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Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux
The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation
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 |
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. |
[2] |
J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609. |
[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). |
[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. |
[5] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701. |
[6] |
G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E., 50 (1994), 1325-1335. |
[7] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E., 47 (1993), 4289-4300. |
[8] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E., 47 (1993), 4301-4312. |
[9] |
G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. |
[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). |
[11] |
G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. |
[12] |
P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. |
[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 |
[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 |
[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 |
[16] |
J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 1972. |
[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. |
[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 |
[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 |
[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 |
[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 |
[22] |
V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian). |
[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). |
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. |
[2] |
J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609. |
[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). |
[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. |
[5] |
K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002), 245701. |
[6] |
G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E., 50 (1994), 1325-1335. |
[7] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E., 47 (1993), 4289-4300. |
[8] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E., 47 (1993), 4301-4312. |
[9] |
G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119. |
[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). |
[11] |
G. Gompper and S. Zschocke, Ginzburg-Landau theory of oil-water-surfactant mixtures, Phys. Rev. A, 46 (1992), 4836-4851. |
[12] |
P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. |
[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 |
[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 |
[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 |
[16] |
J.-L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications," Vol. I, II, Springer Verlag, New York, 1972. |
[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. |
[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 |
[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 |
[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 |
[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 |
[22] |
V. A. Solonnikov, A priori estimates for solutions of second order parabolic equations, Trudy Mat. Inst. Steklov, 70 (1964), 133-212 (in Russian). |
[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). |
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