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Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation
1. | Università degli Studi di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy |
2. | Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany |
The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial $ L^{\infty} $ estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
The convergence of the solution of the Cahn-Hilliard equation to the solution of Hele-Shaw model, Arch. Ration. Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
G. Bankoff, S. H. Davis and A. Oron,
Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.
|
[3] |
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York, 2011. |
[4] |
M. Brokate and J. Sprekels, Hysteresis and Phase Separation, Springer, New York, NY, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[5] |
J. W. Cahn,
On the spinodal decomposition, Acta Metall., 9 (1961), 795-801.
|
[6] |
J. W. Cahn and J. Hilliard,
Free energy of a non uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
[7] |
D. Cohen and J. M. Murray,
A generalized diffusion model for growth and dispersion in population, J. Math. Biol., 12 (1981), 237-248.
doi: 10.1007/BF00276132. |
[8] |
J. Dockery and I. Klapper,
Role of cohesion in the material description of biofilms, Phys. Rev. E, 74 (2006), 0319021-0319028.
doi: 10.1103/PhysRevE.74.031902. |
[9] | |
[10] |
M. Eleuteri, S. Gatti and G. Schimperna,
Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions, Indiana Univ. Math. J., 68 (2019), 1465-1518.
doi: 10.1512/iumj.2019.68.7788. |
[11] |
M. Eleuteri, E. Rocca and G. Schimperna,
On a non-isothermal diffuse interface model for two phase flows of incompressible fluids, DCDS, 35 (2015), 2497-2522.
doi: 10.3934/dcds.2015.35.2497. |
[12] |
M. Eleuteri, E. Rocca and G. Schimperna,
Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 1431-1454.
doi: 10.1016/j.anihpc.2015.05.006. |
[13] |
C. M. Elliot and Z. Songmu,
On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[14] | |
[15] |
M. Grasselli et al.,
Analysis of the Cahn-Hilliard equation with the chemical potential dependent mobility, Commun. Partial Differ. Equ., 36 (2011), 1193-1238.
doi: 10.1080/03605302.2010.543945. |
[16] |
M. E. Gurtin,
Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[17] |
A. Hönig, B. Niethammer and F. Otto,
On first-order corrections to LSW theory I: Infinite systems, J. Stat. Phys., 119 (2005), 61-122.
doi: 10.1007/s10955-004-2057-2. |
[18] |
E. Knobloch and U. Thiele,
Thin liquid films on a slightly inclined heated plate, Physica D, 190 (2004), 213-248.
doi: 10.1016/j.physd.2003.09.048. |
[19] |
E. Ipocoana, Mathematical Modelling for Life, Ph. D. thesis, in progress. |
[20] |
Ph. Laurençot,
Solutions to a Penrose-Fife model of phase-field type, J. Math. Anal. Appl., 185 (1994), 262-274.
doi: 10.1006/jmaa.1994.1247. |
[21] |
I. M. Lifshitz and V. V. Slyozov,
The kinetics of precipitation for supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50.
|
[22] |
S. K. Ma, Statistical Mechanics, World Scientific, 1985.
doi: 10.1142/0073. |
[23] |
A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, Society for Industrial and Applied Mathematics, U.S., 2019.
doi: 10.1137/1.9781611975925. |
[24] |
Y. P. Raizer and Y. B. Zeldovich, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, 1967.
![]() |
[25] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, AMS. Springer, New York, NY, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[26] |
S. Tremaine,
On the origin of irregular Saturn's rings, Astron. J., 125 (2003), 894-901.
|
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
The convergence of the solution of the Cahn-Hilliard equation to the solution of Hele-Shaw model, Arch. Ration. Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
G. Bankoff, S. H. Davis and A. Oron,
Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.
|
[3] |
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York, 2011. |
[4] |
M. Brokate and J. Sprekels, Hysteresis and Phase Separation, Springer, New York, NY, 1996.
doi: 10.1007/978-1-4612-4048-8. |
[5] |
J. W. Cahn,
On the spinodal decomposition, Acta Metall., 9 (1961), 795-801.
|
[6] |
J. W. Cahn and J. Hilliard,
Free energy of a non uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
[7] |
D. Cohen and J. M. Murray,
A generalized diffusion model for growth and dispersion in population, J. Math. Biol., 12 (1981), 237-248.
doi: 10.1007/BF00276132. |
[8] |
J. Dockery and I. Klapper,
Role of cohesion in the material description of biofilms, Phys. Rev. E, 74 (2006), 0319021-0319028.
doi: 10.1103/PhysRevE.74.031902. |
[9] | |
[10] |
M. Eleuteri, S. Gatti and G. Schimperna,
Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions, Indiana Univ. Math. J., 68 (2019), 1465-1518.
doi: 10.1512/iumj.2019.68.7788. |
[11] |
M. Eleuteri, E. Rocca and G. Schimperna,
On a non-isothermal diffuse interface model for two phase flows of incompressible fluids, DCDS, 35 (2015), 2497-2522.
doi: 10.3934/dcds.2015.35.2497. |
[12] |
M. Eleuteri, E. Rocca and G. Schimperna,
Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids, Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016), 1431-1454.
doi: 10.1016/j.anihpc.2015.05.006. |
[13] |
C. M. Elliot and Z. Songmu,
On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[14] | |
[15] |
M. Grasselli et al.,
Analysis of the Cahn-Hilliard equation with the chemical potential dependent mobility, Commun. Partial Differ. Equ., 36 (2011), 1193-1238.
doi: 10.1080/03605302.2010.543945. |
[16] |
M. E. Gurtin,
Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[17] |
A. Hönig, B. Niethammer and F. Otto,
On first-order corrections to LSW theory I: Infinite systems, J. Stat. Phys., 119 (2005), 61-122.
doi: 10.1007/s10955-004-2057-2. |
[18] |
E. Knobloch and U. Thiele,
Thin liquid films on a slightly inclined heated plate, Physica D, 190 (2004), 213-248.
doi: 10.1016/j.physd.2003.09.048. |
[19] |
E. Ipocoana, Mathematical Modelling for Life, Ph. D. thesis, in progress. |
[20] |
Ph. Laurençot,
Solutions to a Penrose-Fife model of phase-field type, J. Math. Anal. Appl., 185 (1994), 262-274.
doi: 10.1006/jmaa.1994.1247. |
[21] |
I. M. Lifshitz and V. V. Slyozov,
The kinetics of precipitation for supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50.
|
[22] |
S. K. Ma, Statistical Mechanics, World Scientific, 1985.
doi: 10.1142/0073. |
[23] |
A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications, Society for Industrial and Applied Mathematics, U.S., 2019.
doi: 10.1137/1.9781611975925. |
[24] |
Y. P. Raizer and Y. B. Zeldovich, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, 1967.
![]() |
[25] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, AMS. Springer, New York, NY, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[26] |
S. Tremaine,
On the origin of irregular Saturn's rings, Astron. J., 125 (2003), 894-901.
|
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