Article Contents
Article Contents

# On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials

• * Corresponding author: Alain Miranville
• Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [24], comparing the nonconserved Caginalp phase-field model with regular and logarithmic nonlinear terms.

Mathematics Subject Classification: 35B45, 35K55, 35L15.

 Citation:

• Figure 1.  Solution $u$ with $f_H = .1$ and logarithmic potential

Figure 2.  Solution $u$ with $f_H = .1$ and cubic potential

Figure 3.  Solution $u$ with $f_H = .1$ and logarithmic potential

Figure 4.  Solution $u$ with $f_H = .1$ and cubic potential

Figure 5.  Solution $u$ with $f_H = 1.5$ and logarithmic potential

Figure 6.  Solution $u$ with $f_H = 1.5$ and cubic potential

Figure 7.  Solution $H$ with $f_H = 1.5$ and logarithmic potential

Figure 8.  Solution $H$ with $f_H = 1.5$ and cubic potential

Table 1.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .1,\ \tau = .03$ and $f(s) = .83(s^3-.5^2s)$

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:02 0.00147 - 0.00124 - 0.0573 - 0.05544 - 1/4 00:00:22 0.00038 0.98 0.0003 0.98 0.02927 0.48 0.02828 0.49 1/16 00:05:58 9.5e-05 0.99 8.1e-05 0.99 0.01472 0.49 0.01422 0.49 1/64 01:50:22 2.4e-05 0.99 2e-05 0.99 0.00737 0.49 0.00711 0.49 1/256 22:06:19 6e-06 1 5e-06 0.99 0.00369 0.5 0.00357 0.49

Table 2.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .1, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:02 0.00146 - 0.00124 - 0.05725 - 0.05544 - 1/4 00:00:25 0.00038 0.98 0.0003 0.98 0.02927 0.48 0.02828 0.49 1/16 00:07:44 9.5e-05 0.99 8.1e-05 0.99 0.01472 0.49 0.01422 0.49 1/64 01:52:09 2.4e-05 0.99 2e-05 0.99 0.00737 0.49 0.00711 0.49 1/256 23:06:55 6e-06 1 5e-06 0.99 0.00369 0.5 0.00357 0.49

Table 3.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .01, \ \tau = .03$ and $f(s) = .83(s^3-.5^2s)$

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:01 0.00042 - 0.00039 - 0.02458 - 0.02397 - 1/4 00:00:22 0.00011 0.98 0.0001 0.98 0.01249 0.49 0.01215 0.49 1/16 00:07:09 2.7e-05 0.99 2.5e-05 0.99 0.00627 0.49 0.00611 0.49 1/64 01:27:20 7e-06 0.99 6e-06 0.99 0.00314 0.49 0.00306 0.49 1/256 22:16:47 2e-06 1 2e-06 0.99 0.00157 0.5 0.00154 0.49

Table 4.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .01, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:01 0.00042 - 0.00039 - 0.02457 - 0.02397 - 1/4 00:00:24 0.00011 0.98 0.0001 0.98 0.01248 0.49 0.01215 0.49 1/16 00:07:41 2.7e-05 0.99 2.5e-05 0.99 0.00627 0.49 0.00611 0.49 1/64 01:26:11 7e-06 0.99 6e-06 0.99 0.00314 0.49 0.00306 0.49 1/256 23:06:30 2e-06 1 2e-06 0.99 0.00157 0.5 0.00154 0.49

Table 5.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .001, \ \tau = .03$ and $f(s) = .83(s^3-.5^2s)$.

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:01 0.00013 - 0.00012 - 0.01227 - 0.01186 - 1/4 00:00:22 3.4e-05 0.98 3.2e-05 0.98 0.00625 0.49 0.00603 0.49 1/16 00:07:19 9e-06 0.99 8e-06 0.99 0.00314 0.49 0.00303 0.49 1/64 01:29:47 2e-06 0.99 2e-06 0.99 0.00157 0.49 0.00152 0.49 1/256 22:23:29 1e-06 1 1e-06 0.99 0.00079 0.5 0.00076 0.49

Table 6.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .001, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.

 $10^{2}\cdot\delta t$ CPU time $N_{L^2}(u)$ rate $N_{L^2}(H)$ rate $N_{H^1}(u)$ rate $N_{H^1}(H)$ rate 1/1 00:00:01 0.00013 - 0.00012 - 0.01226 - 0.011863 - 1/4 00:00:24 3.4e-05 0.98 3.2e-05 0.98 0.00624 0.49 0.00603 0.49 1/16 00:07:44 9e-06 0.99 8e-06 0.99 0.00314 0.49 0.00303 0.49 1/64 01:29:12 2e-06 0.99 2e-06 0.99 0.00157 0.49 0.00152 0.49 1/256 23:22:36 1e-06 1 1e-06 0.99 0.00079 0.5 0.00076 0.49

Table 7.  Comparison of the convergence of the solution $u$ with $\varepsilon = .1$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log explose $-.41$ $[-.37, .37]$ $.41$ .44 .62 $.94$ explose pol explose $-.40$ $[-.37, .37]$ $.40$ .43 .64 explose explose

Table 8.  Comparison of the convergence of the solution $u$ with $\varepsilon = .01$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log $-.76$ $[-.49, .49]$ $[-.49, .49]$ $[-.49, .49]$ .49 .52 $.67$ $.76$ pol $-.86$ $[-.49, .49]$ $[-.49, .49]$ $[-.49, .49]$ $.49$ $.52$ $.70$ $.86$

Table 9.  Comparison of the convergence of the solution $u$ with $\varepsilon = .001$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log $-.56$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $.53$ $.56$ pol $-.57$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $[-.50, .50]$ $.53$ $.57$

Table 10.  Comparison of the convergence of the solution $u$ with $\varepsilon = .1$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log explose explose explose explose .92 .94 .99 explose pol explose explose explose explose explose .94 explose explose

Table 11.  comp_log_eps_0p01_ln6} Comparison of the convergence of the solution $u$ with $\varepsilon = .01$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log -.95 $[-.93, .93]$ $[-.93, .93]$ $[-.93, .93]$ $[-.93, .93]$ explose .94 .95 pol explose explose explose explose explose explose .96 explose

Table 12.  Comparison of the convergence of the solution $u$ with $\varepsilon = .001$.

 $f_H$ $-35$ $-.1$ $0$ $.1$ $.2$ $1.5$ $15$ $35$ log -.93 $[-.93, .93]$ $[-.93, .93]$ $[-.93, .93]$ $[-.93, .93]$ $[-.93, .93]$ explose .93 pol -.94 explose explose explose explose explose explose .94
•  [1] S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Eqns., 1 (2001), 69-84.  doi: 10.1007/PL00001365. [2] S. Aizicovici, E. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287.  doi: 10.1002/mma.215. [3] D. Brochet, X. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212.  doi: 10.1080/00036819108840173. [4] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. [5] G. Caginalp, An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827. [6] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. [7] P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. [8] P. J. Chen, M. E. Gurtin and W. O. Williams, A note on a non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970.  doi: 10.1007/BF01602278. [9] P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112.  doi: 10.1007/BF01591120. [10] L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129. [11] L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6. [12] L. Cherfils, S. Gatti and A. Miranville, A doubly nonlinear parabolic equation with a singular potential, Discrete Contin. Dyn. Systems S, 4 (2011), 51-66.  doi: 10.3934/dcdss.2011.4.51. [13] R. Chill, E Fasangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462.  doi: 10.1002/mana.200410431. [14] C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving medis, Phys. Review Letters, 94 (2005), 154301. [15] B. Doumbé, Etude de modèles de champs de phase de type Caginalp, Université de Poitiers, 2013. [16] A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226.  doi: 10.1080/01495739.2011.608313. [17] C. G. Gal and M. Grasselli, The nonisothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Systems A, 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009. [18] S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in Differential Equations: Inverse and Direct Problems (Proceedings of the Workshop "Evolutiob Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004), A series of Lecture notes in pure and applied mathematics, 251, A. Favini and A. Lorenzi eds., Chapman & Hall, 2006,149–170. doi: 10.1201/9781420011135.ch9. [19] M. Grasselli, A. Miranville, V. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509.  doi: 10.1002/mana.200510560. [20] M. Grasselli, A. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems A, 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67. [21] M. Grasselli and V. Pata, Existence of a universal attractor for a fully hyperbolic phase-field system, J. Evol. Eqns., 4 (2004), 27-51.  doi: 10.1007/s00028-003-0074-2. [22] M. Grasselli, H. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.  doi: 10.4171/ZAA/1277. [23] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194.  doi: 10.1098/rspa.1991.0012. [24] F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, Freefem++ Manual, 2012. [25] J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169.  doi: 10.1016/j.jmaa.2007.09.041. [26] J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182.  doi: 10.1002/mma.1092. [27] A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Systems S, 7 (2014), 271-306.  doi: 10.3934/dcdss.2014.7.271. [28] A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290.  doi: 10.1016/j.na.2009.01.061. [29] A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894.  doi: 10.1080/00036810903042182. [30] A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150.  doi: 10.1007/s00245-010-9114-9. [31] A. Miranville and R. Quintanilla, A type $\rm III$ phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008.  doi: 10.1016/j.aml.2011.01.016. [32] A. Miranville and R. Quintanilla, A generalization of the Allen-Cahn equation, IMA J. Appl. Math., 80 (2015), 410-430.  doi: 10.1093/imamat/hxt044. [33] A. Miranville and R. Quintanilla, A Caginalp phase-field system based on type Ⅲ heat conduction with two temperatures, Quart. Appl. Math., 74 (2016), 375-398.  doi: 10.1090/qam/1430. [34] A. Miranville and R. Quintanilla, On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397.  doi: 10.1002/mma.3867. [35] A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electronic J. Diff. Eqns., 2002 (2002), 1-28. [36] A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590. [37] R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278.  doi: 10.1080/01495730903310599. [38] G. Sadaka, Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case, J. Numer. Math., 20 (2012), 303-324.  doi: 10.1515/jnum-2012-0016. [39] H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390.  doi: 10.1093/imamat/hxh101. [40] Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693.  doi: 10.3934/cpaa.2005.4.683.

Figures(8)

Tables(12)