On the stationary Cahn-Hilliard equation: Interior spike solutions

We study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures and Liapunov-Schmidt reduction

Homogenization of two fluid flow in porous media

The macroscopic behavior of air and water in porous media is often approximated using Richards’ equation for the fluid saturation and pressure. This equation is parametrized by the hydraulic conductivity and water release curve. In this paper, we use homogenization to derive a general model for saturation and pressure in porous media based on an underlying periodic porous structure. Under an appropriate set of assumptions, i.e., constant gas pressure, this model is shown to reduce to the simpler form of Richards’ equation. The starting point for this derivation is the Cahn-Hilliard phase field equation coupled with Stokes equations for fluid flow. This approach allows us, for the first time, to rigorously derive the water release curve and hydraulic conductivities through a series of cell problems. The method captures the hysteresis in the water release curve and ties the macroscopic properties of the porous media to the underlying geometrical and material properties

Quasistatic nonlinear viscoelasticity and gradient flows

We consider the equation of motion for one-dimensional nonlinear viscoelasticity of strain-rate type under the assumption that the stored-energy function is $\lambda$-convex, which allows for solid phase transformations. We formulate this problem as a gradient flow, leading to existence and uniqueness of solutions. By approximating general initial data by those in which the deformation gradient takes only finitely many values, we show that under suitable hypotheses on the stored-energy function the deformation gradient is instantaneously bounded and bounded away from zero. Finally, we discuss the open problem of showing that every solution converges to an equilibrium state as time $t \to \infty$ and prove convergence to equilibrium under a nondegeneracy condition. We show that this condition is satisfied in particular for any real analytic cubic-like stress-strain function.Comment: 40 pages, 1 figur

Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction-diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs. The new capabilities are then applied to the study of the Allen-Cahn equation, the Swift-Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is each replaced by the spectral fractional Laplacian). Our study reveals some common effects, which contributes to a better understanding of fractional diffusion in generic reaction-diffusion systems. In particular, we investigate the changes in snaking bifurcation diagrams and also study the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order can induce very significant qualitative and quantitative changes in global bifurcation structures

Metastable Bubble Solutions For The Allen-Cahn Equation With Mass Conservation

In a multi-dimensional domain, the slow motion behavior of internal layer solutions with spherical interfaces, referred to as bubble solutions, is analyzed for the nonlocal Allen-Cahn equation with mass conservation. This problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint. The bubble is shown to drift exponentially slowly across the domain, without change of shape, towards the closest point on the boundary of the domain. An explicit ordinary differential equation for the motion of the center of the bubble is derived by extending, to a multi-dimensional setting, the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension. An asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point. An analogy between slow bubble motion and ..