Christian Schmeiser

Hypocoercivity for linear kinetic equations conserving mass

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted

Actin branching in the initiation and maintenance of lamellipodia

L^2

MODEL HIERARCHIES FOR CELL AGGREGATION BY CHEMOTAXIS

norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

The Keller--Segel Model with Logistic Sensitivity Function and Small Diffusivity

Using correlated live-cell imaging and electron tomography we found that actin branch junctions in protruding and treadmilling lamellipodia are not concentrated at the front as previously supposed, but link actin filament subsets in which there is a continuum of distances from a junction to the filament plus ends, for up to at least 1 μm. When branch sites were observed closely spaced on the same filament their separation was commonly a multiple of the actin helical repeat of 36 nm. Image averaging of branch junctions in the tomograms yielded a model for the in vivo branch at 2.9 nm resolution, which was comparable with that derived for the in vitro actin–Arp2/3 complex. Lamellipodium initiation was monitored in an intracellular wound-healing model and was found to involve branching from the sides of actin filaments oriented parallel to the plasmalemma. Many filament plus ends, presumably capped, terminated behind the lamellipodium tip and localized on the dorsal and ventral surfaces of the actin network. These findings reveal how branching events initiate and maintain a network of actin filaments of variable length, and provide the first structural model of the branch junction in vivo. A possible role of filament capping in generating the lamellipodium leaflet is discussed and a mathematical model of protrusion is also presented.

The initial time layer problem and the quasineutral limit in the semiconductor drift-diffusion model

We present partial differential equation (PDE) model hierarchies for the chemotactically driven motion of biological cells. Starting from stochastic differential models, we derive a kinetic formulation of cell motion coupled to diffusion equations for the chemoattractants. We also derive a fluid dynamic (macroscopic) Keller–Segel type chemotaxis model by scaling limit procedures. We review rigorous convergence results and discuss finite-time blow-up of Keller–Segel type systems. Finally, recently developed PDE-models for the motion of leukocytes in the presence of multiple chemoattractants and of the slime mold Dictyostelium Discoideum are reviewed.

The Nonlinear Schrödinger Equation with a Strongly Anisotropic Harmonic Potential

The Keller--Segel model is the classical model for chemotaxis of cell populations. It consists of a drift-diffusion equation for the cell density coupled to an equation for the chemoattractant. Here a variant of this model is studied in one-dimensional position space, where the chemotactic drift is turned off for a limiting cell density by a logistic term and where the chemoattractant density solves an elliptic equation modeling a quasi-stationary balance of reaction and diffusion with production of the chemoattractant by the cells. The case of small cell diffusivity is studied by asymptotic and numerical methods. On a time scale characteristic for the convective effects, convergence of solutions to weak entropy solutions of the limiting nonlinear hyperbolic conservation law is proven. Numerical and analytic evidence indicates that solutions of this problem converge to irregular patterns of cell aggregates separated by entropic shocks from vacuum regions as time tends to infinity. Close to each of these p...

Moment Methods for the Semiconductor Boltzmann Equation on Bounded Position Domains

The classical time-dependent drift-diffusion model for semiconductors is considered for small scaled Debye length (which is a singular perturbation parameter). The corresponding limit is carried out on both the dielectric relaxation time scale and the diffusion time scale. The latter is a quasineutral limit, and the former can be interpreted as an initial time layer problem. The main mathematical tool for the analytically rigorous singular perturbation theory of this paper is the (physical) entropy of the system.

A PHASE PLANE ANALYSIS OF TRANSONIC SOLUTIONS FOR THE HYDRODYNAMIC SEMICONDUCTOR MODEL

The nonlinear Schrodinger equation with general nonlinearity of polynomial growth and harmonic confining potential is considered. More precisely, the confining potential is strongly anisotropic; i.e., the trap frequencies in different directions are of different orders of magnitude. The limit as the ratio of trap frequencies tends to zero is carried out. A concentration of mass on the ground state of the dominating harmonic oscillator is shown to be propagated, and the lower-dimensional modulation wave function again satisfies a nonlinear Schrodinger equation. The main tools of the analysis are energy and Strichartz estimates, as well as two anisotropic Sobolev inequalities. As an application, the dimension reduction of the three-dimensional Gross-Pitaevskii equation is discussed, which models the dynamics of Bose-Einstein condensates.

A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion

Galerkin methods for the semiconductor Boltzmann equation based on moment expansions for a discretization in the velocity direction are studied. For the moment equations, boundary conditions are proposed, which are analogues to inflow and, respectively, reflecting boundary conditions for the Boltzmann equation. Stability and an error estimate are proved for an expansion in terms of Hermite polynomials. Finally, an adaptive numerical implementation is introduced and results of numerical experiments are presented.

Arp2/3 complex is essential for actin network treadmilling as well as for targeting of capping protein and cofilin.

We present an analysis of transonic solutions of the steady state 1-dimensional unipolar hydrodynamic model for semiconductors in the isoentropic case. The approach is based on construction of the orbits of the system in the electron density-electric field phase plane and on representation of discontinuous solutions of the hydrodynamic boundary value problem by a union of trajectory pieces. These pieces are related by shocks obeying jump and entropy conditions. A continuation argument in the length of the semiconductor device under consideration is applied to construct a continuum of sub- and transonic solutions, which contains at least one solution for every positive length. We also present numerical results illustrating the various possible solution profiles. For this we use a regularization of the problem, adding artificial diffusion to obtain singularly perturbed problems which are then solved numerically using continuation in the regularization parameter.