Long time evolution of concentrated Euler flows with planar symmetry

We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original proble

Time evolution of concentrated vortex rings

We study the time evolution of an incompressible fluid with axisymmetry without swirl when the vorticity is sharply concentrated. In particular, we consider $N$ disjoint vortex rings of size $\varepsilon$ and intensity of the order of $|\log\varepsilon|^{-1}$. We show that in the limit $\varepsilon\to 0$, when the density of vorticity becomes very large, the movement of each vortex ring converges to a simple translation, at least for a small but positive time.Comment: 24 pages. This updated version provides a new Appendix B, containing the corrected proof of Lemma 3.1. For the sake of clarity, this proof has already been included in arXiv:2102.07807 (where the results of this article have been extended

Molecular Dynamics Simulation of Vascular Network Formation

Endothelial cells are responsible for the formation of the capillary blood vessel network. We describe a system of endothelial cells by means of two-dimensional molecular dynamics simulations of point-like particles. Cells' motion is governed by the gradient of the concentration of a chemical substance that they produce (chemotaxis). The typical time of degradation of the chemical substance introduces a characteristic length in the system. We show that point-like model cells form network resembling structures tuned by this characteristic length, before collapsing altogether. Successively, we improve the non-realistic point-like model cells by introducing an isotropic strong repulsive force between them and a velocity dependent force mimicking the observed peculiarity of endothelial cells to preserve the direction of their motion (persistence). This more realistic model does not show a clear network formation. We ascribe this partial fault in reproducing the experiments to the static geometry of our model cells that, in reality, change their shapes by elongating toward neighboring cells.Comment: 10 pages, 3 figures, 2 of which composite with 8 pictures each. Accepted on J.Stat.Mech. (2009). Appeared at the poster session of StatPhys23, Genoa, Italy, July 13 (2007

Computing the structured pseudospectrum of a Toeplitz matrix and its extreme points

The computation of the structured pseudospectral abscissa and radius (with respect to the Frobenius norm) of a Toeplitz matrix is discussed and two algorithms based on a low rank property to construct extremal perturbations are presented. The algorithms are inspired by those considered in [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for the unstructured case, but their extension to structured pseudospectra and analysis presents several difficulties. Natural generalizations of the algorithms, allowing to draw significant sections of the structured pseudospectra in proximity of extremal points are also discussed. Since no algorithms are available in the literature to draw such structured pseudospectra, the approach we present seems promising to extend existing software tools (Eigtool, Seigtool) to structured pseudospectra representation for Toeplitz matrices. We discuss local convergence properties of the algorithms and show some applications to a few illustrative examples.Comment: 21 pages, 11 figure

From particle systems to the BGK equation

In [Phys. Rev. 94 (1954), 511-525], P.L. Bhatnagar, E.P. Gross and M. Krook introduced a kinetic equation (the BGK equation), effective in physical situations where the Knudsen number is small compared to the scales where Boltzmann's equation can be applied, but not enough for using hydrodynamic equations. In this paper, we consider the stochastic particle system (inhomogeneous Kac model) underlying Bird's direct simulation Monte Carlo method (DSMC), with tuning of the scaled variables yielding kinetic and/or hydrodynamic descriptions. Although the BGK equation cannot be obtained from pure scaling, it does follow from a simple modification of the dynamics. This is proposed as a mathematical interpretation of some arguments in [Phys. Rev. 94 (1954), 511-525], complementing previous results in [Arch. Ration. Mech. Anal. 240 (2021), 785-808] and [Kinet. Relat. Models 16 (2023), 269-293].Comment: 10 page