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February  2021, 20(2): 763-782. doi: 10.3934/cpaa.2020289

Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation

Erica Ipocoana 1,, and  Andrea Zafferi 2,

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

* Corresponding author

Received  April 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Fund Project: The work of E. Ipocoana is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), by MIUR through the project FFABR (M. Eleuteri) and by the University of Modena and Reggio Emilia through the project FAR2017 "Equazioni differenziali: problemi evolutivi, variazionali ed applicazioni" (S. Gatti). A. Zafferi acknowledges the funding by the DFG through grant CRC1114 "Scaling Cascades in Complex Systems", Project Number 235221301, Project (C09) "Dynamics of rock dehydration on multiple scales"

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.

Keywords: Cahn-Hilliard, non-isothermal model, regularity of solutions, uniqueness.
Mathematics Subject Classification: Primary:35Q35;Secondary:35D35, 80A22, 35A02, 35B65.
Citation: Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 763-782. doi: 10.3934/cpaa.2020289
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.  Google Scholar

[2]

G. Bankoff, S. H. Davis and A. Oron, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.   Google Scholar

[3]

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York, 2011.  Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Separation, Springer, New York, NY, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[5]

J. W. Cahn, On the spinodal decomposition, Acta Metall., 9 (1961), 795-801.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Domb, The Critical Point, Taylor and Francis, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. M. Elliot and Z. Songmu, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357.  doi: 10.1007/BF00251803.  Google Scholar

[14]

M. Frémond, Non-smooth Thermomechanics, Springer, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Ipocoana, Mathematical Modelling for Life, Ph. D. thesis, in progress. Google Scholar

[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.  Google Scholar

[21]

I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation for supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50.   Google Scholar

[22]

S. K. Ma, Statistical Mechanics, World Scientific, 1985. doi: 10.1142/0073.  Google Scholar

[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.  Google Scholar

[24] Y. P. Raizer and Y. B. Zeldovich, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, 1967.   Google Scholar
[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.  Google Scholar

[26]

S. Tremaine, On the origin of irregular Saturn's rings, Astron. J., 125 (2003), 894-901.   Google Scholar

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.  Google Scholar

[2]

G. Bankoff, S. H. Davis and A. Oron, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.   Google Scholar

[3]

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext. Springer, New York, 2011.  Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Separation, Springer, New York, NY, 1996. doi: 10.1007/978-1-4612-4048-8.  Google Scholar

[5]

J. W. Cahn, On the spinodal decomposition, Acta Metall., 9 (1961), 795-801.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Domb, The Critical Point, Taylor and Francis, 1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. M. Elliot and Z. Songmu, On the Cahn-Hilliard equation, Arch. Ration. Mech. Anal., 96 (1986), 339-357.  doi: 10.1007/BF00251803.  Google Scholar

[14]

M. Frémond, Non-smooth Thermomechanics, Springer, 2002. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Ipocoana, Mathematical Modelling for Life, Ph. D. thesis, in progress. Google Scholar

[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.  Google Scholar

[21]

I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation for supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50.   Google Scholar

[22]

S. K. Ma, Statistical Mechanics, World Scientific, 1985. doi: 10.1142/0073.  Google Scholar

[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.  Google Scholar

[24] Y. P. Raizer and Y. B. Zeldovich, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, 1967.   Google Scholar
[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.  Google Scholar

[26]

S. Tremaine, On the origin of irregular Saturn's rings, Astron. J., 125 (2003), 894-901.   Google Scholar

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