The motion of a fluid in an open channel
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, p. 77-105

We consider a free boundary value problem for a viscous, incompressible fluid contained in an uncovered three-dimensional rectangular channel, with gravity and surface tension, governed by the Navier-Stokes equations. We obtain existence results for the linear and nonlinear time-dependent problem. We analyse the qualitative behavior of the flow using tools of bifurcation theory. The main result is a Hopf bifurcation theorem with k -symmetry.

Classification:  35Q30,  35P99
@article{ASNSP_2006_5_5_1_77_0,
     author = {Bodea, Simina},
     title = {The motion of a fluid in an open channel},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     pages = {77-105},
     zbl = {1105.35073},
     mrnumber = {2240184},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_1_77_0}
}
Bodea, Simina. The motion of a fluid in an open channel. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, pp. 77-105. http://www.numdam.org/item/ASNSP_2006_5_5_1_77_0/

[1] S. Agmon, A. Douglis and N. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. 17 (1964), 35-92. | MR 162050 | Zbl 0123.28706

[2] T. Beale, Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), 307-352. | MR 721189 | Zbl 0545.76029

[3] T. Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math. 34 (1980), 359-392. | MR 611750 | Zbl 0464.76028

[4] S. Bodea, “Oscillations of a Fluid in a Channel”, Ph.D. Thesis, Preprint 2003-13, SFB 359, Ruprecht-Karls University of Heidelberg. | Zbl 1038.76001

[5] M. Dauge, Stationary Stokes ans Navier-Stokes systems on two- or three-dimensional domains with corners, SIAM J. Math. Anal. 20 (1989), 74-97. | MR 977489 | Zbl 0681.35071

[6] R. Dautray and J.-L. Lions, “Mathematical Analysis and Numerical Methods for Science and Technology”, Vol. 3, Springer-Verlag, 1990. | MR 1064315 | Zbl 0766.47001

[7] T. Kato, “Perturbation Theory for Linear Operators”, Springer Verlag, 1976. | MR 407617 | Zbl 0342.47009

[8] D. Kröner, The flow of a fluid with a free boundary and a dynamic contact angle, Z. Angew. Math. Mech.5 (1987), 304-306. | MR 907629 | Zbl 0632.76038

[9] M. Renardy, An Existence theorem for a free surface problem with open boundaries, Comm. Partial Differential Equations 17 (1992), 1387-1405. | MR 1179291 | Zbl 0767.35061

[10] B. Schweizer, Free boundary fluid systems in a semigroup approach and oscillatory behavior, SIAM J. Math. Anal. 28 (1997), 1135-1157. | MR 1466673 | Zbl 0889.35075

[11] B. Schweizer, A well-posed model for dynamic contact angles, Nonlinear Anal. 43 (2001), 109-125. | MR 1784449 | Zbl 0974.35095

[12] J. Socolowsky, The solvability of a free boundary value problem for the stationary Navier-Stokes equations with a dynamic contact line, Nonlinear Anal. 21 (1993), 763-784. | MR 1246506 | Zbl 0853.35134

[13] V. A. Solonnikov, On some free boundary problems for the Navier-Stokes equations with moving contact points and lines, Math. Ann. 302 (1995), 743-772. | MR 1343648 | Zbl 0926.35116

[14] V. A. Solonnikov, Solvability of two dimensional free boundary value problem for the Navier-Stokes equations for limiting values of contact angle, In: “Recent Developments in Partial Differential Equations”. Rome: Aracne. Quad. Mat. 2, 1998, 163-210. | MR 1688371 | Zbl 0932.35161

[15] R. Temam “Navier-Stokes Equations”, North-Holland, Amsterdam, 1977. | Zbl 0383.35057