American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type
  • CPAA Home
  • This Issue
  • Next Article
    Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux
March  2014, 13(2): 859-880. doi: 10.3934/cpaa.2014.13.859

The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation

Irena Pawłow 1, and  Wojciech M. Zajączkowski 2,

1. 

System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw
2. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw

Received  December 2012 Revised  June 2013 Published  October 2013

We consider again the sixth order Cahn-Hilliard type equation with a nonlinear diffusion, addressed in our previous paper in Commun. Pure Appl. Anal. 10 (2011), 1823--1847. Such PDE arises as a model of oil-water-surfactant mixtures. Applying the approach based on the Bäcklund transformation and the Leray-Schauder fixed point theorem we generalize the existence result of the above mentioned paper by imposing weaker assumptions on the data. Here we prove the global unique solvability of the problem in the Sobolev space $H^{6,1}(\Omega\times(0,T))$ under the assumption that the initial datum is in $H^3(\Omega)$ whereas previously $H^6(\Omega)$-regularity was required. Moreover, we admit a broarder class of nonlinear terms in the free energy potential.
Keywords: global existence., Sixth orfer Cahn-Hilliard type equation with nonlinear diffusion, Bäcklund transformation.
Mathematics Subject Classification: Primary: 35K35, 35K60; Secondary: 35Q72, 35L20.
Citation: Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859
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).   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).   Google Scholar

[3]

O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings,", Nauka, (1975).   Google Scholar

[4]

K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E., 70 (2004).   Google Scholar

[5]

K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth,, Phys. Rev. Lett., 88 (2002).   Google Scholar

[6]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases,, Phys. Rev. E., 50 (1994), 1325.   Google Scholar

[7]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations,, Phys. Rev. E., 47 (1993), 4289.   Google Scholar

[8]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations,, Phys. Rev. E., 47 (1993), 4301.   Google Scholar

[9]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems,, Phys. Rev. Lett., 65 (1990), 1116.   Google Scholar

[10]

G. Gompper and M. Schick, Self-assembling amphiphilic system,, in, 16 (1994).   Google Scholar

[11]

G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures,, Phys. Rev. A, 46 (1992), 4836.   Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. Mod. Phys., 49 (1977), 435.   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.   Google Scholar

[14]

M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation,, SIAM J. Math. Anal., 44 (2012), 3369.   Google Scholar

[15]

M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation,, SIAM J. Appl. Math., 72 (2012), 1343.   Google Scholar

[16]

J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications,", Vol. I, (1972).   Google Scholar

[17]

V. Mitlin, Backlund transformation associated with model B: a new equation describing the evolution of the modulated structure of the order parameter,, Physics Letters A, 327 (2004), 455.   Google Scholar

[18]

I. Pawłow and W. M. Zajączkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,, Commun. Pure Appl. Anal., 10 (2011), 1823.   Google Scholar

[19]

I. Pawłow and W. M. Zajączkowski, On a class of sixth order viscous Cahn-Hilliard type equations,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 517.   Google Scholar

[20]

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).   Google Scholar

[21]

S. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, SIAM J. Math. Anal., 45 (2013), 31.   Google Scholar

[22]

V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations,, Trudy Mat. Inst. Steklov, 70 (1964), 133.   Google Scholar

[23]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type,, Trudy Mat. Inst. Stieklov, 83 (1965), 1.   Google Scholar

show all references

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).   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).   Google Scholar

[3]

O. V. Besov, V. P. Il'in and S. M. Nikolskij, "Integral Representation of Functions and Theorems of Imbeddings,", Nauka, (1975).   Google Scholar

[4]

K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,, Phys. Rev. E., 70 (2004).   Google Scholar

[5]

K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth,, Phys. Rev. Lett., 88 (2002).   Google Scholar

[6]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases,, Phys. Rev. E., 50 (1994), 1325.   Google Scholar

[7]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations,, Phys. Rev. E., 47 (1993), 4289.   Google Scholar

[8]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations,, Phys. Rev. E., 47 (1993), 4301.   Google Scholar

[9]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems,, Phys. Rev. Lett., 65 (1990), 1116.   Google Scholar

[10]

G. Gompper and M. Schick, Self-assembling amphiphilic system,, in, 16 (1994).   Google Scholar

[11]

G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures,, Phys. Rev. A, 46 (1992), 4836.   Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. Mod. Phys., 49 (1977), 435.   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.   Google Scholar

[14]

M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation,, SIAM J. Math. Anal., 44 (2012), 3369.   Google Scholar

[15]

M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation,, SIAM J. Appl. Math., 72 (2012), 1343.   Google Scholar

[16]

J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications,", Vol. I, (1972).   Google Scholar

[17]

V. Mitlin, Backlund transformation associated with model B: a new equation describing the evolution of the modulated structure of the order parameter,, Physics Letters A, 327 (2004), 455.   Google Scholar

[18]

I. Pawłow and W. M. Zajączkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,, Commun. Pure Appl. Anal., 10 (2011), 1823.   Google Scholar

[19]

I. Pawłow and W. M. Zajączkowski, On a class of sixth order viscous Cahn-Hilliard type equations,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 517.   Google Scholar

[20]

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).   Google Scholar

[21]

S. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, SIAM J. Math. Anal., 45 (2013), 31.   Google Scholar

[22]

V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations,, Trudy Mat. Inst. Steklov, 70 (1964), 133.   Google Scholar

[23]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type,, Trudy Mat. Inst. Stieklov, 83 (1965), 1.   Google Scholar

[1]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
[2]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[3]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[4]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[5]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[6]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354
[7]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[8]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267
[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[10]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[11]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[13]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[14]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435
[15]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[16]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[17]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[18]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[19]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[20]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (23)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences