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

Ahmad Makki; Alain Miranville; Georges Sadaka

doi:10.3934/dcdsb.2019019

Article Contents

2019, Volume 24, Issue 3: 1341-1365. Doi: 10.3934/dcdsb.2019019

This issue Previous Article Partial differential inclusions of transport type with state constraints Next Article Asymptotic behavior of the stochastic Keller-Segel equations

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

1.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
2.
Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High Performance Scientific Computing, Xiamen, Fujian, China
3.
Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
4.
Université de Picardie Jules Verne, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR CNRS 7352, Pôle Scientifique, 33, rue Saint Leu, F-80039 Amiens, France

^* Corresponding author: Alain Miranville
^* Corresponding author: Alain Miranville

Received: January 2018

Revised: March 2018

Early access: January 2019

Published: January 2019

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Abstract

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.

Keywords:

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

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

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

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

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

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

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

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

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

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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

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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

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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

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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

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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

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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

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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

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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	$-.86$	$[-.49, .49]$	$[-.49, .49]$	$[-.49, .49]$	$.49$	$.52$	$.70$	$.86$

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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	$-.57$	$[-.50, .50]$	$[-.50, .50]$	$[-.50, .50]$	$[-.50, .50]$	$[-.50, .50]$	$.53$	$.57$

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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

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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

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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

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