Related papers: Ancient shrinking spherical interfaces in the Alle…

Let $M$ be a K\"ahler-Einstein surface with positive scalar curvature. If the initial surface is sufficiently close to a holomorphic curve, we show that the mean curvature flow has a global solution and it converges to a holomorphic curve.

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ \epsilon^2\Delta_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where…

We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres $S^{d_1}\times S^{d_2}$, for conformal field theories whose dual may be approximated by classical…

In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in $\mathbb{R}^2$. We establish a prior gradient estimate for smooth solutions to this…

In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the…

We show that stable solutions $u:\mathbb{R}^4\to (-1,1)$ to the Allen-Cahn equation with bounded energy density (or equivalently, with cubic energy growth) are one-dimensional. This is known to entail important geometric consequences, such…

We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are,…

We develop a new Lagrangian approach --- flow dynamic approach to effectively capture the interface in the Allen-Cahn type equations. The underlying principle of this approach is the Energetic Variational Approach (EnVarA), motivated by…

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the…

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and…

We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the…

We consider the sharp interface limit of a convective Allen-Cahn equation, which can be part of a Navier-Stokes/Allen-Cahn system, for different scalings of the mobility $m_\varepsilon=m_0\varepsilon^\theta$ as $\varepsilon\to 0$. In the…

In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…

We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…

In this note we construct an infinite family of ancient solutions to the Curve Shortening Flow which span the halfplane.

In this paper, we derive a thermodynamically consistent non-isothermal diffuse interface model for phase transition and interface evolution involving heat transfer. This model is constructed by integrating concepts from classical…

We consider bounded solutions of the nonlocal Allen-Cahn equation $$ (-\Delta)^s u=u-u^3\qquad{\mbox{ in }}{\mathbb{R}}^3,$$ under the monotonicity condition $\partial_{x_3}u>0$ and in the genuinely nonlocal regime in…

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic…

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the…

We study the behavior of the solution of a stochastic Allen-Cahn equation $\frac{\partial u_\eps }{\partial t}=\frac 12 \frac{\partial^2 u_\eps }{\partial x^2}+ u_\eps -u_\eps^3+\sqrt\eps\, \dot W$, with Dirichlet boundary conditions on a…