Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

Abstract

where the last term is the contact energy. The modified gradient flow becomes, again after rescaling,As a first attempt to study this problem, we shall consider here the case of constant contact energy, i.e. cV) = 0.Thus, the equation (7) is subject to the Neumann boundary condition |£ (8)In [ conditions about the initial data, prove existence and uniqueness for the linearized system and finally use a fixed point argument to establish a local result for the full problem. Once the linearized system is shown to be well-posed, the rest of the proof follows using standard estimates (see [10]). Thus we shall only show here that, under the assumptions of the theorem, the linearized version of (9)- (12) has a unique solution.After linearizing about the initial conditions we obtain fcri-l,.-,n-2. «=1 i=lHere, and Z7m\ D pm .N n p XiWe want to show that the system (13) subject to the linearized conditions (14) with given right hand sides (which we shall denote by F(t,x)), has a unique solution in C 2+a ([0,T) x 17). For this, let C denote the (diagonal) matrix associated to the system (13) and let B be the matrix of boundary conditions so that (13), (14) 13Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions Markos Katsoulakis* Georgios T. Kossioris* Fernando Reitich* AbstractWe study a sharp-interface model for phase transitions that incorporates the interaction of the phase-boundaries with the walls of a container fi. In this model the interfaces move by their mean curvature and are normal to dQ. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with a transition-layer model. We prove that if fi is convex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in non-convex domains. Mathematics Subject Classifications: 35A05, 53A10

Phase Field Model for Solidification with Boundary Interface Interaction

“…We further illustrate by asymptotic analysis that the relaxational boundary condition leads to a dynamic contact angle relation at the boundary-interface intersection. This relaxational boundary condition and dynamic contact angle relation are more reasonable to study the boundary-interface interaction in phase transition dynamics than Neumann conditions proposed in [10] and stationary contact angle conditions proposed in [4], when the boundary-interface interaction is significant.…”

“…We reformulate (9) in curvilinear coordinates (d, s, t) and rescale d to ρ = d/ε, by using the differential relations (10), to obtain…”

Algorithms for Computing Motion by Mean Curvature

“…Figure 6 shows contour plots of the discrete approximations of (12) obtained with a "good" and "bad" mesh. Note that there are theoretical subtleties associated with Neumann boundary conditions for degenerate second order equations [28]. In general, solutions will either satisfy the Neumann boundary data or satisfy the partial differential equation at a point on the boundary.…”

A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

“…We further illustrate by asymptotic analysis that the relaxational boundary condition leads to a dynamic contact angle relation at the boundary-interface intersection. Compared with the Neumann conditions proposed in [13] or the stationary contact angle conditions proposed in [6], this model with relaxational boundary condition gives the dynamic nature of the contact angle relation in the study of boundary-interface interaction in binary alloy solidification.…”