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Remarks on global solutions of dissipative wave equations with exponential nonlinear terms
Remarks on the full dispersion Davey-Stewartson systems
1. | Laboratoire de Mathématiques, Université Paris-Sud, Bât. 430, 91405 Orsay Cedex, France |
2. | UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, P.O. Box 91405, Orsay |
References:
[1] |
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511623998. |
[2] |
M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.
doi: 10.1017/S0022112079000835. |
[3] |
D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. |
[4] |
C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up, Mathematical Models and Methods in Applied Sciences, 8 (1998), 1363-1386.
doi: 10.1142/S0218202598000640. |
[5] |
T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson system for quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91. |
[6] |
W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), 615-667.
doi: 10.1016/S0294-1449(97)80128-X. |
[7] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522. |
[8] |
W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.
doi: 10.1006/jcph.1993.1164. |
[9] |
A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110. |
[10] |
V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714. |
[11] |
J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. |
[12] |
J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145.
doi: 10.1007/s003329900006. |
[13] |
J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations, in Nonlinear Dispersive Waves (L. Debnath Ed.), World Scientific, 1992, 83-97. |
[14] |
Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Analysis, 254 (2008), 1642-1660.
doi: 10.1016/j.jfa.2007.12.010. |
[15] |
Z. Guo and Y. Wang 2, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, arXiv:1007.4299v3.
doi: 10.1007/s11854-014-0025-6. |
[16] |
N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic- hyperbolic Davey-Stewartson system in the usual Sobolev space, Proc. Edinburgh Math. Soc., 40 (1997), 563-581.
doi: 10.1017/S0013091500024020. |
[17] |
N. Hayashi and H. Hirata, Global existence and scattering of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity, 9 (1996), 1387-1409.
doi: 10.1088/0951-7715/9/6/001. |
[18] |
A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, arXiv:1209.4943v1. |
[19] |
C. Klein, B. Muite and K. Roidot, Numerical study of the blow-up in the Davey-Stewartson system, Discr. Cont. Dyn. Syst. B, 18 (2013), 1361-1387.
doi: 10.3934/dcdsb.2013.18.1361. |
[20] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for Davey-Stewartson II type systems, submitted. |
[21] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, submitted. |
[22] |
Joseph Louis de Lagrange, Mémoire sur la théorie du mouvement des fluides, Oeuvres complètes, tome 4, 695-748. Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, 1781. |
[23] |
D. Lannes, Water Waves : Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.
doi: 10.1090/surv/188. |
[24] |
David Lannes, A stability criterion for two-dimensional interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567.
doi: 10.1007/s00205-012-0604-6. |
[25] |
D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinematics and Related Models, 6 (2013).
doi: 10.3934/krm.2013.6.989. |
[26] |
F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 523-548. |
[27] | |
[28] | |
[29] |
T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc. Roy. Soc. London A, 436 (1992), 345-349.
doi: 10.1098/rspa.1992.0022. |
[30] |
G. C. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves, Physica D, 72 (1994), 61-86.
doi: 10.1016/0167-2789(94)90167-8. |
[31] |
P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$, arXiv:1110.5589v2. |
[32] |
G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov- Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825.
doi: 10.3934/dcds.2005.13.811. |
[33] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139, New York, Berlin 1999. |
[34] |
L. Y. Sung, Long time decay of the solutions of the Davey-Stewartson II equations, J. Nonlinear Sci., 5 (1995), 43-452.
doi: 10.1007/BF01212909. |
[35] |
N. Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys., 335 (2015), 369-443.
doi: 10.1007/s00220-014-2259-7. |
[36] |
N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys., 310 (2012), 817-883.
doi: 10.1007/s00220-012-1422-2. |
[37] |
V. E. Zakharov, Weakly nonlinear waves on surface of ideal finite depth fluid, Amer. Math. Soc. Transl. Ser. 2, 182 (1998), 167-197. |
[38] |
V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., (1972), 84-98. |
[39] |
V. E. Zakharov and E. I Schulman, Degenerate conservation laws, motion invariants and kinetic equations, Physica, 1D (1980), 192-202.
doi: 10.1016/0167-2789(80)90011-1. |
show all references
References:
[1] |
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511623998. |
[2] |
M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech., 92 (1979), 691-715.
doi: 10.1017/S0022112079000835. |
[3] |
D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377-385. |
[4] |
C. Besse and C. H. Bruneau, Numerical study of elliptic-hyperbolic Davey-Stewartson system: dromions simulation and blow-up, Mathematical Models and Methods in Applied Sciences, 8 (1998), 1363-1386.
doi: 10.1142/S0218202598000640. |
[5] |
T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson system for quadratic hyperbolic systems, Asymptotic Analysis, 31 (2002), 69-91. |
[6] |
W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14 (1997), 615-667.
doi: 10.1016/S0294-1449(97)80128-X. |
[7] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522. |
[8] |
W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys., 108 (1993), 73-83.
doi: 10.1006/jcph.1993.1164. |
[9] |
A. Davey and K. Stewartson, One three-dimensional packets of water waves, Proc. Roy. Soc. Lond. A, 338 (1974), 101-110. |
[10] |
V. D. Djordjevic and L. G. Redekopp, On two-dimensional packets of capillary-gravity waves, J. Fluid Mech., 79 (1977), 703-714. |
[11] |
J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. |
[12] |
J.-M. Ghidaglia and J.-C. Saut, Non existence of traveling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci., 6 (1996), 139-145.
doi: 10.1007/s003329900006. |
[13] |
J.-M. Ghidaglia and J.-C. Saut, On the Zakharov-Schulman equations, in Nonlinear Dispersive Waves (L. Debnath Ed.), World Scientific, 1992, 83-97. |
[14] |
Z. Guo, L. Peng and B. Wang, Decay estimates for a class of wave equations, J. Funct. Analysis, 254 (2008), 1642-1660.
doi: 10.1016/j.jfa.2007.12.010. |
[15] |
Z. Guo and Y. Wang 2, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, arXiv:1007.4299v3.
doi: 10.1007/s11854-014-0025-6. |
[16] |
N. Hayashi and H. Hirata, Local existence in time of small solutions to the elliptic- hyperbolic Davey-Stewartson system in the usual Sobolev space, Proc. Edinburgh Math. Soc., 40 (1997), 563-581.
doi: 10.1017/S0013091500024020. |
[17] |
N. Hayashi and H. Hirata, Global existence and scattering of small solutions to the elliptic-hyperbolic Davey-Stewartson system, Nonlinearity, 9 (1996), 1387-1409.
doi: 10.1088/0951-7715/9/6/001. |
[18] |
A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, arXiv:1209.4943v1. |
[19] |
C. Klein, B. Muite and K. Roidot, Numerical study of the blow-up in the Davey-Stewartson system, Discr. Cont. Dyn. Syst. B, 18 (2013), 1361-1387.
doi: 10.3934/dcdsb.2013.18.1361. |
[20] |
C. Klein and J.-C. Saut, A numerical approach to blow-up issues for Davey-Stewartson II type systems, submitted. |
[21] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, submitted. |
[22] |
Joseph Louis de Lagrange, Mémoire sur la théorie du mouvement des fluides, Oeuvres complètes, tome 4, 695-748. Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Berlin, 1781. |
[23] |
D. Lannes, Water Waves : Mathematical Theory and Asymptotics, Mathematical Surveys and Monographs, vol 188 (2013), AMS, Providence.
doi: 10.1090/surv/188. |
[24] |
David Lannes, A stability criterion for two-dimensional interfaces and applications, Arch. Ration. Mech. Anal., 208 (2013), 481-567.
doi: 10.1007/s00205-012-0604-6. |
[25] |
D. Lannes and J.-C. Saut, Remarks on the full dispersion Kadomtsev-Petviashvli equation, Kinematics and Related Models, 6 (2013).
doi: 10.3934/krm.2013.6.989. |
[26] |
F. Linares and G. Ponce, On the Davey-Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 523-548. |
[27] | |
[28] | |
[29] |
T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc. Roy. Soc. London A, 436 (1992), 345-349.
doi: 10.1098/rspa.1992.0022. |
[30] |
G. C. Papanicolaou, C. Sulem, P.-L. Sulem and X. P. Wang, The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves, Physica D, 72 (1994), 61-86.
doi: 10.1016/0167-2789(94)90167-8. |
[31] |
P. A. Perry, Global well-posedness and long time asymptotics for the defocussing Davey-Stewartson II equation in $H^{1,1}(\R^2)$, arXiv:1110.5589v2. |
[32] |
G. Ponce and J.-C. Saut, Well-posedness for the Benney-Roskes-Zakharov- Rubenchik system, Discrete Cont. Dynamical Systems, 13 (2005), 811-825.
doi: 10.3934/dcds.2005.13.811. |
[33] |
C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Applied Mathematical Sciences 139, New York, Berlin 1999. |
[34] |
L. Y. Sung, Long time decay of the solutions of the Davey-Stewartson II equations, J. Nonlinear Sci., 5 (1995), 43-452.
doi: 10.1007/BF01212909. |
[35] |
N. Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys., 335 (2015), 369-443.
doi: 10.1007/s00220-014-2259-7. |
[36] |
N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys., 310 (2012), 817-883.
doi: 10.1007/s00220-012-1422-2. |
[37] |
V. E. Zakharov, Weakly nonlinear waves on surface of ideal finite depth fluid, Amer. Math. Soc. Transl. Ser. 2, 182 (1998), 167-197. |
[38] |
V. E. Zakharov and A. M. Rubenchik, Nonlinear interaction of high-frequency and low frequency waves, Prikl. Mat. Techn. Phys., (1972), 84-98. |
[39] |
V. E. Zakharov and E. I Schulman, Degenerate conservation laws, motion invariants and kinetic equations, Physica, 1D (1980), 192-202.
doi: 10.1016/0167-2789(80)90011-1. |
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