Inviscid limit for free-surface Navier-Stokes equations

Frédéric Rousset[1]

  • [1] IRMAR Université de Rennes 1 campus de Beaulieu 35042 Rennes cedex France

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-11
  • ISSN: 2266-0607

Abstract

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

How to cite

top

Rousset, Frédéric. "Inviscid limit for free-surface Navier-Stokes equations." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-11. <http://eudml.org/doc/275678>.

@article{Rousset2012-2013,
abstract = {The aim of this talk is to present recent results obtained with N. Masmoudi on the free surface Navier-Stokes equations with small viscosity.},
affiliation = {IRMAR Université de Rennes 1 campus de Beaulieu 35042 Rennes cedex France},
author = {Rousset, Frédéric},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {Navier-Stokes equations; free surface},
language = {eng},
pages = {1-11},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Inviscid limit for free-surface Navier-Stokes equations},
url = {http://eudml.org/doc/275678},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Rousset, Frédéric
TI - Inviscid limit for free-surface Navier-Stokes equations
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 11
AB - The aim of this talk is to present recent results obtained with N. Masmoudi on the free surface Navier-Stokes equations with small viscosity.
LA - eng
KW - Navier-Stokes equations; free surface
UR - http://eudml.org/doc/275678
ER -

References

top
  1. Alazard, T., Burq, N., and Zuily C., On the Cauchy problem for gravity water waves, preprint 2012, http://arxiv.org/abs/1212.0626. Zbl1308.35195MR2762387
  2. 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
  3. 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
  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. Zbl0249.35070MR333488
  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. Zbl0485.35010MR645322
  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. Zbl0464.76028MR611750
  7. 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
  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. Zbl1270.35333MR2784899
  9. Bony, J.-M., Calcul symbolique et propagation des singularités pour les équations aux dŽrivées partielles non linéaires. Ann. Sci. Ecole Norm. Sup. (4) 14, 2(1981). Zbl0495.35024MR631751
  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. Zbl1031.35116MR1780703
  11. Clopeau, T., Mikelić, A., and Robert, R., On the vanishing viscosity limit for the 2 D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity 11, 6 (1998), 1625–1636. Zbl0911.76014MR1660366
  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. Zbl1123.35038MR2291920
  13. GŽrard-Varet, D. and Dormy, E., On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23, 2(2010), 591–609 Zbl1197.35204MR2601044
  14. Germain, P., Masmoudi, N., and Shatah, J., Global solutions for the gravity water waves in dimension 3, http://arxiv.org/abs/0906.5343. Zbl1241.35003
  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. Zbl0808.35075MR1289315
  16. Grenier, E., On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 53, 9(2000),1067–1091. Zbl1048.35081MR1761409
  17. 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
  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. Zbl1026.35015MR1846801
  19. Guès, O., Problème mixte hyperbolique quasi-linéaire caractéristique. Comm. Partial Differential Equations 15, 5 (1990), 595–645. Zbl0712.35061MR1070840
  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. Zbl1217.35136MR2646814
  21. Guo, Y. and Nguyen T., A note on the Prandtl boundary layers, http://arxiv.org/abs/1011.0130. Zbl1232.35126MR2849481
  22. Hörmander, L., Pseudo-differential operators and non-elliptic boundary problems. Ann. of Math. (2) 83 (1966), 129–209. Zbl0132.07402MR233064
  23. 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
  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. Zbl1229.35184
  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). Zbl1302.35295MR2217315
  26. Lannes, D., Well-posedness of the water-waves equations, Journal AMS 18 (2005) 605-654. Zbl1069.35056MR2138139
  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. Zbl1025.35017MR1934619
  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. Zbl1095.35021MR2178961
  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. Zbl1286.76026MR2885569
  30. Masmoudi, N. and Rousset F., Vanishing viscosity limit for the free surface compressible Navier-Stokes system, in preparation. Zbl1286.76026
  31. Masmoudi, N. and Rousset F., Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equation, preprint 2012, http://arxiv.org/abs/1202.0657. MR2885569
  32. Métivier, G. and Zumbrun K., Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. 175 2005, 826. Zbl1074.35066MR2130346
  33. Rousset, F., Characteristic boundary layers in real vanishing viscosity limits. J. Differential Equations 210, 1 (2005), 25–64. Zbl1060.35015MR2114123
  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. Zbl0913.35102MR1617542
  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. Zbl1174.76001MR2388661
  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. Zbl0844.76025MR1346360
  37. 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
  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. Zbl0997.35042MR1885683
  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. Zbl1117.35063MR2319054
  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. Zbl0921.76017MR1641609
  41. Wu, S., Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 1(2011), 125–220. Zbl1221.35304MR2782254

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.