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doi: 10.3934/dcds.2021022

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Andreia Chapouto ,

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

* Corresponding author: Andreia Chapouto

Received  June 2020 Revised  December 2020 Published  January 2021

Fund Project: The author is supported by the Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council [grant EP/L016508/01], the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh; and Tadahiro Oh's ERC Starting Grant (grant no. 637995 ProbDynDispEq)

We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This non-existence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside $ H^\frac12(\mathbb{T}) $. Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.

Keywords: Modified Korteweg-de Vries equation, local well-posedness, global well-posedness, Fourier-Lebesgue spaces, momentum.
Mathematics Subject Classification: 35Q53.
Citation: Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021022
References:
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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]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.   Google Scholar

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J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.), 3 (1997), 115-159.  doi: 10.1007/s000290050008.  Google Scholar

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N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett., 9 (2002), 323-335.  doi: 10.4310/MRL.2002.v9.n3.a8.  Google Scholar

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A. Chapouto, A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces, arXiv: 2006.15671. Google Scholar

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M. Christ, Power series solution of a nonlinear Schr¨odinger equation, Mathematical Aspects of Nonlinear Dispersive Equations, 131-155, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.  Google Scholar

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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

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

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A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

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A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

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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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Z. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656-1729. doi: 10.1093/imrn/rnw271.  Google Scholar

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S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., (2006), Article ID 96763, 33 pp. doi: 10.1155/IMRN/2006/96763.  Google Scholar

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A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.  doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

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T. Kappeler and J.-C. Molnar, On the well-posedness of the defocusing mKdV equation below $L^2$, SIAM J. Math. Anal., 49 (2017), 2191-2219.  doi: 10.1137/16M1096979.  Google Scholar

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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

[21]

T. Kappeler and P. Topalov, Global well-posedness of mKdV in $L^2(\Bbb T, \Bbb R)$, Comm. Partial Differential Equations, 30 (2005), 435-449.  doi: 10.1081/PDE-200050089.  Google Scholar

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R. Killip, M. 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

[23]

N. Kishimoto and Y. Tsutsumi, Ill-Posedness of the third order NLS equation with Raman scattering term, Math. Res. Lett., 25 (2018), 1447-1484.  doi: 10.4310/MRL.2018.v25.n5.a5.  Google Scholar

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S. Kwon and T. Oh, On unconditional well-posedness of modified KdV, Int. Math. Res. Not. IMRN, (2012), 3509-3534. doi: 10.1093/imrn/rnr156.  Google Scholar

[25]

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), 649-720.  doi: 10.5802/afst.1643.  Google Scholar

[26]

L. Molinet, Sharp ill-posedness results for KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930.  doi: 10.1016/j.aim.2012.03.026.  Google Scholar

[27]

L. Molinet, D. Pilod and S. Vento, On unconditional well-posedness for the periodic modified Korteweg-de Vries equation, J. Math. Soc. Japan, 71 (2019), 147-201.  doi: 10.2969/jmsj/76977697.  Google Scholar

[28]

J.-C. Mourrat, H. Weber and W. Xu, Construction of $\Phi^4_3$ diagrams for pedestrians, From Particle Systems to Partial Differential Equations, Springer Proc. Math. Stat., Springer, Cham, 209 (2017), 1-46. doi: 10.1007/978-3-319-66839-0_1.  Google Scholar

[29]

K. Nakanishi, H. Takaoka and Y. Tsutsumi, Local well-posedness in low regularity of the mKdV equation with periodic boundary condition, Discrete Contin. Dyn. Syst., 28 (2010), 1635-1654.  doi: 10.3934/dcds.2010.28.1635.  Google Scholar

[30]

T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, (2008), No. 71, 10 pp.  Google Scholar

[31]

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

[32]

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

[33]

R. Schippa, On the existence of periodic solutions to the modified Korteweg-de Vries equation below $H^{1/2}(\Bbb T)$, J. Evol. Equ., 20 (2020), 725-776.  doi: 10.1007/s00028-019-00538-0.  Google Scholar

[34]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

[35]

H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition, Int. Math. Res. Not., (2004), 3009-3040. doi: 10.1155/S1073792804140555.  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]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.   Google Scholar

[3]

J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. (N.S.), 3 (1997), 115-159.  doi: 10.1007/s000290050008.  Google Scholar

[4]

N. Burq, P. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett., 9 (2002), 323-335.  doi: 10.4310/MRL.2002.v9.n3.a8.  Google Scholar

[5]

A. Chapouto, A refined well-posedness result for the modified KdV equation in the Fourier-Lebesgue spaces, arXiv: 2006.15671. Google Scholar

[6]

M. Christ, Power series solution of a nonlinear Schr¨odinger equation, Mathematical Aspects of Nonlinear Dispersive Equations, 131-155, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.  Google Scholar

[7]

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

[8]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[9]

A. de Bouard and A. Debussche, The Korteweg-de Vries equation with multiplicative homogeneous white noise, Stochastic Differential Equations: Theory and Applications, 113-133, Interdiscip. Math. Sci., 2, World Sci. Publ., Hackensack, NJ, 2007. doi: 10.1142/9789812770639_0004.  Google Scholar

[10]

Y. Deng, A. R. Nahmod and H. Yue, Optimal local well-posedness for the periodic derivative nonlinear Schrödinger equation, Commun. Math. Phys., (2020). doi: 10.1007/s00220-020-03898-8.  Google Scholar

[11]

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

[12]

A. Grünrock, An improved local well-posedness result for the modified KdV equation, Int. Math. Res. Not., 61 (2004), 3287-3308.  doi: 10.1155/S1073792804140981.  Google Scholar

[13]

A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

[14]

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

[15]

Z. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656-1729. doi: 10.1093/imrn/rnw271.  Google Scholar

[16]

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

[17]

S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., (2006), Article ID 96763, 33 pp. doi: 10.1155/IMRN/2006/96763.  Google Scholar

[18]

A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.  doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar

[19]

T. Kappeler and J.-C. Molnar, On the well-posedness of the defocusing mKdV equation below $L^2$, SIAM J. Math. Anal., 49 (2017), 2191-2219.  doi: 10.1137/16M1096979.  Google Scholar

[20]

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

[21]

T. Kappeler and P. Topalov, Global well-posedness of mKdV in $L^2(\Bbb T, \Bbb R)$, Comm. Partial Differential Equations, 30 (2005), 435-449.  doi: 10.1081/PDE-200050089.  Google Scholar

[22]

R. Killip, M. 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

[23]

N. Kishimoto and Y. Tsutsumi, Ill-Posedness of the third order NLS equation with Raman scattering term, Math. Res. Lett., 25 (2018), 1447-1484.  doi: 10.4310/MRL.2018.v25.n5.a5.  Google Scholar

[24]

S. Kwon and T. Oh, On unconditional well-posedness of modified KdV, Int. Math. Res. Not. IMRN, (2012), 3509-3534. doi: 10.1093/imrn/rnr156.  Google Scholar

[25]

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), 649-720.  doi: 10.5802/afst.1643.  Google Scholar

[26]

L. Molinet, Sharp ill-posedness results for KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930.  doi: 10.1016/j.aim.2012.03.026.  Google Scholar

[27]

L. Molinet, D. Pilod and S. Vento, On unconditional well-posedness for the periodic modified Korteweg-de Vries equation, J. Math. Soc. Japan, 71 (2019), 147-201.  doi: 10.2969/jmsj/76977697.  Google Scholar

[28]

J.-C. Mourrat, H. Weber and W. Xu, Construction of $\Phi^4_3$ diagrams for pedestrians, From Particle Systems to Partial Differential Equations, Springer Proc. Math. Stat., Springer, Cham, 209 (2017), 1-46. doi: 10.1007/978-3-319-66839-0_1.  Google Scholar

[29]

K. Nakanishi, H. Takaoka and Y. Tsutsumi, Local well-posedness in low regularity of the mKdV equation with periodic boundary condition, Discrete Contin. Dyn. Syst., 28 (2010), 1635-1654.  doi: 10.3934/dcds.2010.28.1635.  Google Scholar

[30]

T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, (2008), No. 71, 10 pp.  Google Scholar

[31]

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

[32]

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

[33]

R. Schippa, On the existence of periodic solutions to the modified Korteweg-de Vries equation below $H^{1/2}(\Bbb T)$, J. Evol. Equ., 20 (2020), 725-776.  doi: 10.1007/s00028-019-00538-0.  Google Scholar

[34]

T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106.  Google Scholar

[35]

H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition, Int. Math. Res. Not., (2004), 3009-3040. doi: 10.1155/S1073792804140555.  Google Scholar

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