Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Diffusion as a singular homogenization of the Frenkel-Kontorova model

International audienceIn this work, we consider a general fully overdamped Frenkel-Kontorova model. This model describes the dynamics of a infinite chain of particles, moving in a periodic landscape. Our aim is to describe the macroscopic behavior of this system. We study a singular limit corresponding to a high density of particles moving in a vanishing periodic landscape. We identify the limit equation which is a nonlinear diffusion equation. Our homogenization approach is done in the framework of viscosity solutions

Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space

On the loss of continuity for super-critical drift-diffusion equations

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local $L^1$ norm of the drift, which is a super-critical quantity.Comment: Minor edit

Hypoxia Inducible Factor Signaling Modulates Susceptibility to Mycobacterial Infection via a Nitric Oxide Dependent Mechanism

Tuberculosis is a current major world-health problem, exacerbated by the causative pathogen, Mycobacterium tuberculosis (Mtb), becoming increasingly resistant to conventional antibiotic treatment. Mtb is able to counteract the bactericidal mechanisms of leukocytes to survive intracellularly and develop a niche permissive for proliferation and dissemination. Understanding of the pathogenesis of mycobacterial infections such as tuberculosis (TB) remains limited, especially for early infection and for reactivation of latent infection. Signaling via hypoxia inducible factor α (HIF-α) transcription factors has previously been implicated in leukocyte activation and host defence. We have previously shown that hypoxic signaling via stabilization of Hif-1α prolongs the functionality of leukocytes in the innate immune response to injury. We sought to manipulate Hif-α signaling in a well-established Mycobacterium marinum (Mm) zebrafish model of TB to investigate effects on the host's ability to combat mycobacterial infection. Stabilization of host Hif-1α, both pharmacologically and genetically, at early stages of Mm infection was able to reduce the bacterial burden of infected larvae. Increasing Hif-1α signaling enhanced levels of reactive nitrogen species (RNS) in neutrophils prior to infection and was able to reduce larval mycobacterial burden. Conversely, decreasing Hif-2α signaling enhanced RNS levels and reduced bacterial burden, demonstrating that Hif-1α and Hif-2α have opposing effects on host susceptibility to mycobacterial infection. The antimicrobial effect of Hif-1α stabilization, and Hif-2α reduction, were demonstrated to be dependent on inducible nitric oxide synthase (iNOS) signaling at early stages of infection. Our findings indicate that induction of leukocyte iNOS by stabilizing Hif-1α, or reducing Hif-2α, aids the host during early stages of Mm infection. Stabilization of Hif-1α therefore represents a potential target for therapeutic intervention against tuberculosis