For the KZK equation 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 and to have a blow-up if .
Pour l'équation KZK 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 et que la solution présente une onde de choc si .
Accepted:
Published online:
Anna Rozanova 1
@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}, }
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] 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] Nonlinear Theory of Sound Beams, American Institute of Physics, New York, 1987 (Nelineinaya Teoriya Zvukovih Puchkov, Nauka, Moscow, 1982)
[3] Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, Springer-Verlag, 2000
[4] 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] Compressible Navier–Stokes model with inflow–outflow boundary conditions, J. Math. Fluid Mech., Volume 7 (2005), pp. 485-514
[7] Long waves in ferromagnetic media, Zabolotskaya–Khokhlov equation, J. Differential Equations, Volume 210 (2005), pp. 263-289
[8] Quasi-planes waves in the nonlinear acoustic of confined beams, Sov. Phys. Acoust., Volume 15 (1969) no. 1, pp. 35-40
Cited by Sources:
Comments - Policy