Related papers: Invariant manifolds for a class of degenerate evol…
We construct stable manifolds for a class of degenerate evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium…
We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary…
We establish an instantaneous smoothing property for decaying solutions on the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann…
This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable…
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic…
We consider the question of exponential decay to equilibrium of solutions of an abstract class of degenerate evolution equations on a Hilbert space modeling the steady Boltzmann and other kinetic equations. Specifically, we provide…
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied as a problem of constructing the slow…
For a class of $(N+1)$-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding…
The article discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.
We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by…
In the paper the possible approaches to the rigorous derivation of the Boltzmann kinetic equation with hard sphere collisions from underlying dynamics are considered. In particular, a formalism for the description of the evolution of…
We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear…
We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
The aim of this paper is to emphasize various concepts of dichotomies for evolution equations in Banach spaces, due to the important role they play in the approach of stable, instable and central manifolds. The asymptotic properties of the…
We develop a rigorous formalism for the description of the kinetic evolution of many-particle systems with the dissipative interaction. The relationships of the evolution of a hard sphere system with inelastic collisions described within…
We consider a new approach to the description of the collective behavior of complex systems of mathematical biology based on the evolution equations for observables of such systems. This representation of the kinetic evolution seems, in…
We consider the development of high order asymptotic-preserving linear multistep methods for kinetic equations and related problems. The methods are first developed for BGK-like kinetic models and then extended to the case of the full…