A stenosis is the narrowing of the artery, this narrowing is usually the result of the formation of an atheromatous plaque infiltrating gradually the artery wall, forming a bump in the ductus arteriosus. This arterial lesion falls within the general context of atherosclerotic arterial disease that can affect the carotid arteries, but also the arteries of the heart (coronary), arteries of the legs (PAD), the renal arteries... It can cause a stroke (hemiplegia, transient paralysis of a limb, speech disorder, sailing before the eye). In this paper we study the bloodplaque and blood-wall interactions using a fluid-structure interaction model. We first propose a 2D analytical study of the generalized Navier-Stokes equations to prove the existence of a weak solution for incompressible non-Newtonian fluids with non standard boundary conditions. Then, coupled, based on the results of the theoretical study approach is given. And to form a realistic model, with high accuracy, additional conditions due to fluid-structure coupling are proposed on the border undergoing inetraction. This coupled model includes (a) a fluid model, where blood is modeled as an incompressible non-Newtonian viscous fluid, (b) a solid model, where the arterial wall and atherosclerotic plaque will be treated as non linear hyperelastic solids, and (c) a fluidstructure interaction (FSI) model where interactions between the fluid (blood) and structures (the arterial wall and atheromatous plaque) are conducted by an Arbitrary Lagrangian Eulerian (ALE) method that allows accurate fluid-structure coupling.

In this paper, we consider the blood flow in a stenosed artery. We give an analytical study of the equations for a non-Newtonian fluid modeling the blood for which the behavior is obeying to Carreau's law. The case we treat is different than classic cases where the total pressure is in the natural boundary conditions. For this, we use the Faedo-Galerkin method to prove the existence of a weak solution for the fluid problem. Then we use a coupled approach between the fluid equations and the solid model of the arterial wall and the atheromatous plaque. A special attention is paid to the effects of the wall motion on the local fluid displacement, on the stresses, and on strains in the diseased arterial wall. These relevant quantities are analyzed extensively through numerical results.

An important task in image processing is the process of filling in missing parts of damaged images based on the information obtained from the surrounding areas. It is called inpainting. The goals of inpainting are numerous such as removing scratches in old photographic image, removing text and logos, restoration of damaged paintings. In this paper we present a nonlinear diffusion model for image inpainting based on a nonlinear partial differential equation as first proposed by Perona and Malik in [8]. In our previous work [3] the existence, uniqueness and regularity of the solution for the proposed mathematical model are established in an Hilbert space. The discretization of the partial differential equation of the proposed model is performed using finite elements method and finite differences method. For finite differences method our model is very simple to implement, fast, and produces nearly identical results to more complex, and usually slower, known methods. However for finite elements method we observe that it requires large computational cost, especially for high-resolution images. To avoid this slowing problem, domain decomposition algorithm has been proposed, aiming to split one large problem into many smaller problems. To illustrate the effective performance of our method, we present some experimental results on several images.

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International audienceA reaction–diffusion system of equations describing the distribution of population density is considered. The existence of pulse solutions is proved by the Leray–Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions. Numerical simulations show that such solutions become stable in the case of global consumption of resources while they are unstable without the integral terms. This model is used to describe human height distribution