• Previous Article
    Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families
  • DCDS Home
  • This Issue
  • Next Article
    Čech cohomology, homoclinic trajectories and robustness of non-saddle sets
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

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  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 $
;
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. AblowitzD. J. KaupA. 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. AlarconJ. 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. BekiranovT. 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. BonaP. 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. ChristJ. 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. GinibreY. 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. GrillakisJ. 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. KenigG. 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. KenigG. 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. KenigG. 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. AblowitzD. J. KaupA. 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. AlarconJ. 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. BekiranovT. 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. BonaP. 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. ChristJ. 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. GinibreY. 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. GrillakisJ. 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. KenigG. 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. KenigG. 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. KenigG. 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
[1]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

[2]

Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022

[3]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[4]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001

[5]

Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021057

[6]

Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227

[7]

Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks & Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005

[8]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018

[9]

Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448

[11]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[12]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056

[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384

[14]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[15]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014

[16]

Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $ O(n) $ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082

[17]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

[18]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[19]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[20]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (74)
  • HTML views (169)
  • Cited by (0)

[Back to Top]