June  2021, 41(6): 2971-2992. doi: 10.3934/dcds.2020393

On global well-posedness of the modified KdV equation in modulation spaces

1. 

School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

2. 

School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom

* Corresponding author: Tadahiro Oh

Received  January 2020 Revised  October 2020 Published  June 2021 Early access  January 2021

Fund Project: The first author was supported by the European Research Council (grant no. 637995 "ProbDynDispEq" and grant no. 864138 "SingStochDispDyn")

We study well-posedness of the complex-valued modified KdV equation (mKdV) on the real line. In particular, we prove local well-posedness of mKdV in modulation spaces $ M^{2,p}_{s}( \mathbb{R}) $ for $ s \ge \frac14 $ and $ 2\leq p < \infty $. For $ s < \frac 14 $, we show that the solution map for mKdV is not locally uniformly continuous in $ M^{2,p}_{s}( \mathbb{R}) $. By combining this local well-posedness with our previous work (2018) on an a priori global-in-time bound for mKdV in modulation spaces, we also establish global well-posedness of mKdV in $ M^{2,p}_{s}( \mathbb{R}) $ for $ s \ge \frac14 $ and $ 2\leq p < \infty $.

Citation: Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393
References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II, The KdV-equation. Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[2]

M. Chen and B. Guo, Local well and ill posedness for the modified KdV equations in subcritical modulation spaces, Commun. Math. Sci., 18 (2020), 909-946.  doi: 10.4310/CMS.2020.v18.n4.a2.

[3]

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.

[4]

M. ChristJ. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math. (N.S.), 32 (2012), 51-75.  doi: 10.14510/lm-ns.v32i1.32.

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[6]

H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Func. Anal., 86 (1989), 307-340.  doi: 10.1016/0022-1236(89)90055-4.

[7]

H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math., 108 (1989), 129-148.  doi: 10.1007/BF01308667.

[8]

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.

[9]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 2004 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.

[10]

A. Grünrock and L. Vega, Local well-posedness for the modified KdV equation in almost critical $ \widehat {H^r_s}$-spaces, Trans. Amer. Math. Soc., 361 (2009), 5681-5694.  doi: 10.1090/S0002-9947-09-04611-X.

[11]

S. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Anal. Appl., 23 (2017), 91-124.  doi: 10.1007/s00041-016-9464-z.

[12]

S. Guo, X. Ren and B. Wang, Local well-posedness for the derivative nonlinear Schrödinger equations with $L^2$ subcritical data,, arXiv: 1608.03136 [math.AP].

[13]

B. Harrop-Griffiths, R. Killip and M. Vişan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011 [math.AP].

[14]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.  doi: 10.1063/1.1666399.

[15]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[16]

T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.

[17]

C. E. KenigG. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.

[18]

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.

[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.

[20]

R. KillipM. Vişan and X. Zhang, Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., 28 (2018), 1062-1090.  doi: 10.1007/s00039-018-0444-0.

[21]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. 

[22]

S. Kwon, T. Oh and H. Yoon, Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line, Ann. Fac. Sci. Toulouse Math. 29 (2020), no. 3,649–720.

[23]

L. MolinetD. Pilod and S. Vento, Unconditional uniqueness for the modified Korteweg-de Vries equation on the line, Rev. Mat. Iberoam., 34 (2018), 1563-1608.  doi: 10.4171/rmi/1036.

[24]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.

[25]

T. Oh and Y. Wang, Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces,, to appear in J. Anal. Math.

[26]

N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear, J. Phys. Soc. Japan, 60 (1991), 409-417.  doi: 10.1143/JPSJ.60.409.

[27]

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.

show all references

References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II, The KdV-equation. Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[2]

M. Chen and B. Guo, Local well and ill posedness for the modified KdV equations in subcritical modulation spaces, Commun. Math. Sci., 18 (2020), 909-946.  doi: 10.4310/CMS.2020.v18.n4.a2.

[3]

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.

[4]

M. ChristJ. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math. (N.S.), 32 (2012), 51-75.  doi: 10.14510/lm-ns.v32i1.32.

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[6]

H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Func. Anal., 86 (1989), 307-340.  doi: 10.1016/0022-1236(89)90055-4.

[7]

H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math., 108 (1989), 129-148.  doi: 10.1007/BF01308667.

[8]

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.

[9]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 2004 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.

[10]

A. Grünrock and L. Vega, Local well-posedness for the modified KdV equation in almost critical $ \widehat {H^r_s}$-spaces, Trans. Amer. Math. Soc., 361 (2009), 5681-5694.  doi: 10.1090/S0002-9947-09-04611-X.

[11]

S. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Anal. Appl., 23 (2017), 91-124.  doi: 10.1007/s00041-016-9464-z.

[12]

S. Guo, X. Ren and B. Wang, Local well-posedness for the derivative nonlinear Schrödinger equations with $L^2$ subcritical data,, arXiv: 1608.03136 [math.AP].

[13]

B. Harrop-Griffiths, R. Killip and M. Vişan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011 [math.AP].

[14]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys., 14 (1973), 805-809.  doi: 10.1063/1.1666399.

[15]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[16]

T. Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.

[17]

C. E. KenigG. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.

[18]

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.

[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.

[20]

R. KillipM. Vişan and X. Zhang, Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., 28 (2018), 1062-1090.  doi: 10.1007/s00039-018-0444-0.

[21]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. 

[22]

S. Kwon, T. Oh and H. Yoon, Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line, Ann. Fac. Sci. Toulouse Math. 29 (2020), no. 3,649–720.

[23]

L. MolinetD. Pilod and S. Vento, Unconditional uniqueness for the modified Korteweg-de Vries equation on the line, Rev. Mat. Iberoam., 34 (2018), 1563-1608.  doi: 10.4171/rmi/1036.

[24]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.  doi: 10.1016/j.jde.2019.12.017.

[25]

T. Oh and Y. Wang, Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces,, to appear in J. Anal. Math.

[26]

N. Sasa and J. Satsuma, New-type of soliton solutions for a higher-order nonlinear, J. Phys. Soc. Japan, 60 (1991), 409-417.  doi: 10.1143/JPSJ.60.409.

[27]

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.

[1]

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

[2]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[3]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[4]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure and Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899

[5]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097

[6]

Xin Yang, Bing-Yu Zhang. Local well-posedness of the coupled KdV-KdV systems on $ \mathbb{R} $. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022002

[7]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078

[8]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[9]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure and Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737

[10]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[11]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087

[12]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[13]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036

[14]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

[15]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469

[16]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527

[17]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[18]

Márcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934/cpaa.2019117

[19]

Cezar Kondo, Ronaldo Pes. Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022063

[20]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (208)
  • HTML views (323)
  • Cited by (0)

Other articles
by authors

[Back to Top]