Mathematical analysis of the stabilization of lamellar phases by a shear stress
ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 239-267.

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.

DOI: 10.1051/cocv:2002010
Classification: 35B35,  35Q35,  76E05,  76U05
Keywords: stabilization, shear stress, Couette system, global solution
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     title = {Mathematical analysis of the stabilization of lamellar phases by a shear stress},
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     pages = {239--267},
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Torri, V. Mathematical analysis of the stabilization of lamellar phases by a shear stress. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 239-267. doi : 10.1051/cocv:2002010. http://archive.numdam.org/articles/10.1051/cocv:2002010/

[1] 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.

[2] 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. | Zbl

[3] 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).

[4] A. De Bouard and J.C. Saut, Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 211-236. | Numdam | MR | Zbl

[5] O. Diat, D. Roux and F. Nallet. J. Phys. II France 3 (1993) 1427.

[6] O. Diat and D. Roux. J. Phys. II France 3 (1993) 9.

[7] I. Gallagher, Asymptotic of solutions of hyperbolic equations with a skew-symmetric perturbation. J. Differential Equations 150 (1998) 363-384. | MR | Zbl

[8] E. Grenier, Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. 76 (1997) 477-498. | Zbl

[9] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach (1963). | MR | Zbl

[10] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod (1969). | MR | Zbl

[11] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Dunod (1968). | Zbl

[12] J. Simon, Compact sets in the Spaces L p (0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl

[13] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, Appl. Math. Sci. 68 (1997). | MR | Zbl

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