Related papers: Stability for the Surface Diffusion Flow
The global existence and boundedness of solutions to volume-surface reaction diffusion systems with a mass control condition are investigated. Such systems arise typically in e.g. cell biology, ecology or fluid mechanics, when some…
We establish the global existence of weak solutions for a two-species cross-diffusion system, set on the 1-dimensional flat torus, in which the evolution of each species is governed by two mechanisms. The first of these is a diffusion which…
We analyze a diffuse interface model for multi-phase flows of $N$ incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space…
We consider the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature),…
This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean…
We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$…
In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted…
It is shown that any three-dimensional periodic configuration that is strictly stable for the area functional is exponentially stable for the surface diffusion flow and for the Mullins-Sekerka or Hele-Shaw flow. The same result holds for…
In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for…
In this paper we consider a layer of incompressible viscous fluid lying above a flat periodic surface in a uniform gravitational field. The upper boundary of the fluid is free and evolves in time. We assume that a mass of surfactants…
A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz…
We prove stability results for nonlinear diffusion equations of the porous medium and fast diffusion types with respect to the nonlinearity power $m$: solutions with fixed data converge in a suitable sense to the solution of the limit…
In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\frac{\alpha}{\beta}}\sigma_k^{\frac{1}{\beta}}$, where $\sigma_k$ is the…
We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space $\mathbb{H}^{n+1}$. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain…
Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented…
We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the…
We consider a locally constrained curvature flow in a static rotationally symmetric space $\mathbf{N}^{n+1}$, which was firstly introduced by Hu and Li in the hyperbolic space. We prove that if the initial hypersurface is graphical, then…
In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and…
For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the…
This study is concerned with the diffusion of a passive scalar $\Theta(\r,t)$ advected by general $n$-dimensional shear flows $\u=u(y,z,...,t)\hat{x}$ having finite mean-square velocity gradients. The unidirectionality of the incompressible…