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

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.

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.

Published online:
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
     author = {Anna Rozanova},
     title = {The {Khokhlov{\textendash}Zabolotskaya{\textendash}Kuznetsov} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {337--342},
     publisher = {Elsevier},
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     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.010},
     language = {en},
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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.

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