Asymptotics of wave models for non star-shaped geometries

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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

Abstract
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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 and 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, 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.

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