Comptes Rendus
Numerical Analysis
The Khokhlov–Zabolotskaya–Kuznetsov equation
[Équation de Khokhlov–Zabolotskaya–Kuznetsov]
Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 337-342.

Pour l'équation KZK (utuuxβuxx)xγΔyu=0 dans la classe des fonctions périodiques en x et de moyennes nulles, on a étudié la dérivation à partir du système de Navier–Stokes isentropique et la validation de son approximation, l'existence, l'unicité et la stabilité de la solution. On a prouvé que la solution est globale en temps pour des données initiales suffisement petites avec β>0 et que la solution présente une onde de choc si β=0.

For the KZK equation (utuuxβuxx)xγΔyu=0 in the class of x-periodic and of zero mean value functions we have analysed the following: the derivation from Navier–Stokes system and the validity of its approximation, the existence, uniqueness and stability of the solution. The solution is proved to be global in time for sufficient small initial data with β>0 and to have a blow-up if β=0.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.01.010

Anna Rozanova 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
@article{CRMATH_2007__344_5_337_0,
     author = {Anna Rozanova},
     title = {The {Khokhlov{\textendash}Zabolotskaya{\textendash}Kuznetsov} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {337--342},
     publisher = {Elsevier},
     volume = {344},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.010},
     language = {en},
}
TY  - JOUR
AU  - Anna Rozanova
TI  - The Khokhlov–Zabolotskaya–Kuznetsov equation
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 337
EP  - 342
VL  - 344
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2007.01.010
LA  - en
ID  - CRMATH_2007__344_5_337_0
ER  - 
%0 Journal Article
%A Anna Rozanova
%T The Khokhlov–Zabolotskaya–Kuznetsov equation
%J Comptes Rendus. Mathématique
%D 2007
%P 337-342
%V 344
%N 5
%I Elsevier
%R 10.1016/j.crma.2007.01.010
%G en
%F CRMATH_2007__344_5_337_0
Anna Rozanova. The Khokhlov–Zabolotskaya–Kuznetsov equation. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 337-342. doi : 10.1016/j.crma.2007.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.010/

[1] S. Alinhac Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions, II, Acta Math., Volume 182 (1999), pp. 1-23

[2] N.S. Bakhvalov; Ya.M. Zhileikin; E.A. Zabolotskaya Nonlinear Theory of Sound Beams, American Institute of Physics, New York, 1987 (Nelineinaya Teoriya Zvukovih Puchkov, Nauka, Moscow, 1982)

[3] C. Dafermos Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer-Verlag, 2000

[4] B. Gustafsson; A. Sundström Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., Volume 35 (1978) no. 2, pp. 343-357

[5] I. Kostin, G. Panasenko, Analysis and homogenization of the Khokhlov–Zabolotskaya–Kuznetsov type equation, Oral communication in the International Conference on Advanced Problems in Mechanics, St Petersburg, June 2005

[6] S. Novo Compressible Navier–Stokes model with inflow–outflow boundary conditions, J. Math. Fluid Mech., Volume 7 (2005), pp. 485-514

[7] D. Sanchez Long waves in ferromagnetic media, Zabolotskaya–Khokhlov equation, J. Differential Equations, Volume 210 (2005), pp. 263-289

[8] E.A. Zabolotskaya; R.V. Khokhlov Quasi-planes waves in the nonlinear acoustic of confined beams, Sov. Phys. Acoust., Volume 15 (1969) no. 1, pp. 35-40

Cité par Sources :

Commentaires - Politique