no. 5
The Korteweg–de Vries equation at H -1 regularity
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 1071-1098

In this paper we will prove the existence of weak solutions to the Korteweg–de Vries initial value problem on the real line with H -1 initial data; moreover, we will study the problem of orbital and asymptotic H s stability of solitons for integers s≥-1; finally, we will also prove new a priori H -1 bound for solutions to the Korteweg–de Vries equation. The paper will utilise the Miura transformation to link the Korteweg–de Vries equation to the modified Korteweg–de Vries equation.

DOI : 10.1016/j.anihpc.2014.05.004
Keywords: Korteweg–de Vries equation, Stability of solitons, Miura map 
@article{AIHPC_2015__32_5_1071_0,
     author = {Buckmaster, Tristan and Koch, Herbert},
     title = {The {Korteweg{\textendash}de} {Vries} equation at $ {H}^{-1}$ regularity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1071--1098},
     year = {2015},
     publisher = {Elsevier},
     volume = {32},
     number = {5},
     doi = {10.1016/j.anihpc.2014.05.004},
     mrnumber = {3400442},
     zbl = {1331.35300},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2014.05.004/}
}
TY  - JOUR
AU  - Buckmaster, Tristan
AU  - Koch, Herbert
TI  - The Korteweg–de Vries equation at $ {H}^{-1}$ regularity
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 1071
EP  - 1098
VL  - 32
IS  - 5
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2014.05.004/
DO  - 10.1016/j.anihpc.2014.05.004
LA  - en
ID  - AIHPC_2015__32_5_1071_0
ER  - 
%0 Journal Article
%A Buckmaster, Tristan
%A Koch, Herbert
%T The Korteweg–de Vries equation at $ {H}^{-1}$ regularity
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 1071-1098
%V 32
%N 5
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2014.05.004/
%R 10.1016/j.anihpc.2014.05.004
%G en
%F AIHPC_2015__32_5_1071_0
Buckmaster, Tristan; Koch, Herbert. The Korteweg–de Vries equation at $ {H}^{-1}$ regularity. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 1071-1098. doi: 10.1016/j.anihpc.2014.05.004

[1] Michael Christ, James Colliander, Terrence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocussing equations, Am. J. Math. 125 no. 6 (2003), 1235 -1293 | MR | Zbl

[2] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋 , J. Am. Math. Soc. 16 no. 3 (2003), 705 -749 | MR | Zbl

[3] P. Deift, R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Commun. Math. Phys. 203 no. 2 (1999), 341 -347 | MR | Zbl

[4] Zihua Guo, Global well-posedness of Korteweg–de Vries equation in H -3/4 (ℝ) , J. Math. Pures Appl. (9) 91 no. 6 (2009), 583 -597 | MR | Zbl

[5] T. Kappeler, P. Topalov, Global wellposedness of KdV in H -1 (𝕋,ℝ) , Duke Math. J. 135 no. 2 (2006), 327 -360 | MR | Zbl

[6] Thomas Kappeler, Peter Perry, Mikhail Shubin, Peter Topalov, The Miura map on the line, Int. Math. Res. Not. no. 50 (2005), 3091 -3133 | MR | Zbl

[7] Tosio Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Studies in Applied Mathematics, Adv. Math. Suppl. Stud. vol. 8 , Academic Press, New York (1983), 93 -128 | MR | Zbl

[8] Carlos E. Kenig, Gustavo Ponce, Luis Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math. 46 no. 4 (1993), 527 -620 | MR | Zbl

[9] Carlos E. Kenig, Gustavo Ponce, Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 no. 3 (2001), 617 -633 | MR | Zbl

[10] Nobu Kishimoto, Well-posedness of the Cauchy problem for the Korteweg–de Vries equation at the critical regularity, Differ. Integral Equ. 22 no. 5–6 (2009), 447 -464 | MR | Zbl

[11] Herbert Koch, Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. no. 16 (2007) | MR | Zbl

[12] Martin D. Kruskal, Robert M. Miura, Clifford S. Gardner, Norman J. Zabusky, Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys. 11 (1970), 952 -960 | MR | Zbl

[13] E. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, The Stability of Matter: From Atoms to Stars (2005), 205 -239

[14] B. Liu, A-priori bounds for KdV equation below H -3/4 , arXiv e-prints, 2011. | MR

[15] Yvan Martel, Frank Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 no. 3 (2001), 219 -254 | MR | Zbl

[16] F. Merle, L. Vega, L 2 stability of solitons for KdV equation, Int. Math. Res. Not. no. 13 (2003), 735 -753 | MR | Zbl

[17] Robert M. Miura, Clifford S. Gardner, Martin D. Kruskal, Korteweg–de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968), 1204 -1209 | MR | Zbl

[18] T. Mizumachi, N. Tzvetkov, Stability of the line soliton of the KP-II equation under periodic transverse perturbations, Math. Ann. (2011), 1 -32 | MR

[19] Molinet Luc, A note on ill-posedness for the KdV equation, Differ. Integral Equ. 24 no. 7–8 (2011), 759 -765 | MR | Zbl

[20] Michael Reed, Barry Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, Harcourt Brace Jovanovich Publishers, New York (1975) | MR | Zbl

[21] Thomas Runst, Winfried Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Ser. Nonlinear Anal. Appl. vol. 3 , Walter de Gruyter & Co., Berlin (1996) | MR | Zbl

[22] Terence Tao, Nonlinear dispersive equations, Local and Global Analysis, CBMS Reg. Conf. Ser. Math. vol. 106 , Published for the Conference Board of the Mathematical Sciences, Washington, DC (2006) | MR | Zbl

[23] Yoshio Tsutsumi, The Cauchy problem for the Korteweg–de Vries equation with measures as initial data, SIAM J. Math. Anal. 20 no. 3 (1989), 582 -588 | MR | Zbl

[24] Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 no. 1 (1986), 51 -67 | MR | Zbl

Cité par Sources :