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.

     

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.     

Convergence of solutions of the weighted Allen–Cahn equations to Brakke type flow

In this paper, we study the parabolic Allen–Cahn equation, which has slow diffusion and fast reaction, with a potential K. In particular, the convergence of solutions to a generalized Brakke’s mean curvature flow is established in the limit of a small parameter \( \varepsilon \rightarrow 0\). More precisely, we show that a sequence of Radon measures, associated to energy density of solutions to the parabolic Allen–Cahn equation, converges to a weight measure of an integral varifold. Moreover, the limiting varifold evolves by a vector which is the difference between the mean curvature vector and the normal part of \({\nabla K}/{2K}\).     

Infinite periodic solutions to a class of second-order Sturm–Liouville neutral differential equations

By means of variational structure and _Z2$Z_2$-group index theory, we obtain infinite periodic solutions to a class second-order Sturm–Liouville neutral delay equations

(p(t)x^′(t−sτ))^′−q(t)x(t−sτ)+f(t,x(t),x(t−τ),x(t−2τ),…,x(t−2sτ))=0.${\left(p\left(t\right)x^{\prime }(t-s\tau )\right)}^{\prime }-q\left(t\right)x(t-s\tau )+f(t,x\left(t\right),x(t-\tau ),x(t-2\tau ),\dots ,x(t-2s\tau ))=0\text{.}$

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

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.     

Existence of solutions for a class of Navier–Stokes equations with infinite delay

In this paper, a class of Navier–Stokes equations with infinite delay is considered. It includes delays in the convective and the forcing terms. We discuss the existence of mild and classical solutions for the problem. We establish the results for an abstract delay problem by using the fact that the Stokes operator is the infinitesimal generator of an analytic semigroup of bounded linear operators. Finally, we apply these abstract results to our particular situation.     

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.     

Global attractor for the Cahn–Hilliard/Allen–Cahn system

We study the coupled Cahn–Hilliard/Allen–Cahn problem with constraints, which describes the isothermal diffusion-driven phase transition phenomena in binary systems. Our aim is to show the existence–uniqueness result and to construct the global attractor for the related dynamical 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.     

Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations

In this paper we study the existence of infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. The proof is based on the variant fountain theorem established by Zou. Recent results from the literature are extended.     

Convergence of Exponential Attractors for a Finite Element Approximation of the Allen–Cahn Equation

Abstract

We consider a space semidiscretization of the Allen–Cahn equation by continuous piecewise linear finite elements. For every mesh parameter h, we build an exponential attractor of the dynamical system associated with the approximate equations. We prove that, as h tends to 0, this attractor converges for the symmetric Hausdorff distance to an exponential attractor of the dynamical system associated with the Allen–Cahn equation. We also prove that the fractal dimension of the exponential attractor and of the global attractor is bounded by a constant independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. Here, the perturbation is a space discretization. The case of a time semidiscretization has been analyzed in a previous paper.     

Spectral analysis for transition front solutions in multidimensional Cahn–Hilliard systems

We consider the spectrum associated with the linear operator obtained when a Cahn–Hilliard system on Rⁿ${\mathbb{R}}^n$ is linearized about a planar transition front solution. In the case of single Cahn–Hilliard equations on Rⁿ${\mathbb{R}}^n$, it's known that under general physical conditions the leading eigenvalue moves into the negative real half plane at a rate |ξ|³${|\xi |}^3$, where ξ is the Fourier transform variable corresponding with components transverse to the wave. Moreover, it has recently been verified that for single equations this spectral behavior implies nonlinear stability. In the current analysis, we establish that the same cubic rate law holds for a broad range of multidimensional Cahn–Hilliard systems. The analysis of nonlinear stability will be carried out separately.     

Boundary interface for the Allen–Cahn equation

We consider the Allen–Cahn equation