# Mathematical analysis of the stabilization of lamellar phases by a shear stress

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 7, page 239-267
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topTorri, V.. " Mathematical analysis of the stabilization of lamellar phases by a shear stress ." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 239-267. <http://eudml.org/doc/90620>.

@article{Torri2010,

abstract = {
We consider a 2D mathematical model describing the motion of a
solution of surfactants submitted to a high shear stress in a
Couette-Taylor system. We are interested in a stabilization process
obtained thanks to the shear. We prove that, if the shear stress is
large enough, there exists global in time solution for small
initial data and that the solution
of the linearized system (controlled by a nonconstant parameter) tends
to 0 as t goes to infinity. This
explains rigorously some experiments.
},

author = {Torri, V.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Stabilization; shear stress; Couette system; global
solution.; global solution; energy estimates; local existence},

language = {eng},

month = {3},

pages = {239-267},

publisher = {EDP Sciences},

title = { Mathematical analysis of the stabilization of lamellar phases by a shear stress },

url = {http://eudml.org/doc/90620},

volume = {7},

year = {2010},

}

TY - JOUR

AU - Torri, V.

TI - Mathematical analysis of the stabilization of lamellar phases by a shear stress

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 7

SP - 239

EP - 267

AB -
We consider a 2D mathematical model describing the motion of a
solution of surfactants submitted to a high shear stress in a
Couette-Taylor system. We are interested in a stabilization process
obtained thanks to the shear. We prove that, if the shear stress is
large enough, there exists global in time solution for small
initial data and that the solution
of the linearized system (controlled by a nonconstant parameter) tends
to 0 as t goes to infinity. This
explains rigorously some experiments.

LA - eng

KW - Stabilization; shear stress; Couette system; global
solution.; global solution; energy estimates; local existence

UR - http://eudml.org/doc/90620

ER -

## References

top- O.V. Besov, V.P. Il'in and S.M. Nikol'skii, Integral representations of functions and embeddings theorems, Vol. 1. V.H. Winston and Sons.
- A. Babin, B. Nicolaenko and A. Mahalov, Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal.15 (1997) 103-150.
- A. Colin, T. Colin, D. Roux and A.S. Wunenburger, Undulation instability under shear: A model to explain the different orientation of a lamellar phase under shear. European J. Soft Condensed Matter (to appear).
- A. de Bouard and J.C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaré Anal. Non Linéaire14 (1997) 211-236.
- O. Diat, D. Roux and F. Nallet. J. Phys. II France3 (1993) 1427.
- O. Diat and D. Roux. J. Phys. II France3 (1993) 9.
- I. Gallagher, Asymptotic of solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations150 (1998) 363-384.
- E. Grenier, Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl.76 (1997) 477-498.
- O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach (1963).
- J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969).
- J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968).
- J. Simon, Compact sets in the Spaces Lp(0,T;B). Ann. Mat. Pura Appl.146 (1987) 65-96.
- R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, Appl. Math. Sci.68 (1997).

## NotesEmbed ?

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