no. 1
The motion of a fluid in an open channel
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 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 ℤ 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},
     pages = {77--105},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     mrnumber = {2240184},
     zbl = {1105.35073},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2006_5_5_1_77_0/}
}
TY  - JOUR
AU  - Bodea, Simina
TI  - The motion of a fluid in an open channel
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 77
EP  - 105
VL  - 5
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2006_5_5_1_77_0/
LA  - en
ID  - ASNSP_2006_5_5_1_77_0
ER  - 
%0 Journal Article
%A Bodea, Simina
%T The motion of a fluid in an open channel
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 77-105
%V 5
%N 1
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2006_5_5_1_77_0/
%G en
%F 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, Série 5, Tome 5 (2006) no. 1, pp. 77-105. http://archive.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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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