An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen--Cahn equation

This paper studies traveling fronts to the Allen–Cahn equation in RN for N ≥ 3. Let (N −2)-dimensional smooth surfaces be the boundaries of compact sets in RN−1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation

Chondroitin sulfates and their binding molecules in the central nervous system

Chondroitin sulfate (CS) is the most abundant glycosaminoglycan (GAG) in the central nervous system (CNS) matrix. Its sulfation and epimerization patterns give rise to different forms of CS, which enables it to interact specifically and with a significant affinity with various signalling molecules in the matrix including growth factors, receptors and guidance molecules. These interactions control numerous biological and pathological processes, during development and in adulthood. In this review, we describe the specific interactions of different families of proteins involved in various physiological and cognitive mechanisms with CSs in CNS matrix. A better understanding of these interactions could promote a development of inhibitors to treat neurodegenerative diseases

Varicella zoster virus glycoprotein C increases chemokine-mediated leukocyte migration

Varicella zoster virus (VZV) is a highly prevalent human pathogen that establishes latency in neurons of the peripheral nervous system. Primary infection causes varicella whereas reactivation results in zoster, which is often followed by chronic pain in adults. Following infection of epithelial cells in the respiratory tract, VZV spreads within the host by hijacking leukocytes, including T cells, in the tonsils and other regional lymph nodes, and modifying their activity. In spite of its importance in pathogenesis, the mechanism of dissemination remains poorly understood. Here we addressed the influence of VZV on leukocyte migration and found that the purified recombinant soluble ectodomain of VZV glycoprotein C (rSgC) binds chemokines with high affinity. Functional experiments show that VZV rSgC potentiates chemokine activity, enhancing the migration of monocyte and T cell lines and, most importantly, human tonsillar leukocytes at low chemokine concentrations. Binding and potentiation of chemokine activity occurs through the C-terminal part of gC ectodomain, containing predicted immunoglobulin-like domains. The mechanism of action of VZV rSgC requires interaction with the chemokine and signalling through the chemokine receptor. Finally, we show that VZV viral particles enhance chemokine-dependent T cell migration and that gC is partially required for this activity. We propose that VZV gC activity facilitates the recruitment and subsequent infection of leukocytes and thereby enhances VZV systemic dissemination in humans

Dislocation dynamics: short time existence and uniqueness of the solution

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. We study the special cases where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton-Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model. AMS Classification: 35F25, 35D05. Keywords: Dislocation dynamics, Peach-Koehler force, eikonal equation, Hamilton-Jacobi equations

Dislocation dynamics described by non-local Hamilton-Jacobi equations

We study a mathematical model describing dislocation dynamics in crystals. This phase-field model is based on the introduction of a core tensor which mollifies the singular field on the core of the dislocation. We present this model in the case of the motion of a single dislocation, without cross-slip. The dynamics of a single dislocation line, moving in its slip plane, is described by an Hamilton-Jacobi equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. Introducing a level sets formulation of this equation, we prove the existence and uniqueness of a continuous viscosity solution when the dislocation stays a graph in one direction. We also propose a numerical scheme for which we prove that the numerical solution converges to the continuous solution