Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations

We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection diffusion equations, and some fractional Porous medium equations. In this paper we define weak entropy solutions for this class of equations and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable functions. Then we introduce a monotone conservative numerical scheme and prove convergence toward an Entropy solution in the class of bounded integrable functions of bounded variation. We then extend the well-posedness results to non-local terms based on general L\'evy type operators, and establish some connections to fully non-linear HJB equations. Finally, we present some numerical experiments to give the reader an idea about the qualitative behavior of solutions of these equations

Continuous dependence estimates for nonlinear fractional convection-diffusion equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link with the results in [51,59

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

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

The Liouville theorem and linear operators satisfying the maximum principle

A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$\mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x)$$ and $$\mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z).$$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short.Comment: This is an independent and substantial update of arXiv:1807.01843. Here we treat general operators which could be both local and nonlocal, symmetric and nonsymmetric. 13 pages. v3: Update according to the suggestions of the referees. To appear in Journal de Math\'ematiques Pures et Appliqu\'ee

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