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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

 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.

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:
 [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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##### 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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