# American Institute of Mathematical Sciences

November  2011, 10(6): 1823-1847. doi: 10.3934/cpaa.2011.10.1823

## A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures

 1 Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland 2 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw, Poland

Received  June 2010 Revised  April 2011 Published  May 2011

An initial-boundary-value problem for the sixth order Cahn-Hilliard type equation in 3-D is studied. The problem describes phase transition dynamics in ternary oil-water-surfactant systems. It is based on the Landau-Ginzburg theory proposed for such systems by G. Gompper et al. We prove that the problem under consideration is well posed in the sense that it admits a unique global smooth solution which depends continuously on the initial datum.
Citation: Irena Pawłow, Wojciech M. Zajączkowski. A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1823-1847. doi: 10.3934/cpaa.2011.10.1823
##### References:
 [1] J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506. Google Scholar [2] J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609. Google Scholar [3] O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings," Nauka, Moscow, 1975 (in Russian). Google Scholar [4] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. Google Scholar [5] K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605. Google Scholar [6] P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110(11). Google Scholar [7] G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335. Google Scholar [8] G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300. Google Scholar [9] G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312. Google Scholar [10] G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. Google Scholar [11] G. Gompper and M. Schick, Self-assembling amphiphilic system, in "Phase Transitions and Critical Phenomena" (C. Domb and J. Lebowitz eds.), vol. 16, pages 1-176, London, 1994, Academic Press. Google Scholar [12] G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-sur-factant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. Google Scholar [13] M. D. Korzec, P. L. Evans, A. Münch and B. Wagner, Stationary solutions of driven fourth-and sixth-order Cahn-Hilliard type equations, SIAM J. Appl. Math., 69 (2008), 348-374. Google Scholar [14] M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, (2011), to appear. Google Scholar [15] J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 1972. Google Scholar [16] T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606. Google Scholar [17] V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian). Google Scholar [18] V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Ste-klov, 83 (1965), 1-162 (in Russian). Google Scholar

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##### References:
 [1] J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506. Google Scholar [2] J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609. Google Scholar [3] O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings," Nauka, Moscow, 1975 (in Russian). Google Scholar [4] C. M. Dafermos and L. Hsiao, Global smooth thermomechanical processes in one-dimensional nonlinear thermoviscoelasticity, Nonlinear Anal., 6 (1982), 435-454. Google Scholar [5] K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004), 051605. Google Scholar [6] P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110(11). Google Scholar [7] G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335. Google Scholar [8] G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300. Google Scholar [9] G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312. Google Scholar [10] G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. Google Scholar [11] G. Gompper and M. Schick, Self-assembling amphiphilic system, in "Phase Transitions and Critical Phenomena" (C. Domb and J. Lebowitz eds.), vol. 16, pages 1-176, London, 1994, Academic Press. Google Scholar [12] G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-sur-factant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. Google Scholar [13] M. D. Korzec, P. L. Evans, A. Münch and B. Wagner, Stationary solutions of driven fourth-and sixth-order Cahn-Hilliard type equations, SIAM J. Appl. Math., 69 (2008), 348-374. Google Scholar [14] M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, (2011), to appear. Google Scholar [15] J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 1972. Google Scholar [16] T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606. Google Scholar [17] V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian). Google Scholar [18] V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Ste-klov, 83 (1965), 1-162 (in Russian). Google Scholar
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