Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 4, 11 p.

The aim of this talk is to present recent results obtained with N. Masmoudi on the free surface Navier-Stokes equations with small viscosity.

@article{SLSEDP_2012-2013____A4_0,
author = {Rousset, Fr\'ed\'eric},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:4},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2012-2013},
doi = {10.5802/slsedp.34},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.34/}
}
Rousset, Frédéric. Inviscid limit for free-surface Navier-Stokes equations. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 4, 11 p. doi : 10.5802/slsedp.34. http://archive.numdam.org/articles/10.5802/slsedp.34/

[1] Alazard, T., Burq, N., and Zuily C., On the Cauchy problem for gravity water waves, preprint 2012, arXiv:1212.0626. | MR 2762387

[2] Alazard, T., Burq, N., and Zuily C., On the Water Waves Equations with Surface Tension, Duke Math. J. 158 3 (2011), 413-499. | MR 2805065 | Zbl 1258.35043

[3] 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

[4] 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

[5] 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

[6] 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

[7] 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

[8] 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. | MR 2784899

[9] Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux drivées partielles non linéaires. Ann. Sci. Ecole Norm. Sup. (4) 14, 2(1981). | EuDML 82073 | Numdam | MR 631751 | Zbl 0495.35024

[10] 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

[11] Clopeau, T., Mikelić, A., and Robert, R., On the vanishing viscosity limit for the $2\mathrm{D}$ incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity 11, 6 (1998), 1625–1636. | MR 1660366 | Zbl 0911.76014

[12] 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

[13] Grard-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

[14] Germain, P., Masmoudi, N., and Shatah, J., Global solutions for the gravity water waves in dimension 3, arXiv:0906.5343.

[15] Gisclon, M. and Serre, D., Étude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique. C. R. Acad. Sci. Paris Sér. I Math. 319, 4 (1994), 377–382. | MR 1289315 | Zbl 0808.35075

[16] 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

[17] 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

[18] 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

[19] Guès, O., Problème mixte hyperbolique quasi-linéaire caractéristique. Comm. Partial Differential Equations 15, 5 (1990), 595–645. | MR 1070840 | Zbl 0712.35061

[20] 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

[21] Guo, Y. and Nguyen T., A note on the Prandtl boundary layers, arXiv:1011.0130. | MR 2849481

[22] Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. (2) 83 (1966), 129–209. | MR 233064 | Zbl 0132.07402

[23] 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

[24] Iftimie, D. and Sueur, F., Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch. Rat. Mech. Analysis, available online. | Zbl 1229.35184

[25] 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

[26] Lannes, D., Well-posedness of the water-waves equations, Journal AMS 18 (2005) 605-654. | MR 2138139 | Zbl 1069.35056

[27] 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

[28] 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

[29] Masmoudi, N. and Rousset F., Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Ration. Mech. Anal. 203 (2012), no. 2, 529Ð575. | MR 2885569

[30] Masmoudi, N. and Rousset F., Vanishing viscosity limit for the free surface compressible Navier-Stokes system, in preparation.

[31] Masmoudi, N. and Rousset F., Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equation, preprint 2012, arXiv: 1202.0657. | MR 2885569

[32] 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

[33] Rousset, F., Characteristic boundary layers in real vanishing viscosity limits. J. Differential Equations 210, 1 (2005), 25–64. | MR 2114123 | Zbl 1060.35015

[34] 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

[35] 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

[36] 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

[37] 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

[38] 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

[39] 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

[40] 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

[41] Wu, S., Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 1(2011), 125–220. | MR 2782254 | Zbl 1221.35304