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

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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 and 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.
[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]	C. Domb, The Critical Point, Taylor and Francis, 1996.
[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]	M. Frémond, Non-smooth Thermomechanics, Springer, 2002.
[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]	C. Domb, The Critical Point, Taylor and Francis, 1996.
[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]	M. Frémond, Non-smooth Thermomechanics, Springer, 2002.
[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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