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Journées équations aux dérivées partielles

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Frederic Rousset
On the free surface Navier-Stokes equation in the inviscid limit
Journées équations aux dérivées partielles (2011), Exp. No. 10, 14 p., doi: 10.5802/jedp.82
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Résumé - Abstract

The aim of this note is to present recent results obtained with N. Masmoudi [29] on the free surface Navier-Stokes equation with small viscosity.

Bibliographie

[1] Alazard, T., Burq, N. and Zuily C. On the Water Waves Equations with Surface Tension, Duke Math. J. 158 3 (2011), 413-499. Article |  MR 2805065 |  Zbl pre05920537
[2] Alinhac, S. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Differential Equations 14, 2(1989), 173–230.  MR 976971 |  Zbl 0692.35063
[3] Bardos, C. Existence et unicité de la solution de l’équation d’Euler en dimension deux. J. Math. Anal. Appl. 40 (1972), 769–790.  MR 333488 |  Zbl 0249.35070
[4] Bardos, C., and Rauch, J. Maximal positive boundary value problems as limits of singular perturbation problems. Trans. Amer. Math. Soc. 270, 2 (1982), 377–408.  MR 645322 |  Zbl 0485.35010
[5] Beale, J. T. The initial value problem for the Navier-Stokes equations with a free surface.Comm. Pure Appl. Math. 34, 3 (1981), 359–392.  MR 611750 |  Zbl 0464.76028
[6] Beirão da Veiga, H. Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 5, 4 (2006), 907–918.  MR 2246015 |  Zbl 1132.35067
[7] Beirão da Veiga, H., and Crispo, F. Concerning the ${W}^{k,p}$ -inviscid limit for 3-d flows under a slip boundary condition. J. Math. Fluid Mech.
[8] Bony, J.-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sci. Ecole Norm. Sup. (4) 14, 2(1981). Numdam |  MR 631751 |  Zbl 0495.35024
[9] Christodoulou, D. and Lindblad, H. On the motion of the free surface of a liquid. Comm. Pure Appl. Math. 53, 12(2000), 1536–1602.  MR 1780703 |  Zbl 1031.35116
[10] Clopeau, T., Mikelić, A., and Robert, R. On the vanishing viscosity limit for the $2{\rm D}$ incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity 11, 6 (1998), 1625–1636.  MR 1660366 |  Zbl 0911.76014
[11] Coutand, D. and Shkoller S., Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007),829–930.  MR 2291920 |  Zbl 1123.35038
[12] Gérard-Varet, D. and Dormy, E. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23, 2(2010), 591–609  MR 2601044 |  Zbl 1197.35204
[13] Germain, P., Masmoudi, N. and Shatah, J. Global solutions for the gravity water waves in dimension 3, arXiv:0906.5343.
[14] Gisclon, M., and Serre, D. Étude des conditions aux limites pour un système strictement hyberbolique via l’approximation parabolique. C. R. Acad. Sci. Paris Sér. I Math. 319, 4 (1994), 377–382.  MR 1289315 |  Zbl 0808.35075
[15] Grenier, E. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 9(2000),1067–1091.  MR 1761409 |  Zbl 1048.35081
[16] Grenier, E., and Guès, O. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143, 1 (1998), 110–146.  MR 1604888 |  Zbl 0896.35078
[17] Grenier, E., and Rousset, F. Stability of one-dimensional boundary layers by using Green’s functions. Comm. Pure Appl. Math. 54, 11 (2001), 1343–1385.  MR 1846801 |  Zbl 1026.35015
[18] Guès, O. Problème mixte hyperbolique quasi-linéaire caractéristique. Comm. Partial Differential Equations 15, 5 (1990), 595–645.  MR 1070840 |  Zbl 0712.35061
[19] Guès, O., Métivier, G., Williams, M. and Zumbrun, K., Existence and stability of noncharacteristic boundary layers for the compressible Navier-Stokes and viscous MHD equations. Arch. Ration. Mech. Anal. 197, 1(2010), 1–87.  MR 2646814 |  Zbl 1217.35136
[20] Guo, Y. and Nguyen T. A note on the Prandtl boundary layers, arXiv:1011.0130.  MR 2849481
[21] Hörmander, L. Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. (2) 83 (1966), 129–209.  MR 233064 |  Zbl 0132.07402
[22] Iftimie, D., and Planas, G. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity 19, 4 (2006), 899–918.  MR 2214949 |  Zbl 1169.35365
[23] Iftimie, D., and Sueur, F. Viscous boundary layers for the Navier-Stokes equations with the navier slip conditions. Arch. Rat. Mech. Analysis, available online.
[24] Kelliher, J. P. Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38, 1 (2006), 210–232 (electronic).  MR 2217315 |  Zbl pre05029418
[25] Lannes, D.Well-posedness of the water-waves equations, Journal AMS 18 (2005) 605-654.  MR 2138139 |  Zbl 1069.35056
[26] Lindblad, H. Well-posedness for the linearized motion of an incompressible liquid with free surface boundary. Comm. Pure Appl. Math. 56. 2(2003), 153–197.  MR 1934619 |  Zbl 1025.35017
[27] Lindblad, H. Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. of Math. (2) 162. 1 (2005), 109–194.  MR 2178961 |  Zbl 1095.35021
[28] Masmoudi, N. and Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition, preprint 2010, arXiv:1008.1678.  MR 2885569
[29] Masmoudi, N. and Rousset F. Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equation, preprint 2011.
[30] Métivier, G. and Zumbrun K., Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175 2005, 826.  MR 2130346 |  Zbl 1074.35066
[31] Rousset, F. Characteristic boundary layers in real vanishing viscosity limits. J. Differential Equations 210, 1 (2005), 25–64.  MR 2114123 |  Zbl 1060.35015
[32] Sammartino, M., and Caflisch, R. E. Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm. Math. Phys. 192, 2 (1998), 433–461.  MR 1617542 |  Zbl 0913.35102
[33] Shatah, J. and Zeng, C. Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008),698–744.  MR 2388661 |  Zbl 1174.76001
[34] Tani, A., and Tanaka, N. Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Rational Mech. Anal. 130, 4(1995), 303–314.  MR 1346360 |  Zbl 0844.76025
[35] Tartakoff, D. S. Regularity of solutions to boundary value problems for first order systems. Indiana Univ. Math. J. 21 (1971/72), 1113–1129.  MR 440182 |  Zbl 0235.35019
[36] Temam, R., and Wang, X. Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differential Equations 179, 2 (2002), 647–686.  MR 1885683 |  Zbl 0997.35042
[37] Xiao, Y., and Xin, Z. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm. Pure Appl. Math. 60, 7 (2007), 1027–1055.  MR 2319054 |  Zbl 1117.35063
[38] Wu, S. Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc.,12 (1999), 445–495.  MR 1641609 |  Zbl 0921.76017
[39] Wu, S. Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 1(2011), 125–220.  MR 2782254 |  Zbl pre05883623
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