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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), 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]

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

[5]

K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701. Google Scholar

[6]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E., 50 (1994), 1325-1335. Google Scholar

[7]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E., 47 (1993), 4289-4300. Google Scholar

[8]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E., 47 (1993), 4301-4312. Google Scholar

[9]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. Google Scholar

[10]

G. Gompper and M. Schick, Self-assembling amphiphilic system, in "Phase Transitions and Critical Phenomena" (eds. C. Domb and J. Lebowitz), Academic Press, London, 16 (1994). Google Scholar

[11]

G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. 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. Available from: http://dx.doi.org/10.1137/070710949 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), 3369-3387. Available from: http://dx.doi.org/10.1137/100817590 Google Scholar

[15]

M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72 (2012), 1343-1360. Available from: http://dx.doi.org/10.1137/110834123 Google Scholar

[16]

J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 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), 455-460. 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), 1823-1847. Available from: http://dx.doi.org/10.3934/cpaa.2011.10.1823 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), 517-546. Available from: http://dx.doi.org/10.3934/dcdss.2013.6.517 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), 021606. Available from: http://dx.doi.org/10.1137/110835608 Google Scholar

[21]

S. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. Available from: http://dx.doi.org/10.1137/110835608 Google Scholar

[22]

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

[23]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Stieklov, 83 (1965), 1-162 (in Russian). 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), 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]

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

[5]

K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701. Google Scholar

[6]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E., 50 (1994), 1325-1335. Google Scholar

[7]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E., 47 (1993), 4289-4300. Google Scholar

[8]

G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E., 47 (1993), 4301-4312. Google Scholar

[9]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. Google Scholar

[10]

G. Gompper and M. Schick, Self-assembling amphiphilic system, in "Phase Transitions and Critical Phenomena" (eds. C. Domb and J. Lebowitz), Academic Press, London, 16 (1994). Google Scholar

[11]

G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. 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. Available from: http://dx.doi.org/10.1137/070710949 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), 3369-3387. Available from: http://dx.doi.org/10.1137/100817590 Google Scholar

[15]

M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72 (2012), 1343-1360. Available from: http://dx.doi.org/10.1137/110834123 Google Scholar

[16]

J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 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), 455-460. 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), 1823-1847. Available from: http://dx.doi.org/10.3934/cpaa.2011.10.1823 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), 517-546. Available from: http://dx.doi.org/10.3934/dcdss.2013.6.517 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), 021606. Available from: http://dx.doi.org/10.1137/110835608 Google Scholar

[21]

S. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. Available from: http://dx.doi.org/10.1137/110835608 Google Scholar

[22]

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

[23]

V. A. Solonnikov, Boundary value problems for linear parabolic systems of differential equations of general type, Trudy Mat. Inst. Stieklov, 83 (1965), 1-162 (in Russian). Google Scholar

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