An efficient reproducing kernel method for solving the Allen–Cahn equation

In this paper, an efficient reproducing kernel method combined with the finite difference method and the Quasi-Newton method is proposed to solve the Allen–Cahn equation. Based on the Legendre polynomials, we construct a new reproducing kernel function with polynomial form. We prove that the semi-scheme can preserve the energy dissipation property unconditionally. Numerical experiments are given to show the efficiency and validity of the proposed scheme.     

Large deviation principle for a stochastic Allen–Cahn equation

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.     

Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation

In this paper, we use finite difference methods for solving the Allen–Cahn equation that contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three-level scheme in time and a second-order finite difference approach in space, and establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.     

Boundary interface for the Allen–Cahn equation

We consider the Allen–Cahn equation

Layered solutions for a fractional inhomogeneous Allen–Cahn equation

We consider the problem

$$\varepsilon^{2s} (-\partial_{xx})^s \tilde{u}(\tilde{x}) -V(\tilde{x})\tilde{u}(\tilde{x})(1-\tilde{u}^2(\tilde{x}))=0 \quad{\rm in} \mathbb{R},$$

where \({(-\partial_{xx})^s}\) denotes the usual fractional Laplace operator, \({\varepsilon > 0}\) is a small parameter and the smooth bounded function V satisfies \({{\rm inf}_{\tilde{x} \in \mathbb{R}}V(\tilde{x}) > 0}\). For \({s\in(\frac{1}{2},1)}\), we prove the existence of separate multi-layered solutions for any small \({\varepsilon}\), where the layers are located near any non-degenerate local maximal points and non-degenerate local minimal points of function V. We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of V.

     

Global solution branches for a nonlocal Allen–Cahn equation

We consider the Neumann problem of a 1D stationary Allen–Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen–Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.     

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation \( \frac{\partial f}{\partial t}=\triangle f-(f^3-f)\) on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.     

Error control for the approximation of Allen–Cahn and Cahn–Hilliard equations with a logarithmic potential

A fully computable upper bound for the finite element approximation error of Allen–Cahn and Cahn–Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.     

Minimisers of the Allen–Cahn equation on hyperbolic graphs

We investigate minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic behaviours. For a phase field model on a hyperbolic graph, such solutions describe energy-minimising steady-state phase transitions that converge towards prescribed phases given by the asymptotic directions on the graph.     

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to \(-\varepsilon ^2\Delta u + W'(u)=0\) on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if \(\varepsilon \) is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as \(\varepsilon \rightarrow 0\) and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed \(\varepsilon \), as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as \(\varepsilon \rightarrow 0\), around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with \(\mathrm{Ric}_M>0\), the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.     

Two-step discretization method for 2D/3D Allen–Cahn equation based on RBF-FD scheme

In this article, a two-step discretization method based on multi-quadrics (MQ) radial basis function (RBF) is presented for solving Allen–Cahn (AC) equation with integer derivative for time and space. In the first step, backward Euler formula with Newton iterative method is used to discrete the time direction of AC equation. And RBF method is applied in space for solving semi-discrete linearized problem on a coarse mesh. In the second step, finite difference (FD) and radial basis function-finite difference (RBF-FD) methods are used to solve the problem on a fine mesh, respectively. Numerical tests for the equation are obtained to verify the feasibility and computational efficiency of the considered process. In addition, the comparison between FD and RBF-FD shows that solutions obtained by RBF-FD are higher accuracy.     

The approximation of a Crank–Nicolson scheme for the stochastic Navier–Stokes equations

In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized _L2$L_2$-projection operator. Optimal strong convergence error estimates in the _L2$L_2$ norm are obtained.     

Interface foliation for an inhomogeneous Allen–Cahn equation in Riemannian manifolds

Let ${(\mathcal{M}, \tilde{g})}$ be an N-dimensional smooth compact Riemannian manifold. We consider the problem ${\varepsilon^2 \triangle_{\tilde{g}} \tilde{u} + V(\tilde{z})\tilde{u}(1-\tilde{u}^2)=0\; {\rm in}\; \mathcal{M}}$ , where ${\varepsilon > 0}$ is a small parameter and V is a positive, smooth function in ${\mathcal{M}}$ . Let ${\kappa \subset \mathcal{M}}$ be an (N ? 1)-dimensional smooth submanifold that divides ${\mathcal{M}}$ into two disjoint components ${\mathcal{M}_{\pm}}$ . We assume κ is stationary and non-degenerate relative to the weighted area functional ${\int_{\kappa}V^{\frac{1}{2}}}$ . For each integer m ≥ 2, we prove the existence of a sequence ${\varepsilon = \varepsilon_\ell \rightarrow 0}$ , and two opposite directional solutions with m-transition layers near κ, whose mutual distance is ${{\rm O}(\varepsilon | \log \varepsilon | )}$ . Moreover, the interaction between neighboring layers is governed by a type of Jacobi–Toda system.     

Singular stochastic Allen–Cahn equations with dynamic boundary conditions

We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented.