119 research outputs found
Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt
We show existence and uniqueness of a stationary state for a kinetic
Fokker-Planck equation modelling the fibre lay-down process in the production
of non-woven textiles. Following a micro-macro decomposition, we use
hypocoercivity techniques to show exponential convergence to equilibrium with
an explicit rate assuming the conveyor belt moves slow enough. This work is an
extension of (Dolbeault et al., 2013), where the authors consider the case of a
stationary conveyor belt. Adding the movement of the belt, the global Gibbs
state is not known explicitly. We thus derive a more general hypocoercivity
estimate from which existence, uniqueness and exponential convergence can be
derived. To treat the same class of potentials as in (Dolbeault et al., 2013),
we make use of an additional weight function following the Lyapunov functional
approach in (Kolb et al., 2013)
Nonlinear stability of chemotactic clustering with discontinuous advection
We perform the nonlinear stability analysis of a chemotaxis model of
bacterial self-organization, assuming that bacteria respond sharply to chemical
signals. The resulting discontinuous advection speed represents the key
challenge for the stability analysis. We follow a perturbative approach, where
the shape of the cellular profile is clearly separated from its global motion,
allowing us to circumvent the discontinuity issue. Further, the homogeneity of
the problem leads to two conservation laws, which express themselves in
differently weighted functional spaces. This discrepancy between the weights
represents another key methodological challenge. We derive an improved
Poincar\'e inequality that allows to transfer the information encoded in the
conservation laws to the appropriately weighted spaces. As a result, we obtain
exponential relaxation to equilibrium with an explicit rate. A numerical
investigation illustrates our results
Nonlinear stability of chemotactic clustering with discontinuous advection
We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results
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Emergence in biology and social sciences
Mathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf venation, opinion dynamics, and network formation, to name a few. In this article, we explore how mathematics is able to traverse scales, and in particular its application in modelling collective motion of bacteria driven by chemical signalling
Geometric structure of graph Laplacian embeddings
We analyze the spectral clustering procedure for identifying coarse structure in a data set xâ,âŠ,x_n, and in particular study the geometry of graph Laplacian embeddings which form the basis for spectral clustering algorithms. More precisely, we assume that the data is sampled from a mixture model supported on a manifold M embedded in R^d, and pick a connectivity length-scale Δ>0 to construct a kernelized graph Laplacian. We introduce a notion of a well-separated mixture model which only depends on the model itself, and prove that when the model is well separated, with high probability the embedded data set concentrates on cones that are centered around orthogonal vectors. Our results are meaningful in the regime where Δ=Δ(n) is allowed to decay to zero at a slow enough rate as the number of data points grows. This rate depends on the intrinsic dimension of the manifold on which the data is supported
Uniqueness of stationary states for singular Keller-Segel type models
We consider a generalised KellerâSegel model with non-linear porous medium type diffusion and non-local attractive power law interaction, focusing on potentials that are more singular than Newtonian interaction. We show uniqueness of stationary states (if they exist) in any dimension both in the diffusion-dominated regime and in the fair-competition regime when attraction and repulsion are in balance. As stationary states are radially symmetric decreasing, the question of uniqueness reduces to the radial setting. Our key result is a sharp generalised HardyâLittlewoodâSobolev type functional inequality in the radial setting
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