On the free surface Navier-Stokes equation in the inviscid limit

Rousset, Frederic

Journées équations aux dérivées partielles, (2011), p. 1-14 / Harvested from Numdam

Résumé

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.

Détail

Publié le : 2011-01-01
DOI : https://doi.org/10.5802/jedp.82

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. | 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