Vuk Milisic

Kinetic approximation of a boundary value problem for conservation laws

Summary.We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.

Mathematical modeling of lymphocytes selection in the germinal center

Lymphocyte selection is a fundamental process of adaptive immunity. In order to produce B-lymphocytes with a target antigenic profile, mutation selection and division occur in the germinal center, a specific part of lymph nodes. We introduce in this article a simplified mathematical model of this phenomenon, taking into account the main mechanisms. This model is written as a non-linear, non-local, inhomogeneous second order partial differential equation, for which we develop a mathematical analysis. We assess, mathematically and numerically, in the case of piecewise-constant coefficients, the performance of the biological function by evaluating the duration of this production process as a function of several parameters such as the mutation rate or the selection profile, in various asymptotic regimes.

ON A STRUCTURED MODEL FOR LOAD-DEPENDENT REACTION KINETICS OF TRANSIENT ELASTIC LINKAGES MEDIATING NONLINEAR FRICTION ∗

We consider a microscopic model for friction mediated by transient elastic linkages introduced in [D. Oelz and C. Schmeiser, Arch. Ration. Mech. Anal., 198 (2010), pp. 963--980; D. Oelz, C. Schmeiser, and V. Small, Cell Adh. Migr., 2 (2008), pp. 117--126]. In this study we extend results and the general approach employed in [V. Milisic and D. Oelz, J. Math. Pures Appl. (9), 96 (2011), pp. 484--501]. We introduce a new unknown and reformulate the model. Based on this framework, we derive new a priori estimates. In a first step this approach allows us to reproduce results of our earlier paper concerning the convergence of the system to a macroscopic friction law in the semicoupled case, but under weaker assumptions. Furthermore, we consider the fully coupled case and prove existence and uniqueness of the solution.