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.
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.
Accepté le :
Publié le :
Mots clés : Euler–Korteweg system, solitary waves, nonlinear instability
Matthew Paddick 1
CC-BY 4.0
@article{AFST_2017_6_26_1_23_0,
author = {Matthew Paddick},
title = {Transverse nonlinear instability of {Euler{\textendash}Korteweg} solitons},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {23--48},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {1},
year = {2017},
doi = {10.5802/afst.1525},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1525/}
}
TY - JOUR AU - Matthew Paddick TI - Transverse nonlinear instability of Euler–Korteweg solitons JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 23 EP - 48 VL - 26 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1525/ DO - 10.5802/afst.1525 LA - en ID - AFST_2017_6_26_1_23_0 ER -
%0 Journal Article %A Matthew Paddick %T Transverse nonlinear instability of Euler–Korteweg solitons %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 23-48 %V 26 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1525/ %R 10.5802/afst.1525 %G en %F AFST_2017_6_26_1_23_0
Matthew Paddick. Transverse nonlinear instability of Euler–Korteweg solitons. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 23-48. doi : 10.5802/afst.1525. https://afst.centre-mersenne.org/articles/10.5802/afst.1525/
[1] On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation, Phys. Lett. A, Volume 226 (1997) no. 3-4, pp. 187-192 | DOI
[2] 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] Global well-posedness of the Euler-Korteweg system for small irrotational data (2016) (https://hal.archives-ouvertes.fr/hal-01278163, 45 pages)
[4] Impulse, flow force and variational principles, IMA J. Appl. Math., Volume 32 (1984) no. 1-3, pp. 3-68 | DOI
[5] Linear stability of propagating phase boundaries in capillary fluids, Phys. D, Volume 155 (2001) no. 3-4, pp. 235-273 | DOI
[6] Spectral transverse instability of solitary waves in Korteweg fluids, J. Math. Anal. Appl., Volume 361 (2010) no. 2, pp. 338-357 | DOI
[7] 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
[8] Structure of Korteweg models and stability of diffuse interfaces, Interfaces Free Bound., Volume 7 (2005) no. 4, pp. 371-414 | DOI
[9] Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 20 (2003) no. 1, pp. 87-106 | DOI
[10] Well/ill posedness for the Euler-Korteweg-Poisson system and related problems, Comm. Partial Differential Equations, Volume 40 (2015) no. 7, pp. 1314-1335 | DOI
[11] 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] On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., Volume 53 (2000) no. 9, pp. 1067-1091 | DOI
[13] Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Volume 74 (1987) no. 1, pp. 160-197 | DOI
[14] Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, 1981, iv+348 pages
[15] Stability of large- and small-amplitude solitary waves in the generalized Korteweg-de Vries and Euler-Korteweg/Boussinesq equations, J. Diff. Equations, Volume 251 (2011) no. 9, pp. 2515-2533 | DOI
[16] Standing solitary Euler-Korteweg waves are unstable, Z. Anal. Anwend., Volume 33 (2014) no. 4, pp. 441-445 | DOI
[17] Stability of oscillating boundary layers in rotating fluids, Ann. Sci. Éc. Norm. Supér., Volume 41 (2008) no. 6, pp. 955-1002
[18] Stability and instability of Navier boundary layers, Differ. Integral Equ., Volume 27 (2014) no. 9-10, pp. 893-930
[19] Ordinary differential equations in , Applied Mathematical Sciences, 39, Springer-Verlag, New York, 1984, xii+385 pages (Problems and methods, Translated from the Italian by A. LoBello) | DOI
[20] Stability of large Ekman boundary layers in rotating fluids, Arch. Ration. Mech. Anal., Volume 172 (2004) no. 2, pp. 213-245 | DOI
[21] Transverse nonlinear instability of solitary waves for some Hamiltonian PDE’s, J. Math. Pures Appl., Volume 90 (2008) no. 6, pp. 550-590 | DOI
[22] Transverse nonlinear instability for two-dimensional dispersive models, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 2, pp. 477-496 | DOI
[23] A simple criterion of transverse linear instability for solitary waves, Math. Res. Lett., Volume 17 (2010) no. 1, pp. 157-169 | DOI
[24] Transverse instability of the line solitary water-waves, Invent. Math., Volume 184 (2011) no. 2, pp. 257-388 | DOI
[25] Stability and instability of the KdV solitary wave under the KP-I flow, Commun. Math. Phys., Volume 313 (2012) no. 1, pp. 155-173 | DOI
[26] Asymptotic approximations of integrals, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989, xii+543 pages
Cité par Sources :