logo AFST
Transverse nonlinear instability of Euler–Korteweg solitons
Matthew Paddick
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 1, p. 23-48

We show that solitary waves for the 2D Euler– Korteweg model for capillary fluids display nonlinear orbital instability when subjected to transverse perturbations, based on their linear instability.

On montre que les solitons de l’équation d’Euler– Korteweg 2D, un modèle pour les fluides avec capillarité, sont orbitalement instables lorsqu’ils sont soumis à des perturbations transverses, en partant de leur instabilité linéaire.

Received : 2015-06-30
Accepted : 2016-03-01
Published online : 2017-02-07
DOI : https://doi.org/10.5802/afst.1525
Classification:  35C08,  35Q35,  37K45
Keywords: Euler–Korteweg system, solitary waves, nonlinear instability
@article{AFST_2017_6_26_1_23_0,
     author = {Matthew Paddick},
     title = {Transverse nonlinear instability of Euler--Korteweg solitons},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {1},
     year = {2017},
     pages = {23-48},
     doi = {10.5802/afst.1525},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_1_23_0}
}
Paddick, Matthew. Transverse nonlinear instability of Euler–Korteweg solitons. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 1, pp. 23-48. doi : 10.5802/afst.1525. afst.centre-mersenne.org/item/AFST_2017_6_26_1_23_0/

[1] James Crew Alexander; Robert L. Pego; Robert L. Sachs On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A, Tome 226 (1997) no. 3-4, pp. 187-192 | Article

[2] Corentin Audiard; Boris Haspot From Gross-Pitaevskii equation to Euler Korteweg system, existence of global strong solutions with small irrotational initial data (2014) (https://hal.archives-ouvertes.fr/hal-01077281, working paper or preprint)

[3] Corentin Audiard; Boris Haspot Global well-posedness of the Euler-Korteweg system for small irrotational data (2016) (https://hal.archives-ouvertes.fr/hal-01278163, 45 pages)

[4] T. Brooke Benjamin Impulse, flow force and variational principles, IMA J. Appl. Math., Tome 32 (1984) no. 1-3, pp. 3-68 | Article

[5] Sylvie Benzoni-Gavage Linear stability of propagating phase boundaries in capillary fluids, Phys. D, Tome 155 (2001) no. 3-4, pp. 235-273 | Article

[6] Sylvie Benzoni-Gavage Spectral transverse instability of solitary waves in Korteweg fluids, J. Math. Anal. Appl., Tome 361 (2010) no. 2, pp. 338-357 | Article

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

[8] Sylvie Benzoni-Gavage; Raphaël Danchin; Stéphane Descombes; Didier Jamet Structure of Korteweg models and stability of diffuse interfaces, Interfaces Free Bound., Tome 7 (2005) no. 4, pp. 371-414 | Article

[9] Benoît Desjardins; Emmanuel Grenier Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 20 (2003) no. 1, pp. 87-106 | Article

[10] Donatella Donatelli; Eduard Feireisl; Pierangelo Marcati Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Comm. Partial Differential Equations, Tome 40 (2015) no. 7, pp. 1314-1335 | Article

[11] Nelson Dunford; Jacob T. Schwartz Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963, ix+pp. 859–1923+7 pages

[12] Emmanuel Grenier On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., Tome 53 (2000) no. 9, pp. 1067-1091 | Article

[13] Manoussos Grillakis; Jalal Shatah; Walter Strauss Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Tome 74 (1987) no. 1, pp. 160-197 | Article

[14] Daniel Henry Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Tome 840, Springer-Verlag, 1981, iv+348 pages

[15] Johannes Höwing Stability of large- and small-amplitude solitary waves in the generalized Korteweg-de Vries and Euler-Korteweg/Boussinesq equations, J. Diff. Equations, Tome 251 (2011) no. 9, pp. 2515-2533 | Article

[16] Johannes Höwing Standing solitary Euler-Korteweg waves are unstable, Z. Anal. Anwend., Tome 33 (2014) no. 4, pp. 441-445 | Article

[17] Nader Masmoudi; Frédéric Rousset Stability of oscillating boundary layers in rotating fluids, Ann. Sci. Éc. Norm. Supér., Tome 41 (2008) no. 6, pp. 955-1002

[18] Matthew Paddick Stability and instability of Navier boundary layers, Differ. Integral Equ., Tome 27 (2014) no. 9-10, pp. 893-930

[19] Livio Clemente Piccinini; Guido Stampacchia; Giovanni Vidossich Ordinary differential equations in R n , Applied Mathematical Sciences, Tome 39, Springer-Verlag, New York, 1984, xii+385 pages (Problems and methods, Translated from the Italian by A. LoBello) | Article

[20] Frédéric Rousset Stability of large Ekman boundary layers in rotating fluids, Arch. Ration. Mech. Anal., Tome 172 (2004) no. 2, pp. 213-245 | Article

[21] Frédéric Rousset; Nikolay Tzvetkov Transverse nonlinear instability of solitary waves for some Hamiltonian PDE’s, J. Math. Pures Appl., Tome 90 (2008) no. 6, pp. 550-590 | Article

[22] Frédéric Rousset; Nikolay Tzvetkov Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 26 (2009) no. 2, pp. 477-496 | Article

[23] Frédéric Rousset; Nikolay Tzvetkov A simple criterion of transverse linear instability for solitary waves, Math. Res. Lett., Tome 17 (2010) no. 1, pp. 157-169 | Article

[24] Frédéric Rousset; Nikolay Tzvetkov Transverse instability of the line solitary water-waves, Invent. Math., Tome 184 (2011) no. 2, pp. 257-388 | Article

[25] Frédéric Rousset; Nikolay Tzvetkov Stability and instability of the KdV solitary wave under the KP-I flow, Commun. Math. Phys., Tome 313 (2012) no. 1, pp. 155-173 | Article

[26] Roderick Wong Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989, xii+543 pages