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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). 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 |
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