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June  2021, 41(6): 2699-2723. doi: 10.3934/dcds.2020382

On local well-posedness and ill-posedness results for a coupled system of mkdv type equations

Xavier Carvajal 1, , Liliana Esquivel 2, and  Raphael Santos 3,

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

Received  March 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations
$ \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*} $
and prove the local well-posedness results for a given data in low regularity Sobolev spaces
$ H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R}) $
,
$ s,k> -\frac12 $
 and
$ |s-k|\leq 1/2 $
, for
$ \alpha\neq 0,1 $
. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be
$ C^3 $
 at the origin, when
$ s<-1/2 $
 or
$ k<-1/2 $
 or
$ |s-k|>2 $
; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a)
$ s-2k>1 $
 or
$ k<-1/2 $
 (b)
$ k-2s>1 $
 or
$ s<-1/2 $
; (c)
$ s = k = -1/2 $
;
Keywords: Modified Korteweg-de Vries equation, Cauchy problem, local well-posedness, ill-posedness.
Mathematics Subject Classification: 35A01, 35M31, 35Q53.
Citation: Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Domingues, Sharp well-posedness results for the Schrödinger-Benjamin-Ono system, Adv. Differential Equations, 21 (2016), 31-54.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. Domingues, Sharp well-posedness results for the Schrödinger-Benjamin-Ono system, Adv. Differential Equations, 21 (2016), 31-54.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Theorem 1.4
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