On the well-posedness of a quasi-linear Korteweg-de Vries equation
Annales mathématiques Blaise Pascal, Volume 24 (2017) no. 1, pp. 83-114.

The Korteweg-de Vries equation (KdV) and various generalized, most often semilinear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamiltonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major interest being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim & Ponce [22] and developed later by Kenig, Ponce & Vega [20] and S. Benzoni-Gavage, R. Danchin & S. Descombes [4]. The existence of a solution is obtained as a limit from regularized parabolic problems. Uniqueness and continuity with respect to the initial data are proven using a Bona–Smith regularization technique.

Published online:
DOI: 10.5802/ambp.365
Classification: 35Q53
Keywords: quasilinear dispersive equation, energy estimates, gauging technique, parabolic regularization, Bona–Smith technique
Keywords: quasilinear dispersive equation, energy estimates, gauging technique, parabolic regularization, Bona–Smith technique
Colin Mietka 1

1 Université de Lyon, CNRS UMR 5208 Institut Camille Jordan 43 bd 11 novembre 1918 69622 Villeurbanne cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AMBP_2017__24_1_83_0,
     author = {Colin Mietka},
     title = {On the well-posedness of a quasi-linear {Korteweg-de} {Vries} equation},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {83--114},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {24},
     number = {1},
     year = {2017},
     doi = {10.5802/ambp.365},
     language = {en},
     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.365/}
}
TY  - JOUR
AU  - Colin Mietka
TI  - On the well-posedness of a quasi-linear Korteweg-de Vries equation
JO  - Annales mathématiques Blaise Pascal
PY  - 2017
SP  - 83
EP  - 114
VL  - 24
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.365/
DO  - 10.5802/ambp.365
LA  - en
ID  - AMBP_2017__24_1_83_0
ER  - 
%0 Journal Article
%A Colin Mietka
%T On the well-posedness of a quasi-linear Korteweg-de Vries equation
%J Annales mathématiques Blaise Pascal
%D 2017
%P 83-114
%V 24
%N 1
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.365/
%R 10.5802/ambp.365
%G en
%F AMBP_2017__24_1_83_0
Colin Mietka. On the well-posedness of a quasi-linear Korteweg-de Vries equation. Annales mathématiques Blaise Pascal, Volume 24 (2017) no. 1, pp. 83-114. doi : 10.5802/ambp.365. https://ambp.centre-mersenne.org/articles/10.5802/ambp.365/

[1] Mark J. Ablowitz Nonlinear dispersive waves. Asymptotic analysis and solitons, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2011, xiv+348 pages | DOI | MR | Zbl

[2] Timur Akhunov Local well-posedness of quasi-linear systems generalizing KdV, Commun. Pure Appl. Anal., Volume 12 (2013) no. 2, pp. 899-921 | DOI | MR | Zbl

[3] Sylvie Benzoni-Gavage Local well-posedness of nonlocal Burgers equations, Differ. Integral Equ., Volume 22 (2009) no. 3-4, pp. 303-320 | MR | Zbl

[4] Sylvie Benzoni-Gavage; Raphaël Danchin; Stéphane Descombes Well-posedness of one-dimensional Korteweg models, Electron. J. Differ. Equ., Volume 2006 (2006) No. 59, 35 pp. (electronic) | MR | Zbl

[5] Sylvie Benzoni-Gavage; Raphaël Danchin; Stéphane Descombes On the well-posedness for the Euler-Korteweg model in several space dimensions, Indiana Univ. Math. J., Volume 56 (2007) no. 4, pp. 1499-1579 | DOI | MR | Zbl

[6] Sylvie Benzoni-Gavage; Colin Mietka; L. Miguel Rodrigues Co-periodic stability of periodic waves in some Hamiltonian PDEs (2015) (https://arxiv.org/abs/1505.01382)

[7] Jerry L. Bona; R. Smith The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London, Volume 278 (1975) no. 1287, pp. 555-601 | DOI | MR | Zbl

[8] Jean Bourgain Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., Volume 3 (1993) no. 3, pp. 209-262 | DOI | MR | Zbl

[9] Michael Christ; James Colliander; Terence Tao Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., Volume 125 (2003) no. 6, pp. 1235-1293 http://muse.jhu.edu/journals/american_journal_of_mathematics/v125/125.6christ.pdf | DOI | MR | Zbl

[10] James Colliander; Markus Keel; Gigliola Staffilani; Hideo Takaoka; Terence Tao Sharp global well-posedness for KdV and modified KdV on and 𝕋, J. Am. Math. Soc., Volume 16 (2003) no. 3, pp. 705-749 | DOI | Zbl

[11] Walter Craig; Thomas Kappeler; Walter Alexander Strauss Gain of regularity for equations of KdV type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 9 (1992) no. 2, pp. 147-186 | DOI | Zbl

[12] Boris A. Dubrovin Hamiltonian partial differential equations and Painlevé transcendents, 2015 (Conference Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications, at Banff International Research Station)

[13] Clifford S. Gardner; John M. Greene; Martin D. Kruskal; Robert M. Miura Method for Solving the Korteweg-de Vries Equation, Phys. Rev. Lett., Volume 19 (1967), pp. 1095-1097 | DOI | Zbl

[14] Clifford S. Gardner; John M. Greene; Martin D. Kruskal; Robert M. Miura Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Commun. Pure Appl. Math., Volume 27 (1974), pp. 97-133 | DOI | MR | Zbl

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

[16] Tosio Kato Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974) (Lecture Notes in Mathematics), Volume 448 (1975), pp. 25-70 | MR | Zbl

[17] Carlos E. Kenig; Gustavo Ponce; Luis Vega The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., Volume 71 (1993) no. 1, pp. 1-21 | DOI | MR | Zbl

[18] 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., Volume 46 (1993) no. 4, pp. 527-620 | DOI | MR | Zbl

[19] Carlos E. Kenig; Gustavo Ponce; Luis Vega A bilinear estimate with applications to the KdV equation, J. Am. Math. Soc., Volume 9 (1996) no. 2, pp. 573-603 | DOI | MR | Zbl

[20] Carlos E. Kenig; Gustavo Ponce; Luis Vega The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math., Volume 158 (2004) no. 2, pp. 343-388 | DOI | MR | Zbl

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

[22] Wee Keong Lim; Gustavo Ponce On the initial value problem for the one dimensional quasi-linear Schrödinger equations, SIAM J. Math. Anal., Volume 34 (2002) no. 2, pp. 435-459 | DOI | MR | Zbl

[23] Felipe Linares; Gustavo Ponce; Derek L. Smith On the regularity of solutions to a class of nonlinear dispersive equations (2015) (https://arxiv.org/abs/1510.02512)

[24] Alessandra Lunardi Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Birkhäuser, 1995, xviii+424 pages | MR | Zbl

[25] Ammon Pazy Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1983, viii+279 pages | DOI | MR | Zbl

[26] L. Miguel Rodrigues; Kevin Zumbrun Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow, SIAM J. Math. Anal., Volume 48 (2016) no. 1, pp. 268-280 | DOI | Zbl

[27] Steven H. Schochet; Michael I. Weinstein The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence, Commun. Math. Phys., Volume 106 (1986) no. 4, pp. 569-580 http://projecteuclid.org/euclid.cmp/1104115852 | DOI | MR | Zbl

[28] Denis Serre Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves, Cambridge University Press, 1999, xxii+263 pages (translated from the 1996 French original by I. N. Sneddon) | DOI | MR | Zbl

Cited by Sources: