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On local well-posedness and ill-posedness results for a coupled system of mkdv type equations
1. | Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ, Ilha do Fundão, 21945-970. Rio de Janeiro-RJ, Brazil |
2. | Gran Sasso Science Institute, CP 67100, L' Aquila, Italia |
3. | Universidade Federal do Rio de Janeiro, Campus Macaé/RJ, Brazil |
$ \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) = 0,&v(x,0) = \phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) = 0,& w(x,0) = \psi(x), \end{cases} \end{equation*} $ |
$ H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R}) $ |
$ s,k> -\frac12 $ |
$ |s-k|\leq 1/2 $ |
$ \alpha\neq 0,1 $ |
$ C^3 $ |
$ s<-1/2 $ |
$ k<-1/2 $ |
$ |s-k|>2 $ |
$ s-2k>1 $ |
$ k<-1/2 $ |
$ k-2s>1 $ |
$ s<-1/2 $ |
$ s = k = -1/2 $ |
References:
[1] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,
Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
[2] |
E. Alarcon, J. Angulo and J. F. Montenegro,
Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.
doi: 10.1016/S0362-546X(97)00724-4. |
[3] |
I. Bejenaru and T. Tao,
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259.
doi: 10.1016/j.jfa.2005.08.004. |
[4] |
D. Bekiranov, T. Ogawa and G. Ponce,
Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.
doi: 10.1090/S0002-9939-97-03941-5. |
[5] |
J. L. Bona, P. E. Souganidis and W. A. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type equation, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.
|
[6] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[7] |
X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), No. 13, 10 pp. |
[8] |
X. Carvajal,
Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.
doi: 10.1007/s00041-005-5028-3. |
[9] |
X. Carvajal and M. Panthee,
Sharp well-posedness for a coupled system of mKdV-type equations, J. Evol. Equ., 19 (2019), 1167-1197.
doi: 10.1007/s00028-019-00508-6. |
[10] |
M. Christ, J. Colliander and T. Tao,
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
[11] |
A. J. Corcho and M. Panthee,
Global well-posedness for a coupled modified KdV system, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 27-57.
doi: 10.1007/s00574-012-0004-4. |
[12] |
L. Domingues,
Sharp well-posedness results for the Schrödinger-Benjamin-Ono system, Adv. Differential Equations, 21 (2016), 31-54.
|
[13] |
L. Domingues and R. Santos, A note on $C^2$ ill-posedness results for the Zakharov system in arbitrary dimension, 2019, arXiv: 1910.06486. Google Scholar |
[14] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 1994-1995 (1996), 163-187.
|
[15] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[16] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[17] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[18] |
C. E. Kenig, G. Ponce and L. Vega,
A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[19] |
C. E. Kenig, G. Ponce and L. Vega,
On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.
doi: 10.1215/S0012-7094-01-10638-8. |
[20] |
J. F. Montenegro, Sistemas de Equações de Evolução não Lineares; Estudo Local, Global e Estabilidade de Ondas Solitárias, Ph.D. thesis, IMPA, Rio de Janeiro, Brazil, 1995. Google Scholar |
[21] |
H. Takaoka,
Well-posedness for the higher order nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 10 (2000), 149-171.
|
[22] |
T. Tao,
Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
[23] |
N. Tzvetkov,
Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 1043-1047.
doi: 10.1016/S0764-4442(00)88471-2. |
show all references
References:
[1] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur,
Nonlinear-evolution equations of physical significance, Phys. Rev. Lett., 31 (1973), 125-127.
doi: 10.1103/PhysRevLett.31.125. |
[2] |
E. Alarcon, J. Angulo and J. F. Montenegro,
Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035.
doi: 10.1016/S0362-546X(97)00724-4. |
[3] |
I. Bejenaru and T. Tao,
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259.
doi: 10.1016/j.jfa.2005.08.004. |
[4] |
D. Bekiranov, T. Ogawa and G. Ponce,
Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.
doi: 10.1090/S0002-9939-97-03941-5. |
[5] |
J. L. Bona, P. E. Souganidis and W. A. Strauss,
Stability and instability of solitary waves of Korteweg-de Vries type equation, Proc. Roy. Soc. London Ser. A, 411 (1987), 395-412.
|
[6] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[7] |
X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), No. 13, 10 pp. |
[8] |
X. Carvajal,
Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.
doi: 10.1007/s00041-005-5028-3. |
[9] |
X. Carvajal and M. Panthee,
Sharp well-posedness for a coupled system of mKdV-type equations, J. Evol. Equ., 19 (2019), 1167-1197.
doi: 10.1007/s00028-019-00508-6. |
[10] |
M. Christ, J. Colliander and T. Tao,
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
[11] |
A. J. Corcho and M. Panthee,
Global well-posedness for a coupled modified KdV system, Bull. Braz. Math. Soc. (N.S.), 43 (2012), 27-57.
doi: 10.1007/s00574-012-0004-4. |
[12] |
L. Domingues,
Sharp well-posedness results for the Schrödinger-Benjamin-Ono system, Adv. Differential Equations, 21 (2016), 31-54.
|
[13] |
L. Domingues and R. Santos, A note on $C^2$ ill-posedness results for the Zakharov system in arbitrary dimension, 2019, arXiv: 1910.06486. Google Scholar |
[14] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Séminaire Bourbaki, 1994-1995 (1996), 163-187.
|
[15] |
J. Ginibre, Y. Tsutsumi and G. Velo,
On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.
doi: 10.1006/jfan.1997.3148. |
[16] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[17] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[18] |
C. E. Kenig, G. Ponce and L. Vega,
A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[19] |
C. E. Kenig, G. Ponce and L. Vega,
On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.
doi: 10.1215/S0012-7094-01-10638-8. |
[20] |
J. F. Montenegro, Sistemas de Equações de Evolução não Lineares; Estudo Local, Global e Estabilidade de Ondas Solitárias, Ph.D. thesis, IMPA, Rio de Janeiro, Brazil, 1995. Google Scholar |
[21] |
H. Takaoka,
Well-posedness for the higher order nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 10 (2000), 149-171.
|
[22] |
T. Tao,
Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
doi: 10.1353/ajm.2001.0035. |
[23] |
N. Tzvetkov,
Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 1043-1047.
doi: 10.1016/S0764-4442(00)88471-2. |
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