Inventiones mathematicae

, Volume 173, Issue 2, pp 265–304 | Cite as

Global well-posedness of the KP-I initial-value problem in the energy space

Article

Abstract

We prove that the KP-I initial-value problem
$$\begin{cases} \partial_tu+\partial_x^3u-\partial_x^{-1}\partial_y^2u+\partial_x(u^2/2)=0 \,\text{ on }\,\mathbb{R}^2_{x,y}\times\mathbb{R}_t;\\ u(0)=\phi, \end{cases}$$
is globally well-posed in the energy space
$$\mathbf{E}^1(\mathbb{R}^2)=\big\{\phi:\mathbb{R}^2\to\mathbb{R}: \|\phi\|_{\mathbf{E}^1(\mathbb{R}^2)}\approx\|\phi\|_{L^2}+\|\partial_x\phi\|_{L^2}+\big\|\partial_x^{-1}\partial_y\phi\big\|_{L^2}<\infty\big\}.$$

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. Lond. Math. Soc. Lect. Note Ser., vol. 149. Cambridge University Press, Cambridge (1991)MATHGoogle Scholar
  2. 2.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM Stud. Appl. Math., vol. 4. Soc. Ind. Appl. Math. (SIAM), Philadelphia, Pa. (1981)MATHGoogle Scholar
  3. 3.
    Alexander, J.C., Pego, R.L., Sachs, R.L.: On the transverse instability of solitary waves in the Kadomtsev–Petviashvili equation. Phys. Lett., A 226, 187–192 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ben-Artzi, M., Saut, J.-C.: Uniform decay estimates for a class of oscillatory integrals and applications. Differ. Integral Equ. 12, 137–145 (1999)MATHMathSciNetGoogle Scholar
  5. 5.
    Bona, J.L., Smith, R.: The initial-value problem for the Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond., Ser. A 278, 555–601 (1975)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bona, J.L., Souganidis, P.E., Strauss, W.A.: Stability and instability of solitary waves of Korteweg–de Vries type. Proc. R. Soc. Lond., Ser. A 411, 395–412 (1987)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    De Bouard, A., Saut, J.-C.: Solitary waves of generalized Kadomtsev–Petviashvili equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 211–236 (1997)MATHCrossRefGoogle Scholar
  8. 8.
    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV-equation. Geom. Funct. Anal. 3, 209–262 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bourgain, J.: On the Cauchy problem for the Kadomtsev–Petviashvili equation. Geom. Funct. Anal. 3, 315–341 (1993)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cazenave, T., Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Christ, M., Colliander, J., Tao, T.: A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. Preprint (2006)Google Scholar
  12. 12.
    Colliander, J., Ionescu, A.D., Kenig, C.E., Staffilani, G.: Weighted low-regularity solutions of the KP-I initial-value problem. Preprint (2007)Google Scholar
  13. 13.
    Ionescu, A.D., Kenig, C.E.: Local and global well-posedness of periodic KP-I equations. In: Mathematical Aspects of Nonlinear Dispersive Equations. Ann. Math. Stud., vol. 163, pp.181–211. Princeton Univ. Press, Princeton, NJ (2007)Google Scholar
  14. 14.
    Iorio, R.J., Nunes, W.V.L.: On equations of KP-type. Proc. R. Soc. Edinb., Sect. A, Math. 128, 725–743 (1998)MATHMathSciNetGoogle Scholar
  15. 15.
    Kenig, C.E., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kenig, C.E., Ponce, G., Vega, L.: The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kenig, C.E.: On the local and global well-posedness theory for the KP-I equation. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, 827–838 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kenig, C.E., Martel, Y.: Asymptotic stability of solitons for the Benjamin–Ono equation. Preprint (2008)Google Scholar
  19. 19.
    Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46, 1221–1268 (1993)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Koch, H., Tataru, D.: A-priori bounds for the 1-d cubic NLS in negative Sobolev spaces. Preprint (2006)Google Scholar
  21. 21.
    Martel, Y., Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157, 219–254 (2001)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Martel, Y., Merle, F.: Asymptotic stability of solitons of the subcritical gKdV equations revisited. Nonlinearity 18, 55–80 (2005)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Molinet, L., Saut, J.-C., Tzvetkov, N.: Well-posedness and ill-posedness results for the Kadomtsev–Petviashvili-I equation. Duke Math. J. 115, 353–384 (2002)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Molinet, L., Saut, J.-C., Tzvetkov, N.: Global well-posedness for the KP-I equation. Math. Ann. 324, 255–275 (2002)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Molinet, L., Saut, J.-C., Tzvetkov, N.: Correction: Global well-posedness for the KP-I equation. Math. Ann. 328, 707–710 (2004)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Rousset, F., Tzvetkov, N.: Transverse nonlinear instability for two-dimensional dispersive models. Preprint (2007)Google Scholar
  27. 27.
    Saut, J.-C.: Remarks on the generalized Kadomtsev–Petviashvili equations. Indiana Univ. Math. J. 42, 1011–1026 (1993)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Takaoka, H., Tzvetkov, N.: On the local regularity of the Kadomtsev–Petviashvili-II equation. Int. Math. Res. Not. 2001, 77–114 (2001)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Tataru, D.: Local and global results for wave maps I. Commun. Partial Differ. Equations 23, 1781–1793 (1998)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math. 39, 51–67 (1986)MATHCrossRefGoogle Scholar
  31. 31.
    Weinstein, M.I.: Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Commun. Partial Differ. Equations 12, 1133–1173 (1987)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin–MadisonMadisonUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Department of MathematicsUniversity of California–BerkeleyBerkeleyUSA

Personalised recommendations