On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition

Hamid Bellout; Jiří Neustupa; Patrick Penel

doi:10.1016/j.crma.2009.09.007

Partial Differential Equations/Mathematical Problems in Mechanics

On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition
[Solutions continues en la viscosité pour les équations d'Euler ou de Navier–Stokes avec des conditions aux limites de type Navier]

Présenté par : Gérard Iooss

Hamid Bellout ¹ ; Jiří Neustupa ² ; Patrick Penel ³

¹ Northern Illinois University, Department of Mathematical Sciences, De Kalb, IL 60115, USA
² Czech Technical University, Faculty of Mechanical Engineering, Department of Technical Mathematics, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
³ Université du Sud Toulon-Var, département de mathématique et laboratoire SNC, BP 20132, 83957 La Garde cedex, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1141-1146.

Résumé
Abstract

Pourvu que les données soient suffisamment régulières, il existe $T_{0} > 0$ , $ν^{*} > 0$ et ${u^{ν}}_{0 ⩽ ν < ν^{*}}$ famille unique de solutions fortes, locales en temps sur $(0, T_{0})$ et dépendant continûment de ν, pour les problèmes d'Euler ou de Navier–Stokes. Ces solutions vérifient des conditions aux limites de type celles de Navier.

Provided the initial velocity and the external body force are sufficiently smooth, there exist $T_{0} > 0$ , $ν^{*} > 0$ and a unique continuous family of strong solutions $u^{ν}$ ( $0 ⩽ ν < ν^{*}$ ) of the Euler or Navier–Stokes initial–boundary value problem on the time interval $(0, T_{0})$ . The solutions of the Navier–Stokes problem satisfy a Navier-type boundary condition.

Reçu le : 2009-06-06
Accepté le : 2009-09-09
Publié le : 2009-09-30

DOI : 10.1016/j.crma.2009.09.007

Affiliations des auteurs :

Hamid Bellout ¹ ; Jiří Neustupa ² ; Patrick Penel ³

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     author = {Hamid Bellout and Ji\v{r}{\'\i} Neustupa and Patrick Penel},
     title = {On viscosity-continuous solutions of the {Euler} and {Navier{\textendash}Stokes} equations with a {Navier-type} boundary condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1141--1146},
     publisher = {Elsevier},
     volume = {347},
     number = {19-20},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.007},
     language = {en},
}

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Hamid Bellout; Jiří Neustupa; Patrick Penel. On viscosity-continuous solutions of the Euler and Navier–Stokes equations with a Navier-type boundary condition. Comptes Rendus. Mathématique, Volume 347 (2009) no. 19-20, pp. 1141-1146. doi : 10.1016/j.crma.2009.09.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.007/

Bibliographie
Cité par

[1] C. Bardos Existence et unicité de la solution de l'équation d'Euler en dimension deux, J. Math. Anal. Appl., Volume 40 (1972), pp. 769-790

[2] H. Beirão da Veiga, F. Crispo, Sharp inviscid limit results under Navier type boundary condition. An $L^{p}$ theory, J. Math. Fluid Mech., in press

[3] H. Bellout; J. Neustupa A Navier–Stokes approximation of the 3D Euler equation with the zero flux on the boundary, J. Math. Fluid Mech., Volume 10 (2008), pp. 531-553

[4] H. Bellout; J. Neustupa; P. Penel On the Navier–Stokes equations with boundary conditions based on vorticity, Math. Nachr., Volume 269–270 (2004), pp. 59-72

[5] H. Bellout, J. Neustupa, P. Penel, On a ν-continuous family of strong solutions to the Euler or Navier–Stokes equations with the Navier-type boundary condition, preprint, 2009

[6] J.P. Bourguignon; H. Brezis Remarks on the Euler equation, J. Funct. Anal., Volume 15 (1974), pp. 341-363

[7] G.Q. Chen; D. Osborne; Z. Qian The Navier–Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries, December 2008 (preprint) | arXiv

[8] T. Kato, Remarks on zero viscosity limit for non-stationary Navier–Stokes flows with boundary, in: S.S. Chern (Ed.), Seminar on Nonlinear PDE, MSRI, 1984

[9] A. Mahalov; B. Nicolaenko; C. Bardos; F. Golse Non blow-up of the 3D Euler equations for a class of three dimensional initial data in cylindrical domains, Methods Appl. Anal., Volume 11 (2004) no. 4, pp. 605-634

[10] R. Temam On the Euler equations of incompressible perfect fluids, J. Funct. Anal., Volume 20 (1975), pp. 32-43

[11] Y.L. Xiao; Z.P. Xin On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition, Comm. Pure Appl. Math., Volume 60 (2007), pp. 1027-1055

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