| Citation: |
Alain Miranville, Zahraa Taha. ALLEN-CAHN EQUATION BASED ON AN UNCONSTRAINED ORDER PARAMETER WITH SOURCE TERM AND ITS CAHN-HILLIARD LIMIT[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1330-1359. doi: 10.11948/20230128
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ALLEN-CAHN EQUATION BASED ON AN UNCONSTRAINED ORDER PARAMETER WITH SOURCE TERM AND ITS CAHN-HILLIARD LIMIT
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Abstract
Our aim in this paper is to study an Allen-Cahn equation based on a microforce balance and unconstrained order parameter, with the introduction of a source term. We first consider the source term, $ g(s)=\beta s $, and obtain the existence, uniqueness and regularity of solutions. We prove that, on finite time intervals, the solutions converge to those of the Cahn-Hilliard-Oono equation as a small parameter goes to zero and then to those of the original Cahn-Hilliard equation as $ \beta \rightarrow 0^+ $. Then, we consider another source term and obtain similar results. In this case, we prove that the solutions converge to those of a Cahn-Hilliard equation on finite time intervals as a small parameter goes to zero. We finally give some numerical simulations which confirm the theoretical results.
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References
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Alain Miranville, Zahraa Taha. ALLEN-CAHN EQUATION BASED ON AN UNCONSTRAINED ORDER PARAMETER WITH SOURCE TERM AND ITS CAHN-HILLIARD LIMIT[J]. Journal of Applied Analysis & Computation, 2024, 14(3): 1330-1359. doi: 10.11948/20230128
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Figure 1.
$ \beta=0.05 $ -
Figure 2.
$ \beta=0.5 $ -
Figure 3.
$ g(s)=\beta s $ -
Figure 4.
$ \varphi_0(x, y) = 1-\sin^2(x+y) $ -
Figure 5.
$ \varphi_0(x, y) =\dfrac{\cos(x-y)}{x^6+1} $ -
Figure 6.
$ \varepsilon=0 $ -
Figure 7.
$ \varepsilon=0.00001 $ -
Figure 8.
$ \varepsilon=0.0001 $ -
Figure 9.
$ \varepsilon=0.001 $ -
Figure 10.
$ \varphi_0(x, y) = \dfrac{1}{5(x^6+1)(y^5+0.5)} $ -
Figure 11.
$ (\varepsilon, \beta)=(0, 0) $ -
Figure 12.
$ (\varepsilon, \beta)=(0.00001, 0.001) $ -
Figure 13.
$ (\varepsilon, \beta)=(0.0001, 0.01) $ -
Figure 14.
$ (\varepsilon, \beta)=(0.001, 0.1) $ -
Figure 15.
Random initial datum between
$ 0 $ $ 0.5 $ -
Figure 16.
$ \varepsilon=0 $ -
Figure 17.
$ \varepsilon=0.00001 $ -
Figure 18.
$ \varepsilon=0.0001 $ -
Figure 19.
$ \varepsilon=0.001 $ -
Figure 20.
$ \varphi_0(x, y) = 1-\sin^2(x+y) $ -
Figure 21.
$ \varepsilon=0 $ -
Figure 22.
$ \varepsilon=0.0000001 $ -
Figure 23.
$ \varepsilon=0.000001 $ -
Figure 24.
$ \varepsilon=0.00001 $ -
Figure 25.
$ \varepsilon=0.0001 $ -
Figure 26.
$ \varepsilon=0.001 $ -
Figure 27.
Random initial datum between
$ 0 $ $ 0.5 $