Figure 1.
(a) plots of $A z(s)$ and $K v(s)$ ; (b) parametric plot of $s\mapsto (A z(s),Kv(s))$ . Here $s=t/\tau$ is the rescaled time.
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2017, 10(6): 1519-1537.
doi: 10.3934/dcdss.2017078
Smooth and non-smooth regularizations of the nonlinear diffusion equation
Roma Tre University, Engineering Department -Civil Engineering Section, Via Vito Volterra 62, Rome -00152, Italy |
We illustrate an alternative derivation of the viscous regularization of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc., 324:331-351]. We propose and discuss a new ''non-smooth'' variant of the viscous regularization and we offer an heuristic argument that indicates that this variant should display interesting hysteretic effects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approximation scheme.
Mathematics Subject Classification:
Primary: 35K65, 74N25; Secondary: 74N30.
Citation:
Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete and Continuous Dynamical Systems - S,
2017, 10
(6)
: 1519-1537.
doi: 10.3934/dcdss.2017078
![]()
References:
| [1] |
P. Atkins and J. de Paula,
Atkins' Physical Chemistry W. H. Freeman and Company, New York, 2006. |
| [2] |
G. Barenblatt, M. Bertsch, R. D. Passo and M. Ughi,
A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.
doi: 10.1137/0524082. |
| [3] |
M. Bertsch, P. Podio-Guidugli and V. Valente,
On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pur. Appl., 179 (2001), 331-360.
doi: 10.1007/BF02505962. |
| [4] |
M. Bertsch, F. Smarrazzo and A. Tesei,
Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Analysis PDE, 6 (2013), 1719-1754.
doi: 10.2140/apde.2013.6.1719. |
| [5] |
E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward
parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661, arXiv: 1508.03225.
doi: 10.1142/S0218202517500129. |
| [6] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1002/9781118788295.ch4. |
| [7] |
B. D. Coleman and W. Noll,
The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963), 167-178.
doi: 10.1007/BF01262690. |
| [8] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.
doi: 10.1137/110828526. |
| [9] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Diff. Eq., 254 (2013), 4217-4244.
doi: 10.1016/j.jde.2013.02.014. |
| [10] |
F. P. Duda and G. Tomassetti,
On the effect of elastic distortions on the kinetics of diffusion-induced phase transformations, J. Elasticity, 122 (2016), 179-195.
doi: 10.1007/s10659-015-9539-0. |
| [11] |
C. Elliot and S. Luckhaus, Generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy,
Preprint 887, Institute for Mathematics and its Applications, Minneapolis, 1991. |
| [12] |
C. Elliott and A. Stuart,
Viscous Cahn-Hilliard Equation Ⅱ. Analysis, J. Diff. Eq., 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
| [13] |
C. M. Elliott and H. Garcke,
On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
| [14] |
L. C. Evans and M. Portilheiro,
Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.
doi: 10.1142/S0218202504003763. |
| [15] |
P. J. Flory, Thermodynamics of high polymer solutions,
J. Chem. Phys. 10 (1942), 51. |
| [16] |
E. Fried and M. Gurtin,
Coherent solid-state phase transitions with atomic diffusion: A thermomechanical treatment, J. Stat. Phys., 95 (1999), 1361-1427.
doi: 10.1023/A:1004535408168. |
| [17] |
E. Fried and S. Sellers,
Microforces and the theory of solute transport, Z. angew. Math. Phys, 51 (2000), 732-751.
doi: 10.1007/PL00001517. |
| [18] |
M. E. Gurtin,
Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
| [19] |
M. E. Gurtin, E. Fried and L. Anand,
The Mechanics and Thermodynamics of Continua Cambridge University Press, New York, 2010. |
| [20] |
J. -L. Lions,
Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires Dunod, 1969. |
| [21] |
A. Lucantonio, P. Nardinocchi and L. Teresi,
Transient analysis of swelling-induced large deformations in polymer gels, J. Mech. Phys. Solids, 61 (2013), 205-218.
doi: 10.1016/j.jmps.2012.07.010. |
| [22] |
A. Miranville,
A model of Cahn-Hilliard equation based on a microforce balance, Compt. Rend. Acad. Sci. -Ser. I-Math., 328 (1999), 1247-1252.
doi: 10.1016/S0764-4442(99)80448-0. |
| [23] |
A. Miranville,
Some generalizations of the Cahn-Hilliard equation, Asympt. Anal., 22 (2000), 235-259.
|
| [24] |
A. Miranville, A. Pietrus and J.-M. Rakotoson,
Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asympt. Anal., 16 (1998), 315-345.
|
| [25] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, In Material instabilities in continuum mechanics (Edinburgh, 1985-1986), pages 329-342. Oxford University Press, 1988. |
| [26] |
A. Novick-Cohen and R. L. Pego,
Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.
doi: 10.1090/S0002-9947-1991-1015926-7. |
| [27] |
P. I. Plotnikov,
Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Eq., 30 (1994), 614-622.
|
| [28] |
P. Podio-Guidugli,
Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.
doi: 10.1007/s11587-006-0008-8. |
| [29] |
M. M. Porzio, F. Smarrazzo and A. Tesei,
Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.
doi: 10.1007/s00205-013-0666-0. |
| [30] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Springer Basel, 2013. |
| [31] |
T. Roubíček and G. Tomassetti,
Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst.-B, 19 (2014), 2313-2333.
doi: 10.3934/dcdsb.2014.19.2313. |
| [32] |
T. Roubíček and G. Tomassetti,
Thermomechanics of damageable materials under diffusion: Modelling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572.
doi: 10.1007/s00033-015-0566-2. |
| [33] |
B. E. Sar, S. Fréour, P. Davies and F. Jacquemin,
Accounting for differential swelling in the multi-physics modelling of the diffusive behaviour of polymers, ZAMM Z. Angew. Math. Mech., 94 (2014), 452-460.
doi: 10.1002/zamm.201200272. |
| [34] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [35] |
B. L. T. Thanh, F. Smarrazzo and A. Tesei,
Passage to the limit over small parameters in the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 420 (2014), 1265-1300.
doi: 10.1016/j.jmaa.2014.06.036. |
| [36] |
B. L. T. Thanh, F. Smarrazzo and A. Tesei,
Sobolev regularization of a class of forward-backward parabolic equations, J. Diff. Eq., 257 (2014), 1403-1456.
doi: 10.1016/j.jde.2014.05.004. |
| [37] |
P. Victor,
Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Part. Diff. Eq., 23 (1998), 457-486.
doi: 10.1080/03605309808821353. |
| [38] |
A. Visintin,
Differential Models of Hysteresis Springer Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
| [39] |
A. Visintin,
Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Diff. Eq., 15 (2002), 115-132.
doi: 10.1007/s005260100120. |
show all references
Dedicated to Tomáš Roubíček on the occasion of his 60th birthday
References:
| [1] |
P. Atkins and J. de Paula,
Atkins' Physical Chemistry W. H. Freeman and Company, New York, 2006. |
| [2] |
G. Barenblatt, M. Bertsch, R. D. Passo and M. Ughi,
A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24 (1993), 1414-1439.
doi: 10.1137/0524082. |
| [3] |
M. Bertsch, P. Podio-Guidugli and V. Valente,
On the dynamics of deformable ferromagnets. I. Global weak solutions for soft ferromagnets at rest, Ann. Mat. Pur. Appl., 179 (2001), 331-360.
doi: 10.1007/BF02505962. |
| [4] |
M. Bertsch, F. Smarrazzo and A. Tesei,
Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity, Analysis PDE, 6 (2013), 1719-1754.
doi: 10.2140/apde.2013.6.1719. |
| [5] |
E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward
parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661, arXiv: 1508.03225.
doi: 10.1142/S0218202517500129. |
| [6] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1002/9781118788295.ch4. |
| [7] |
B. D. Coleman and W. Noll,
The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal., 13 (1963), 167-178.
doi: 10.1007/BF01262690. |
| [8] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.
doi: 10.1137/110828526. |
| [9] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity, J. Diff. Eq., 254 (2013), 4217-4244.
doi: 10.1016/j.jde.2013.02.014. |
| [10] |
F. P. Duda and G. Tomassetti,
On the effect of elastic distortions on the kinetics of diffusion-induced phase transformations, J. Elasticity, 122 (2016), 179-195.
doi: 10.1007/s10659-015-9539-0. |
| [11] |
C. Elliot and S. Luckhaus, Generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy,
Preprint 887, Institute for Mathematics and its Applications, Minneapolis, 1991. |
| [12] |
C. Elliott and A. Stuart,
Viscous Cahn-Hilliard Equation Ⅱ. Analysis, J. Diff. Eq., 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
| [13] |
C. M. Elliott and H. Garcke,
On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423.
doi: 10.1137/S0036141094267662. |
| [14] |
L. C. Evans and M. Portilheiro,
Irreversibility and hysteresis for a forward-backward diffusion equation, Math. Mod. Meth. Appl. Sci., 14 (2004), 1599-1620.
doi: 10.1142/S0218202504003763. |
| [15] |
P. J. Flory, Thermodynamics of high polymer solutions,
J. Chem. Phys. 10 (1942), 51. |
| [16] |
E. Fried and M. Gurtin,
Coherent solid-state phase transitions with atomic diffusion: A thermomechanical treatment, J. Stat. Phys., 95 (1999), 1361-1427.
doi: 10.1023/A:1004535408168. |
| [17] |
E. Fried and S. Sellers,
Microforces and the theory of solute transport, Z. angew. Math. Phys, 51 (2000), 732-751.
doi: 10.1007/PL00001517. |
| [18] |
M. E. Gurtin,
Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
| [19] |
M. E. Gurtin, E. Fried and L. Anand,
The Mechanics and Thermodynamics of Continua Cambridge University Press, New York, 2010. |
| [20] |
J. -L. Lions,
Quelques Méthodes de Résolution des Problémes Aux Limites Non Linéaires Dunod, 1969. |
| [21] |
A. Lucantonio, P. Nardinocchi and L. Teresi,
Transient analysis of swelling-induced large deformations in polymer gels, J. Mech. Phys. Solids, 61 (2013), 205-218.
doi: 10.1016/j.jmps.2012.07.010. |
| [22] |
A. Miranville,
A model of Cahn-Hilliard equation based on a microforce balance, Compt. Rend. Acad. Sci. -Ser. I-Math., 328 (1999), 1247-1252.
doi: 10.1016/S0764-4442(99)80448-0. |
| [23] |
A. Miranville,
Some generalizations of the Cahn-Hilliard equation, Asympt. Anal., 22 (2000), 235-259.
|
| [24] |
A. Miranville, A. Pietrus and J.-M. Rakotoson,
Dynamical aspect of a generalized Cahn-Hilliard equation based on a microforce balance, Asympt. Anal., 16 (1998), 315-345.
|
| [25] |
A. Novick-Cohen, On the viscous Cahn-Hilliard equation, In Material instabilities in continuum mechanics (Edinburgh, 1985-1986), pages 329-342. Oxford University Press, 1988. |
| [26] |
A. Novick-Cohen and R. L. Pego,
Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.
doi: 10.1090/S0002-9947-1991-1015926-7. |
| [27] |
P. I. Plotnikov,
Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Diff. Eq., 30 (1994), 614-622.
|
| [28] |
P. Podio-Guidugli,
Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.
doi: 10.1007/s11587-006-0008-8. |
| [29] |
M. M. Porzio, F. Smarrazzo and A. Tesei,
Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.
doi: 10.1007/s00205-013-0666-0. |
| [30] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, second edition, Springer Basel, 2013. |
| [31] |
T. Roubíček and G. Tomassetti,
Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discr. Cont. Dyn. Syst.-B, 19 (2014), 2313-2333.
doi: 10.3934/dcdsb.2014.19.2313. |
| [32] |
T. Roubíček and G. Tomassetti,
Thermomechanics of damageable materials under diffusion: Modelling and analysis, Zeit. angew. Math. Phys., 66 (2015), 3535-3572.
doi: 10.1007/s00033-015-0566-2. |
| [33] |
B. E. Sar, S. Fréour, P. Davies and F. Jacquemin,
Accounting for differential swelling in the multi-physics modelling of the diffusive behaviour of polymers, ZAMM Z. Angew. Math. Mech., 94 (2014), 452-460.
doi: 10.1002/zamm.201200272. |
| [34] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [35] |
B. L. T. Thanh, F. Smarrazzo and A. Tesei,
Passage to the limit over small parameters in the viscous Cahn-Hilliard equation, J. Math. Anal. Appl., 420 (2014), 1265-1300.
doi: 10.1016/j.jmaa.2014.06.036. |
| [36] |
B. L. T. Thanh, F. Smarrazzo and A. Tesei,
Sobolev regularization of a class of forward-backward parabolic equations, J. Diff. Eq., 257 (2014), 1403-1456.
doi: 10.1016/j.jde.2014.05.004. |
| [37] |
P. Victor,
Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Comm. Part. Diff. Eq., 23 (1998), 457-486.
doi: 10.1080/03605309808821353. |
| [38] |
A. Visintin,
Differential Models of Hysteresis Springer Berlin, 1994.
doi: 10.1007/978-3-662-11557-2. |
| [39] |
A. Visintin,
Forward-backward parabolic equations and hysteresis, Calc. Var. Partial Diff. Eq., 15 (2002), 115-132.
doi: 10.1007/s005260100120. |
Figure 2.
Plots of the free energy in (105) and of its derivative (respectively, solid and dashed line).
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