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Strict separation and numerical approximation for a non–local Cahn–Hilliard equation with single–well potential

  • *Corresponding author: Abramo Agosti

    *Corresponding author: Abramo Agosti 

Dedicated to our dear colleague Pierluigi Colli with friendship and admiration

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  • In this paper we study a non–local Cahn–Hilliard equation with singular single–well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction–diffusion type evolution for the nutrient concentration and a non–local convective Cahn–Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn–Hilliard equation only. For the non–local Cahn–Hilliard equation with singular single–well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $ d\leq 3 $. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $ d\leq 3 $, which preserves the physical properties of the continuous solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system.

    Mathematics Subject Classification: Primary: 35K55, 45K05, 65M60, 65R20; Secondary: 35Q92, 92B05.

    Citation:

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  • Figure 1.  Plot of the single well potential (2) corresponding to the value $ \bar{\phi} = 0.6 $

    Figure 2.  Plots of the Lyapunov functionals $ \mathcal{L}_{NL} $ and $ \mathcal{L}_L $ vs time for the non-local and the local model with $ \alpha = 1 $ and $ \phi_0 = 0.3\pm 0.15\iota $, together with inserts showing the plots of $ \phi_h^n $ at different time points

    Figure 3.  Plot of $ \phi_h^n $ at different time points in the case $ \phi_0 = 0.05\pm 0.025\iota $ and $ \alpha = 1 $, for the non-local model (first column) and the local model (second column)

    Figure 4.  Plot of $ \phi_h^n $ at different time points at late times for the non-local models with $ \alpha = 1 $ (left column) and $ \alpha = 2 $ (right column), in the case $ \phi_0 = 0.05\pm 0.025\iota $, together with plots over vertical lines of $ \phi_h^n $. The vertical lines are indicated as dotted lines in the two dimensional plots

    Figure 5.  Plot of $ \phi_h^n $ at different time points in the case $ \phi_0 = 0.3\pm 0.15\iota $ and $ \alpha = 1 $, for the non-local model (first column) and the local model (second column)

    Figure 6.  Plot of $ \phi_h^n $ at different time points at late times for the non-local models with $ \alpha = 1 $ (left column) and $ \alpha = 2 $ (right column), in the case $ \phi_0 = 0.3\pm 0.15\iota $, together with plots over vertical lines of $ \phi_h^n $. The vertical lines are indicated as dotted lines in the two dimensional plots

    Figure 7.  Plot of $ \phi_h^n $ at different time points in the case $ \phi_0 = 0.36\pm 0.072\iota $ and $ \alpha = 1 $, for the non-local model (first column) and the local model (second column)

    Figure 8.  Plot of $ \phi_h^n $ at different time points at late times for the non-local models with $ \alpha = 1 $ (left column) and $ \alpha = 2 $ (right column), in the case $ \phi_0 = 0.36\pm 0.072\iota $, together with plots over vertical lines of $ \phi_h^n $. The vertical lines are indicated as dotted lines in the two dimensional plots

    Figure 9.  Plot of the quantity $ \frac{1}{|\Omega|}\left(\delta_0(\phi_h^n),1\right) $ over time for the non-local models with $ \alpha = 1 $ (left column) and $ \alpha = 2 $ (right column), in the case $ \phi_0 = 0.3\pm 0.15\iota $

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