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 & 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.  Google Scholar

[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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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]. Google Scholar

[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]. Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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]. Google Scholar

[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]. Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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