.467. |
[12] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation in $\mathbb{R}^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[13] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[14] |
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 64 (1985), 363-401. |
[15] |
M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.
doi: 10.2307/1971427. |
[16] |
M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure App. Math., 45 (1992), 749-774.
doi: 10.1002/cpa.3160450604. |
[17] |
O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293.
doi: 10.2140/apde.2010.3.261. |
[18] |
O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177.
doi: 10.1016/j.anihpc.2010.04.001. |
[19] |
O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains, arXiv:0812.2733, (2009). |
[20] |
V. Ja. Ivrii, Exponential decay of the solution of the wave equation outside an almost star-shaped region, (Russian) Dokl. Akad. Nauk SSSR, 189 (1969), 938-940. |
[21] |
L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), 211-223. |
[22] |
R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arXiv:1208.4904, (2012). |
[23] |
De-Fu Liu, Local energy decay for hyperbolic systems in exterior domains, J. Math. Anal. Appl., 128 (1987), 312-331.
doi: 10.1016/0022-247X(87)90185-5. |
[24] |
C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568.
doi: 10.1002/cpa.3160140327. |
[25] |
C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.
doi: 10.1002/cpa.3160280204. |
[26] |
C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math., 15 (1962), 349-362.
doi: 10.1002/cpa.3160150303. |
[27] |
C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math., 30 (1977), 447-508.
doi: 10.1002/cpa.3160300405. |
[28] |
K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spacial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[29] |
F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290. |
[30] |
F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897. |
[31] |
J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math., 138 (1993), 503-518.
doi: 10.2307/2946554. |
[32] |
J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, 7 (1994), 303-309.
doi: 10.1155/S1073792894000346. |
[33] |
H. F. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., 8 (1995), 879-916.
doi: 10.1090/S0894-0347-1995-1308407-1. |
[34] |
W. A. Strauss, Dispersal of waves vanishing on the boundary of an exterior domain, Comm. Pure Appl. Math., 28 (1975), 265-278.
doi: 10.1002/cpa.3160280205. |
[35] |
T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dynamics of PDE, 3 (2006), 93-110. |
show all references
References:
[1] |
F. Abou Shakra, Asymptotics of the critical non-linear wave equation for a class of non star-shaped obstacles, to appear in JHDE, arXiv:1206.0272. |
[2] |
F. Abou Shakra, On 2D NLS on non-trapping exterior domains, preprint, arXiv:1304.7628. |
[3] |
H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.
doi: 10.1353/ajm.1999.0001. |
[4] |
H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. Henri Poinaré, 15 (1998), 783-789.
doi: 10.1016/S0294-1449(99)80005-5. |
[5] |
M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430.
doi: 10.1007/s00208-011-0772-y. |
[6] |
M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Annales de l'Institut Henri Poincare, 26 (2009), 1817-1829.
doi: 10.1016/j.anihpc.2008.12.004. |
[7] |
C. O. Bloom and N. D. Kazarinoff, Local energy decay for a class of nonstar-shaped bodies, Arch. Rat. Mech. Anal., 55 (1974), 73-85. |
[8] |
C. O. Bloom and N. D. Kazarinoff, "Short wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions," Lecture Notes in Mathematics, 522, Springer-Verlag, Berlin-New York, 1976. |
[9] |
N. Burq, G. Lebeau and F. Planchon, Global existence for energy critical wave in 3-D domains, J. Amer. Math. Soc., 21 (2008), 831-845.
doi: 10.1090/S0894-0347-08-00596-1. |
[10] |
J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 62 (2009), 920-968.
doi: 10.1002/cpa.20278. |
[11] |
J. Colliander, J. Holmer, M. Visan and X. Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\mathbb{R}$, Commun. Pure Appl. Anal., 7 (2008), 467-489.
doi: 10.3934/cpaa.2008.7.467. |
[12] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation in $\mathbb{R}^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[13] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^3$, Ann. of Math. (2), 167 (2008), 767-865.
doi: 10.4007/annals.2008.167.767. |
[14] |
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 64 (1985), 363-401. |
[15] |
M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., 132 (1990), 485-509.
doi: 10.2307/1971427. |
[16] |
M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure App. Math., 45 (1992), 749-774.
doi: 10.1002/cpa.3160450604. |
[17] |
O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293.
doi: 10.2140/apde.2010.3.261. |
[18] |
O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in 3D non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177.
doi: 10.1016/j.anihpc.2010.04.001. |
[19] |
O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains, arXiv:0812.2733, (2009). |
[20] |
V. Ja. Ivrii, Exponential decay of the solution of the wave equation outside an almost star-shaped region, (Russian) Dokl. Akad. Nauk SSSR, 189 (1969), 938-940. |
[21] |
L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), 211-223. |
[22] |
R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arXiv:1208.4904, (2012). |
[23] |
De-Fu Liu, Local energy decay for hyperbolic systems in exterior domains, J. Math. Anal. Appl., 128 (1987), 312-331.
doi: 10.1016/0022-247X(87)90185-5. |
[24] |
C. S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math., 14 (1961), 561-568.
doi: 10.1002/cpa.3160140327. |
[25] |
C. S. Morawetz, Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264.
doi: 10.1002/cpa.3160280204. |
[26] |
C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math., 15 (1962), 349-362.
doi: 10.1002/cpa.3160150303. |
[27] |
C. S. Morawetz, J. V. Ralston and W. A. Strauss, Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math., 30 (1977), 447-508.
doi: 10.1002/cpa.3160300405. |
[28] |
K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spacial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225.
doi: 10.1006/jfan.1999.3503. |
[29] |
F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290. |
[30] |
F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897. |
[31] |
J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math., 138 (1993), 503-518.
doi: 10.2307/2946554. |
[32] |
J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices, 7 (1994), 303-309.
doi: 10.1155/S1073792894000346. |
[33] |
H. F. Smith and C. D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc., 8 (1995), 879-916.
doi: 10.1090/S0894-0347-1995-1308407-1. |
[34] |
W. A. Strauss, Dispersal of waves vanishing on the boundary of an exterior domain, Comm. Pure Appl. Math., 28 (1975), 265-278.
doi: 10.1002/cpa.3160280205. |
[35] |
T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dynamics of PDE, 3 (2006), 93-110. |
[1] |
Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589 |
[2] |
Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919 |
[3] |
Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803 |
[4] |
Ahmed Bchatnia, Amina Boukhatem. Stability of a damped wave equation on an infinite star-shaped network. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022024 |
[5] |
Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070 |
[6] |
Deyue Zhang, Yue Wu, Yinglin Wang, Yukun Guo. A direct imaging method for the exterior and interior inverse scattering problems. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022025 |
[7] |
Soichiro Katayama, Hideo Kubo, Sandra Lucente. Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2331-2360. doi: 10.3934/cpaa.2013.12.2331 |
[8] |
Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631 |
[9] |
Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286 |
[10] |
Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129 |
[11] |
Amin Boumenir, Vu Kim Tuan. Reconstruction of the coefficients of a star graph from observations of its vertices. Inverse Problems and Imaging, 2018, 12 (6) : 1293-1308. doi: 10.3934/ipi.2018054 |
[12] |
Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507 |
[13] |
Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113 |
[14] |
Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems and Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397 |
[15] |
Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Controllability for Schrödinger type system with mixed dispersion on compact star graphs. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022019 |
[16] |
Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759 |
[17] |
P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029 |
[18] |
Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825 |
[19] |
Takahiro Hashimoto. Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains. Conference Publications, 2007, 2007 (Special) : 487-494. doi: 10.3934/proc.2007.2007.487 |
[20] |
Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]