In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $Bbb R^n+1$ with positive mean curvature is $kappa$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.

This work is a continuation of [3]; it deals with rough boundaries in the simplified context of a Poisson equation. We impose Dirichlet boundary conditions on the periodic microscopic perturbation of a flat edge on one side and natural homogeneous Neumann boundary conditions are applied on the inlet/outlet of the domain. To prevent oscillations on the Neumann-like boundaries, we introduce a microscopic vertical corrector defined in a rough quarter-plane. In [3] we studied a priori estimates in this setting; here we fully develop very weak estimates à la Nečas [17] in the weighted Sobolev spaces on an unbounded domain. We obtain optimal estimates which improve those derived in [3]. We validate these results numerically, proving first order results for boundary layer approximation including the vertical correctors and a little less for the averaged wall-law introduced in the literature [13, 18].

We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate linear parabolic and elliptic equations admit estimates of spatial derivatives up to any given order independent of the mesh size.

In this article, we are interested in the behavior of the FENE (Finitely Extensible Non-Linear Elastic), a constitutive relation within an almost shear flow. As this relation results from a microscopic analysis of the polymer chains, the study presented here is based on new results from the Fokker-Planck equation. Thin flows are generally considered like almost one-dimensional flows and almost shear flows. The study of the FENE model makes it possible to understand which could be the dominant terms in the equations coupling this rheology (the FENE model) and the hydrodynamic (via the conservation equations). Some possible examples of applications are given at the end of this article, for example, in the field of lubrication, the phenomena of boundary layers, the industry of the nanotechnology, biology or Shallow-Water equations.

Using the strong maximum principle, we obtain a constant rank theorem for the Hermitian $k$–convex solutions of complex Laplace equation.