Related papers: Two cases of squares evolving by anisotropic diffu…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and…
We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.
Regularizing effects of surface tension are studied for interfacial waves between a two-dimensional, infinitely-deep and irrotational flow of water and vacuum. The water wave problem under the influence of surface tension is formulated as a…
An exact rotating anisotropic fluid solution and a family of exact rotating anisotropic fluid solutions are presented which satisfy all energy conditions for certain values of their parameters. The components of the Ricci tensor the…
Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical…
A nonstandard system of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krejci and Sprekels as the diffusion coefficient…
We consider the Riemann problem for relativistic flows of polytropic fluids and find relations for the flow characteristics. Evolution of physical quantities take especially simple form for the case of cold magnetized plasmas. We find…
We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale…
We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining…
We show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard…
Enhanced diffusion, which describes the accelerated spread of passive scalars due to the interaction between advection and molecular diffusion, has been extensively studied in simplified geometries, such as uniform shear and radial flows.…
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…
We present a numerical study of classical particles diffusing on a solid surface. The particles' motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particle-surface interaction is described by a periodic…
In this paper a rotating two-fluid model for the propagation of internal waves is introduced. The model can be derived from a rotating-fluid problem by including gravity effects or from a nonrotating one by adding rotational forces in the…
We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching…
In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the…
Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various…
We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the $n$--dimensional flat torus. More precisely, we show that if…
There are three fundamental physical processes that gives rise to the morphology of a surface: deposition, surface diffusion and desorption. The characteristics of the interfaces generated by the combination of deposition and surface…
In this paper, we determine an exact solution to the governing equations in spherical coordinates for an inviscid, incompressible fluid. This solution describes a steady, purely azimuthal equatorial flow with an associated free surface.…