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

Rousset, Frederic

Journées équations aux dérivées partielles (2011), article no. 10, 14 p.

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.

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[10] 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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[15] Grenier, E. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 9(2000),1067–1091. | MR | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[28] Masmoudi, N. and Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition, preprint 2010, arXiv:1008.1678. | MR

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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