arXiv subjects

Ewelina Zatorska

Publications and source records attributed to Ewelina Zatorska.

49 records · Page 3Linked to original sources

arXiv2017

Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions

We provide a numerical study of the macroscopic model of [3] derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodelling process is very fast, the macroscopic model takes the form of a single aggregation diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the 1-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the 2-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modelling of dynamical networks at a large scale.

cond-mat.stat-mech↗

arXiv2016

Transport of congestion in two-phase compressible/incompressible flows

We study the existence of weak solutions to the two-phase model of crowd motion. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime $\varrho^*(t,x)$ achieves a threshold value $\varrho^*(t,x)$ that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier-Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.

math.AP↗

arXiv2016

Kinetic theory of particle interactions mediated by dynamical networks

We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset.

math.AP↗

arXiv2016

On the pressureless damped Euler-Poisson equations with non-local forces: Critical thresholds and large-time behavior

We analyse the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.

math.AP↗

arXiv2016

Existence of weak solutions for compressible Navier-Stokes equations with entropy transport

We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the Navier-Stokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory.

math.AP↗

arXiv2015

From the highly compressible Navier-Stokes equations to the Porous Medium equation - rate of convergence

We consider the one-dimensional Cauchy problem for the Navier-Stokes equations with degenerate viscosity coefficient in highly compressible regime. It corresponds to the compressible Navier-Stokes system with large Mach number equal to $\frac{1}{\sqrt{\varepsilon}}$ for $\varepsilon$ going to $0$. When the initial velocity is related to the gradient of the initial density, a solution to the continuity equation-$ρ_\varepsilon$ converges to the unique solution to the porous medium equation [13,14]. For viscosity coefficient $μ(ρ_\varepsilon)=ρ_\varepsilon^α$ with $α>1$, we obtain a rate of convergence of $ρ_\varepsilon$ in $L^\infty(0,T; H^{-1}(\mathbb{R}))$; for $1<α\leq\frac{3}{2}$ the solution $ρ_\varepsilon$ converges in $L^\infty(0,T;L^2(\mathbb{R}))$. For compactly supported initial data, we prove that most of the mass corresponding to solution $ρ_\varepsilon$ is located in the support of the solution to the porous medium equation. The mass outside this support is small in terms of $\varepsilon$.

math.AP↗

arXiv2015

Existence of stationary weak solutions for the heat conducting flows

The steady compressible Navier--Stokes--Fourier system is considered, with either Dirichlet or Navier boundary conditions for the velocity and the heat flux on the boundary proportional to the difference of the temperature inside and outside. In dependence on several parameters, i.e. the adiabatic constant $γ$ appearing in the pressure law $p(\vr,\vt) \sim \vr^γ+ \vr \vt$ and the growth exponent in the heat conductivity, i.e. $κ(\vt) \sim (1+ \vt^m)$, and without any restriction on the size of the data, the main ideas of the construction of weak and variational entropy solutions for the three-dimensional flows with temperature dependent viscosity coefficients are explained. Further, the case when it is possible to prove existence of solutions with bounded density is reviewed. The main changes in the construction of solutions for the two-dimensional flows are mentioned and finally, results for more complex systems are reviewed, where the steady compressible Navier--Stokes--Fourier equations play an important role.

math.AP↗

arXiv2015

On singular limits arising in the scale analysis of stratified fluid flows

We study the low Mach low Freude numbers limit in the compressible Navier-Stokes equations and the transport equation for evolution of an entropy variable -- the potential temperature $Θ$. We consider the case of well-prepared initial data on "flat" tours and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier-Stokes system and the transport equation for the second order variation of $Θ$

math.AP↗

arXiv2015

Two-velocity hydrodynamics in fluid mechanics: Part I Well posedness for zero Mach number systems

In this paper we prove global in time existence of weak solutions to zero Mach number systems arising in fluid mechanics. Relaxing a certain algebraic constraint between the viscosity and the conductivity introduced in [D. Bresch, E.H. Essoufi, and M. Sy, J. Math. Fluid Mech. 2007] gives a more complete answer to an open question formulated in [P.-L. Lions, Oxford 1998]. A new mathematical entropy shows clearly the existence of two-velocity hydrodynamics with a fixed mixture ratio. As an application of our result we first discuss a model of gaseous mixture extending the results of [P. Embid, Comm. Partial Diff. Eqs. 1987] to the global weak solutions framework. Second, we present the ghost effect system studied by [C.D. Levermore, W. Sun, K. Trivisa, SIAM J. Math. Anal. 2012] and discuss a contribution of the density-dependent heat-conductivity coefficient to the issue of existence of weak solutions.

math.AP↗

arXiv2015

Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global $κ$-entropy solutions to compressible Navier-Stokes systems with degenerate viscosities

This paper addresses the issue of global existence of so-called $κ$-entropy solutions to the Navier-Stokes equations for viscous compressible and barotropic fluids with degenerate viscosities. We consider the three dimensional space domain with periodic boundary conditions. Our solutions satisfy the weak formulation of the mass and momentum conservation equations and also a generalization of the BD-entropy identity called: $κ$-entropy. This new entropy involves a mixture parameter $κ\in (0,1)$ between the two velocities $\bf{u}$ and $\bf{u} + 2\nablaφ(\varrho)$ (the latter was introduced by the first two authors in [C. R. Acad. Sci. Paris 2004]), where $\bf{u}$ is the velocity field and $φ$ is a function of the density $\varrho$ defined by $φ'(s)=μ'(s)/s$. As a byproduct of the existence proof, we show that two-velocity hydrodynamics (in the spirit of S.C. Shugrin 1994) is a possible formulation of a model of barotropic compressible flow with degenerate viscosities. It is used in construction of approximate solutions based on a conservative augmented approximate scheme.

math.AP↗

arXiv2014

Free/Congested Two-Phase Model from Weak Solutions to Multi-Dimensional Compressible Navier-Stokes Equations

We approximate a two--phase model by the compressible Navier-Stokes equations with a singular pressure term. Up to a subsequence, these solutions are shown to converge to a global weak solution of the compressible system with the congestion constraint studied for instance by P.L. Lions and N. Masmoudi [ Annales I.H.P., 1999]. The paper is an extension of the previous result obtained in one-dimensional setting by D. Bresch et al. [ C. R. Acad. Sciences Paris, 2014] to the multi-dimensional case with heterogeneous barrier for the density.

math.AP↗

arXiv2014

Heat-conducting, compressible mixtures with multicomponent diffusion: construction of a weak solution

We investigate a coupling between the compressible Navier-Stokes-Fourier system and the full Maxwell-Stefan equations. This model describes the motion of chemically reacting heat-conducting gaseous mixture. The viscosity coefficients are density-dependent functions vanishing on vacuum and the internal pressure depends on species concentrations. By several levels of approximation we prove the global-in-time existence of weak solutions on the three-dimensional torus.

math.AP↗

arXiv2013

Mixtures: sequential stability of variational entropy solutions

The purpose of this work is to analyze the mathematical model governing motion of $n$-component, heat conducting reactive mixture of compressible gases. We prove sequential stability of weak variational entropy solutions when the state equation essentially depends on the species concentration and the species diffusion fluxes depend on gradients of partial pressures. Of crucial importance for our analysis is the fact that viscosity coefficients vanish on vacuum and the source terms enjoy the admissibility condition dictated by the second law of thermodynamics.

math.AP↗