Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvergence of the Allen-Cahn Equation to multi-phase mean-curvature flow
123
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and a volume constraint.
rate research
Read More
We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.
This paper is concerned with a fully nonlinear variant of the Allen-Cahn equation with strong irreversibility, where each solution is constrained to be non-decreasing in time. Main purposes of the paper are to prove the well-posedness, smoothing effect and comparison principle, to provide an equivalent reformulation of the equation as a parabolic obstacle problem and to reveal long-time behaviors of solutions. More precisely, by deriving emph{partial} energy-dissipation estimates, a global attractor is constructed in a metric setting, and it is also proved that each solution $u(x,t)$ converges to a solution of an elliptic obstacle problem as $t to +infty$.
In this paper we prove the uniqueness of the saddle-shaped solution to the semilinear nonlocal elliptic equation $(-Delta)^gamma u = f(u)$ in $mathbb R^{2m}$, where $gamma in (0,1)$ and $f$ is of Allen-Cahn type. Moreover, we prove that this solution is stable whenever $2mgeq 14$. As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone ${(x, x) in mathbb R^{m}times mathbb R^m : |x| = |x|}$ is a stable nonlocal $(2gamma)$-minimal surface in dimensions $2mgeq 14$. Saddle-shaped solutions of the fractional Allen-Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions $2$, $4$ and $6$. Thus, after our result, the stability remains an open problem only in dimensions $8$, $10$, and $12$. The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.
We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.
This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Omegasubsetmathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Delta)^s$ with $sin(0,1/2)$, as defined in Fourier space. In the singular limit $varepsilonto 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Omega$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^prime$-perimeter in $Omega$ for every $s^primein(0,1/2)$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
