We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discre...

Our aim in this paper is to study generalizations of the Allen-Cahn equation based on a modification of the Ginzburg-Landau free energy proposed in [25]. In particular, the free energy contains an additional term called Willmore regularization. We prove the existence, uniqueness and regularity of solutions, as well as the existence of the global attractor. Furthermore, we study the convergence ...

This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling motion an incompressible two-phase fluid mixture in presence capillarity effects a bounded smooth domain $\Omega\subset \mathbb{R}^d$, $d=2,3$. We focus on dissipative mixing originating from mass-conserving Allen-Cahn dynamics with physically relevant Flory-Huggins potential. More precisely, we study N...

process was impaired by inflammation.3,9 These observations were confirmed by means of gas analysis by Lindskog and Van Allen.A Collateral respiration was also demonstrated in relation to experimental segmental emphysema produced by valvular bronchial obstruction. Lobar emphysema followed valvular obstruction of lobar bronchi, but no emphysema resulted from this type of obstruction in the prima...

Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-NewtonSteihaug-cg method. In case of an obstacle potential firs...

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We study an Allen–Cahn equation perturbed by a multiplicative stochastic noise that is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive a uniform bound for the diffuse surface area, prove the tightness of solutions in the sharp interface limit, and show the convergence to phase-indicator functions.