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

Asymptotics of wave models for non star-shaped geometries

Farah Abou Shakra 1,

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.
Keywords: scattering, illuminated from interior, illuminated from exterior, Wave and Schrödinger equations, almost star-shaped..
Mathematics Subject Classification: 35L70, 35L20, 35B33, 35Q5.
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, ().   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 467-489. doi: 10.3934/cpaa.2008.7.467.  Google Scholar

[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), 987-1014. doi: 10.1002/cpa.20029.  Google Scholar

[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), 767-865. doi: 10.4007/annals.2008.167.767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.  Google Scholar

[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.  Google Scholar

[19]

O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains, arXiv:0812.2733, (2009). Google Scholar

[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.  Google Scholar

[21]

L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), 211-223.  Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arXiv:1208.4904, (2012). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math., 15 (1962), 349-362. doi: 10.1002/cpa.3160150303.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290.  Google Scholar

[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.  Google Scholar

[31]

J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math., 138 (1993), 503-518. doi: 10.2307/2946554.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dynamics of PDE, 3 (2006), 93-110.  Google Scholar

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, ().   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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), 467-489. doi: 10.3934/cpaa.2008.7.467.  Google Scholar

[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), 987-1014. doi: 10.1002/cpa.20029.  Google Scholar

[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), 767-865. doi: 10.4007/annals.2008.167.767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Anal. PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.  Google Scholar

[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.  Google Scholar

[19]

O. Ivanovici and F. Planchon, Square functions and heat flow estimates on domains, arXiv:0812.2733, (2009). Google Scholar

[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.  Google Scholar

[21]

L. V. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), 211-223.  Google Scholar

[22]

R. Killip, M. Visan and X. Zhang, Quintic NLS in the exterior of a strictly convex obstacle, arXiv:1208.4904, (2012). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

C. S. Morawetz, The limiting amplitude principle, Comm. Pure Appl. Math., 15 (1962), 349-362. doi: 10.1002/cpa.3160150303.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290.  Google Scholar

[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.  Google Scholar

[31]

J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math., 138 (1993), 503-518. doi: 10.2307/2946554.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dynamics of PDE, 3 (2006), 93-110.  Google Scholar

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