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

 Access to full text

 Full (PDF)

1. Alazard, T., Burq, N. and Zuily C. On the Water Waves Equations with Surface Tension, Duke Math. J. 158 3 (2011), 413-499. Zbl1258.35043MR2805065
2. Alinhac, S. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Differential Equations 14, 2(1989), 173–230. Zbl0692.35063MR976971
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. Zbl0249.35070MR333488
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. Zbl0485.35010MR645322
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. Zbl0464.76028MR611750
6. Beirão da Veiga, H. Vorticity and regularity for flows under the Navier boundary condition. Commun. Pure Appl. Anal. 5, 4 (2006), 907–918. Zbl1132.35067MR2246015
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. Zbl1270.35333
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). Zbl0495.35024MR631751
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. Zbl1031.35116MR1780703
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. Zbl0911.76014MR1660366
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. Zbl1123.35038MR2291920
12. Gérard-Varet, D. and Dormy, E. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23, 2(2010), 591–609 Zbl1197.35204MR2601044
13. Germain, P., Masmoudi, N. and Shatah, J. Global solutions for the gravity water waves in dimension 3, arXiv:0906.5343. Zbl1241.35003
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. Zbl0808.35075MR1289315
15. Grenier, E. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 9(2000),1067–1091. Zbl1048.35081MR1761409
16. Grenier, E., and Guès, O. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differential Equations 143, 1 (1998), 110–146. Zbl0896.35078MR1604888
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. Zbl1026.35015MR1846801
18. Guès, O. Problème mixte hyperbolique quasi-linéaire caractéristique. Comm. Partial Differential Equations 15, 5 (1990), 595–645. Zbl0712.35061MR1070840
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. Zbl1217.35136MR2646814
20. Guo, Y. and Nguyen T. A note on the Prandtl boundary layers, arXiv:1011.0130. Zbl1232.35126MR2849481
21. Hörmander, L. Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. (2) 83 (1966), 129–209. Zbl0132.07402MR233064
22. Iftimie, D., and Planas, G. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity 19, 4 (2006), 899–918. Zbl1169.35365MR2214949
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. Zbl1229.35184
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). Zbl1302.35295MR2217315
25. Lannes, D.Well-posedness of the water-waves equations, Journal AMS 18 (2005) 605-654. Zbl1069.35056MR2138139
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. Zbl1025.35017MR1934619
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. Zbl1095.35021MR2178961
28. Masmoudi, N. and Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition, preprint 2010, arXiv:1008.1678. Zbl1286.76026MR2885569
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. Zbl1074.35066MR2130346
31. Rousset, F. Characteristic boundary layers in real vanishing viscosity limits. J. Differential Equations 210, 1 (2005), 25–64. Zbl1060.35015MR2114123
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. Zbl0913.35102MR1617542
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. Zbl1174.76001MR2388661
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. Zbl0844.76025MR1346360
35. Tartakoff, D. S. Regularity of solutions to boundary value problems for first order systems. Indiana Univ. Math. J. 21 (1971/72), 1113–1129. Zbl0235.35019MR440182
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. Zbl0997.35042MR1885683
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. Zbl1117.35063MR2319054
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. Zbl0921.76017MR1641609
39. Wu, S. Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 1(2011), 125–220. Zbl1221.35304MR2782254