Advertisement

Selecta Mathematica

, Volume 24, Issue 2, pp 1479–1526 | Cite as

On the wellposedness of the KdV equation on the space of pseudomeasures

  • Thomas Kappeler
  • Jan Molnar
Article
  • 47 Downloads

Abstract

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudomeasures, also referred to as the Fourier Lebesgue space \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\), where \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space \(\mathscr {F}\ell ^{s,\infty }(\mathbb {T},\mathbb {R})\) with \(-1/2 < s \le 0\) and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space \(H^{-1}(\mathbb {T},\mathbb {R})\) to be in \(\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})\) in terms of asymptotic behavior of spectral quantities of the Hill operator \(-\partial _{x}^{2} + q\). In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

Keywords

KdV equation Well-posedness Birkhoff coordinates 

Mathematics Subject Classification

Primary 37K10 Secondary 35Q53 35D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgain, J.: Periodic Korteweg–de Vries equation with measures as initial data. Sel. Math. (New Ser.) 3(2), 115–159 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buckmaster, T., Koch, H.: The Korteweg–de Vries equation at H\(^{-1}\) regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(5), 1071–1098 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christ, M., Colliander, J., Tao, T.: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Am. J. Math. 125(6), 1235–1293 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Sharp global well-posedness for KdV and modified KdV on R and T. J. Am. Math. Soc. 16(3), 705–749 (2003). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Djakov, P., Mityagin, B.: Instability zones of periodic 1-dimensional Schrödinger and Dirac operators. Russ. Math. Surv. 61, 663–766 (2006)CrossRefzbMATHGoogle Scholar
  7. 7.
    Djakov, P., Mityagin, B.: Spectral gaps of Schrödinger operators with periodic singular potentials. Dyn. Partial Differ. Equ. 6(2), 95–165 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grébert, B., Kappeler, T.: The Defocusing NLS Equation and Its Normal Form. European Mathematical Society (EMS), Zürich (2014)CrossRefzbMATHGoogle Scholar
  9. 9.
    Guo, Z.: Global well-posedness of Korteweg–de Vries equation in \(H^{-3/4}({\mathbb{R}})\). J. Math. Pures Appl. 91(6), 583–597 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kappeler, T.: Solutions to the Korteweg–de Vries equation with irregular initial profile. Commun. Partial Diff. Equ. 11(9), 927–945 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kappeler, T., Maspero, A., Molnar, J.C., Topalov, P.: On the convexity of the KdV Hamiltonian. Commun. Math. Phys. 346(1), 191–236 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kappeler, T., Mityagin, B.: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator. SIAM J. Math. Anal. 33(1), 113–152 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kappeler, T., Möhr, C.: Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator with singular potentials. J. Funct. Anal. 186(1), 62–91 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kappeler, T., Möhr, C., Topalov, P.: Birkhoff coordinates for KdV on phase spaces of distributions. Sel. Math. (New Ser.) 11(1), 37–98 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kappeler, T., Molnar, J.C.: On the wellposedness of the KdV/KdV2 equations and their frequency maps. Ann. Inst. H. Poincaré Anal. Non Linéaire (online) (2017). doi: 10.1016/j.anihpc.2017.03.003
  16. 16.
    Kappeler, T., Perry, P., Topalov, P.: The Miura map on the line. Int. Math. Res. Not. 50, 3091–3133 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kappeler, T., Pöschel, J.: KdV and KAM. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kappeler, T., Pöschel, J.: On the periodic KdV equation in weighted Sobolev spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(3), 841–853 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kappeler, T., Schaad, B., Topalov, P.: Qualitative features of periodic solutions of KdV. Commun. Partial Diff. Equ. 38(9), 1626–1673 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kappeler, T., Serier, F., Topalov, P.: On the symplectic phase space of KdV. Proc. Am. Math. Soc. 136(5), 1691–1698 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kappeler, T., Topalov, P.: Riccati representation for elements in \(H^{-1}(\mathbb{T})\) and its applications. Pliska Stud. Math. Bulg. 15, 171–188 (2003)MathSciNetGoogle Scholar
  22. 22.
    Kappeler, T., Topalov, P.: Global wellposedness of KdV in \(H^{-1}(\mathbb{T},\mathbb{R})\). Duke Math. J. 135(2), 327–360 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kenig, C.E., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9(2), 573–603 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kenig, C.E., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106(3), 617–633 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Korotyaev, E.: Characterization of the spectrum of Schrödinger operators with periodic distributions. Int. Math. Res. Not. 2003(37), 2019–2031 (2003)CrossRefzbMATHGoogle Scholar
  26. 26.
    Molinet, L.: A note on ill posedness for the KdV equation. Differ. Integral Equ. 24(7–8), 759–765 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Molinet, L.: Sharp ill-posedness results for the KdV and mKdV equations on the torus. Adv. Math. 230(4–6), 1895–1930 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pöschel, J.: Hill’s potentials in weighted Sobolev spaces and their spectral gaps. Math. Ann. 349(2), 433–458 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Savchuk, A.M., Shkalikov, A.A.: Sturm–Liouville operators with distribution potentials. Tr. Mosk. Math. Obs. 64, 159–212 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Savchuk, A.M., Shkalikov, A.A.: Inverse problem for Sturm–Liouville operators with distribution potentials: reconstruction from two spectra. Russ. J. Math. Phys. 12(4), 507–514 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Tao, T.: Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2006)Google Scholar
  32. 32.
    Tsutsumi, Y.: The Cauchy problem for the Korteweg–de Vries equation with measures as initial data. SIAM J. Math. Anal. 20(3), 582–588 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, University of ZurichZurichSwitzerland

Personalised recommendations