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On the existence of periodic solutions to the modified Korteweg–de Vries equation below \(H^{1/2}({\mathbb {T}})\)

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Abstract

Existence and a priori estimates for real-valued periodic solutions to the modified Korteweg–de Vries equation with initial data in \(H^s\) are established for \(s>0\). The short-time Fourier restriction norm method is employed to overcome the derivative loss. Further, non-existence of solutions below \(L^2\) is proved conditional upon conjectured linear Strichartz estimates.

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Notes

  1. For the actually more involved energy estimate, see Sect. 6.

  2. Strictly speaking, we had to consider \(f_{m,k_i}\) or \(f_{m,k_i,j_i}\), respectively, tracking the additional dependence on m. Since all the estimates below are uniform in m, we choose to drop dependence on m for the sake of brevity.

References

  1. Abdelouhab, L., Bona, J.L., Felland, M., Saut, J.C.: Nonlocal models for nonlinear, dispersive waves. Phys. D 40(3), 360–392 (1989). https://doi.org/10.1016/0167-2789(89)90050-X.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bona, J.L., Smith, R.: The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. 278(1287), 555–601 (1975). https://doi.org/10.1098/rsta.1975.0035.

  3. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993). https://doi.org/10.1007/BF01896020.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Bourgain, J.: Periodic Korteweg de Vries equation with measures as initial data. Selecta Math. (N.S.) 3(2), 115–159 (1997). https://doi.org/10.1007/s000290050008.

  6. Burq, N., Gérard, P., Tzvetkov, N.: An instability property of the nonlinear Schrödinger equation on \(S^d\). Math. Res. Lett. 9(2-3), 323–335 (2002). https://doi.org/10.4310/MRL.2002.v9.n3.a8.

    Article  MathSciNet  MATH  Google Scholar 

  7. Christ, M., Colliander, J., Tao, T.: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125(6), 1235–1293 (2003). URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.6christ.pdf

  8. Christ, M., Colliander, J., Tao, T.: A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. J. Funct. Anal. 254(2), 368–395 (2008). https://doi.org/10.1016/j.jfa.2007.09.005

    Article  MathSciNet  MATH  Google Scholar 

  9. Christ, M., Holmer, J., Tataru, D.: Low regularity a priori bounds for the modified Korteweg-de Vries equation. Lib. Math. (N.S.) 32(1), 51–75 (2012). https://doi.org/10.14510/lm-ns.v32i1.32.

  10. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on \({\mathbb{R}}\) and \({\mathbb{T}}\). J. Amer. Math. Soc. 16(3), 705–749 (2003). https://doi.org/10.1090/S0894-0347-03-00421-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211(1), 173–218 (2004). https://doi.org/10.1016/S0022-1236(03)00218-0.

    Article  MathSciNet  MATH  Google Scholar 

  12. Dinh, V.D.: Strichartz estimates for the fractional Schrödinger and wave equations on compact manifolds without boundary. J. Differential Equations 263(12), 8804–8837 (2017). https://doi.org/10.1016/j.jde.2017.08.045.

    Article  MathSciNet  MATH  Google Scholar 

  13. Guo, Z., Oh, T.: Non-existence of solutions for the periodic cubic NLS below \(L^2\). Int. Math. Res. Not. IMRN (6), 1656–1729 (2018). https://doi.org/10.1093/imrn/rnw271.

    Article  MATH  Google Scholar 

  14. Hu, Y., Li, X.: Discrete Fourier restriction associated with KdV equations. Anal. PDE 6(4), 859–892 (2013). https://doi.org/10.2140/apde.2013.6.859.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ionescu, A.D., Kenig, C.E., Tataru, D.: Global well-posedness of the KP-I initial-value problem in the energy space. Invent. Math. 173(2), 265–304 (2008). https://doi.org/10.1007/s00222-008-0115-0.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kappeler, T., Molnar, J.C.: On the well-posedness of the defocusing mKdV equation below \(L^2\). SIAM J. Math. Anal. 49(3), 2191–2219 (2017). https://doi.org/10.1137/16M1096979.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kappeler, T., Topalov, P.: Global well-posedness of mKdV in \(L^{2}({\mathbb{T}},{\mathbb{R}})\). Comm. Partial Differential Equations 30(1-3), 435–449 (2005). https://doi.org/10.1081/PDE-200050089.

    Article  MathSciNet  MATH  Google Scholar 

  18. Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46(4), 527–620 (1993). https://doi.org/10.1002/cpa.3160460405.

    Article  MathSciNet  MATH  Google Scholar 

  19. Killip, R., Vişan, M., Zhang, X.: Low regularity conservation laws for integrable PDE. Geom. Funct. Anal. 28(4), 1062–1090 (2018). https://doi.org/10.1007/s00039-018-0444-0.

    Article  MathSciNet  MATH  Google Scholar 

  20. Killip, R., Visan, M.: KdV is well-posed in \({H}^{-1}\). Ann. of Math. (2) 190(1), 249–305 (2019). https://doi.org/10.4007/annals.2019.190.1.4.

  21. Koch, H., Tataru, D.: A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Int. Math. Res. Not. IMRN (16), Art. ID rnm053, 36 (2007). https://doi.org/10.1093/imrn/rnm053.

  22. Koch, H., Tzvetkov, N.: Nonlinear wave interactions for the Benjamin-Ono equation. Int. Math. Res. Not. (30), 1833–1847 (2005). https://doi.org/10.1155/IMRN.2005.1833.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kwon, S., Oh, T.: On unconditional well-posedness of modified KdV. Int. Math. Res. Not. IMRN (15), 3509–3534 (2012). https://doi.org/10.1093/imrn/rnr156.

    Article  MathSciNet  MATH  Google Scholar 

  24. Miura, R.M.: Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Mathematical Phys. 9, 1202–1204 (1968). https://doi.org/10.1063/1.1664700.

  25. Molinet, L.: Sharp ill-posedness results for the KdV and mKdV equations on the torus. Adv. Math. 230(4-6), 1895–1930 (2012). https://doi.org/10.1016/j.aim.2012.03.026.

    Article  MathSciNet  MATH  Google Scholar 

  26. Molinet, L., Pilod, D., Vento, S.: On unconditional well-posedness for the periodic modified Korteweg–de Vries equation. J. Math. Soc. Japan 71(1), 147–201 (2019). https://doi.org/10.2969/jmsj/76977697.

  27. Nakanishi, K., Takaoka, H., Tsutsumi, Y.: Local well-posedness in low regularity of the mKdV equation with periodic boundary condition. Discrete Contin. Dyn. Syst. 28(4), 1635–1654 (2010). https://doi.org/10.3934/dcds.2010.28.1635.

    Article  MathSciNet  MATH  Google Scholar 

  28. Oh, T., Wang, Y.: Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces. Forum Math. Sigma 6, e5, 80 (2018). https://doi.org/10.1017/fms.2018.4.

  29. Staffilani, G.: On solutions for periodic generalized KdV equations. Internat. Math. Res. Notices (18), 899–917 (1997). https://doi.org/10.1155/S1073792897000585.

    Article  MathSciNet  MATH  Google Scholar 

  30. Takaoka, H., Tsutsumi, Y.: Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. Int. Math. Res. Not. (56), 3009–3040 (2004). https://doi.org/10.1155/S1073792804140555.

    Article  MathSciNet  MATH  Google Scholar 

  31. Tao, T.: Multilinear weighted convolution of \(L^2\)-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123(5), 839–908 (2001). URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v123/123.5tao.pdf

  32. Tao, T.: Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2006). https://doi.org/10.1090/cbms/106. Local and global analysis

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Acknowledgements

Financial support by the German Science Foundation (IRTG 2235) is gratefully acknowledged. I would like to thank the anonymous referee for a careful reading and commenting on an earlier manuscript, which significantly improved the presentation.

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  1. Universität Bielefeld, 33501, Bielefeld, Germany

    Robert Schippa

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

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