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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, (). 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.
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). 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.
|
[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). Google Scholar |
[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, (). 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.
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). 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.
|
[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). Google Scholar |
[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.
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