April  2014, 7(2): 347-362. doi: 10.3934/dcdss.2014.7.347

Asymptotics of wave models for non star-shaped geometries

1. 

Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS, UMR 7539, F-93430, Villetaneuse, France

Received  April 2013 Revised  May 2013 Published  September 2013

In this paper, we provide a detailed study and interpretation of various non star-shaped geometries linking them to recent results for the 3D critical wave equation and the 2D Schrödinger equation. These geometries date back to the 1960's and 1970's and they were previously studied only in the setting of the linear wave equation.
Citation: Farah Abou Shakra. Asymptotics of wave models for non star-shaped geometries. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 347-362. doi: 10.3934/dcdss.2014.7.347
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, ().

[2]

F. Abou Shakra, On 2D NLS on non-trapping exterior domains,, preprint, ().

[3]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, Amer. J. Math., 121 (1999), 131. 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. 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. 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. 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.

[8]

C. O. Bloom and N. D. Kazarinoff, "Short wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions,", Lecture Notes in Mathematics, 522 (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. 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. 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 $\mathbbR$,, Commun. Pure Appl. Anal., 7 (2008), 467. 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 $\mathbbR^3$,, Comm. Pure Appl. Math., 57 (2004), 987. 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 $\mathbbR^3$,, Ann. of Math. (2), 167 (2008), 767. 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.

[15]

M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity,, Ann. of Math., 132 (1990), 485. doi: 10.2307/1971427.

[16]

M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity,, Comm. Pure App. Math., 45 (1992), 749. doi: 10.1002/cpa.3160450604.

[17]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles,, Anal. PDE, 3 (2010), 261. 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. doi: 10.1016/j.anihpc.2010.04.001.

[19]

O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains,, , (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.

[21]

L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations,, Math. Res. Lett., 1 (1994), 211.

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle,, , (2012).

[23]

De-Fu Liu, Local energy decay for hyperbolic systems in exterior domains,, J. Math. Anal. Appl., 128 (1987), 312. 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. 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. doi: 10.1002/cpa.3160280204.

[26]

C. S. Morawetz, The limiting amplitude principle,, Comm. Pure Appl. Math., 15 (1962), 349. 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. 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. 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.

[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.

[31]

J. Shatah and M. Struwe, Regularity results for nonlinear wave equations,, Ann. of Math., 138 (1993), 503. 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. 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. 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. 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.

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, ().

[2]

F. Abou Shakra, On 2D NLS on non-trapping exterior domains,, preprint, ().

[3]

H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations,, Amer. J. Math., 121 (1999), 131. 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. 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. 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. 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.

[8]

C. O. Bloom and N. D. Kazarinoff, "Short wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions,", Lecture Notes in Mathematics, 522 (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. 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. 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 $\mathbbR$,, Commun. Pure Appl. Anal., 7 (2008), 467. 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 $\mathbbR^3$,, Comm. Pure Appl. Math., 57 (2004), 987. 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 $\mathbbR^3$,, Ann. of Math. (2), 167 (2008), 767. 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.

[15]

M. G. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity,, Ann. of Math., 132 (1990), 485. doi: 10.2307/1971427.

[16]

M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity,, Comm. Pure App. Math., 45 (1992), 749. doi: 10.1002/cpa.3160450604.

[17]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles,, Anal. PDE, 3 (2010), 261. 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. doi: 10.1016/j.anihpc.2010.04.001.

[19]

O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains,, , (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.

[21]

L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations,, Math. Res. Lett., 1 (1994), 211.

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle,, , (2012).

[23]

De-Fu Liu, Local energy decay for hyperbolic systems in exterior domains,, J. Math. Anal. Appl., 128 (1987), 312. 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. 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. doi: 10.1002/cpa.3160280204.

[26]

C. S. Morawetz, The limiting amplitude principle,, Comm. Pure Appl. Math., 15 (1962), 349. 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. 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. 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.

[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.

[31]

J. Shatah and M. Struwe, Regularity results for nonlinear wave equations,, Ann. of Math., 138 (1993), 503. 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. 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. 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. 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.

[1]

Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803

[4]

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

[5]

Soichiro Katayama, Hideo Kubo, Sandra Lucente. Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2331-2360. doi: 10.3934/cpaa.2013.12.2331

[6]

Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631

[7]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129

[8]

Amin Boumenir, Vu Kim Tuan. Reconstruction of the coefficients of a star graph from observations of its vertices. Inverse Problems & Imaging, 2018, 12 (6) : 1293-1308. doi: 10.3934/ipi.2018054

[9]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507

[10]

Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113

[11]

Nurlan Dairbekov, Gunther Uhlmann. Reconstructing the metric and magnetic field from the scattering relation. Inverse Problems & Imaging, 2010, 4 (3) : 397-409. doi: 10.3934/ipi.2010.4.397

[12]

Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759

[13]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825

[14]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

[15]

Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139

[16]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[17]

Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089

[18]

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

[19]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[20]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]