MODERATE DEVIATIONS FROM HYDRODYNAMIC LIMIT OF A GINZBURG-LANDAU MODEL

The authors consider the moderate deviations of hydrodynamic limit for Ginzburg-Landau models. The moderate deviation principle of hydrodynamic limit for a specific Ginzburg-Landau model is obtained and an explicit formula of the rate function is derived.

Hydrodynamic Limit for Exclusion Processes

The exclusion process,sometimes called Kawasaki dynamics or lattice gas model,describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each site.We mostly discuss the symmetric and reversible case.The weakly asymmetric case recently attracts attention related to KPZ equation;cf.Bertini and Giacomin(Commun Math Phys 183:571–607,1995)for a simple exclusion case and Gonçalves and Jara(Arch Ration Mech Anal 212:597–644,2014)for an exclusion process with speed change,see also Gonçalves et al.(Ann Probab 43:286–338,2015),Gubinelli and Perkowski(J Am Math Soc 31:427–471,2018).In Sect.1,as a warm-up,we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time scaling.From this model,one can derive a linear heat equation and a stochastic partial differential equation(SPDE)in the limit,respectively.Section 2 is devoted to the entropy method originally invented by Guo et al.(Commun Math Phys 118:31–59,1988).We consider the exclusion process with speed change,in which the jump rate of a particle depends on the configuration nearby the particle.This gives a non-trivial interaction among particles.We study only the case that the jump rate satisfies the so-called gradient condition.The hydrodynamic limit,which leads to a nonlinear diffusion equation,follows from the local ergodicity or the local equilibrium of the system,and this is shown by establishing one-block and twoblock estimates.We also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs principle.Section 3 explains the relative entropy method originally due to Yau(Lett Math Phys 22:63–80,1991).This is a variant of GPV method and gives another proof for the hydrodynamic limit.The difference between these two methods is as follows.Let N^(d)be the volume of the domain on which the system is defined(typically,d-dimensional discrete box with side length N)and denote the(relative)entropy by H.Then,H relative to a global equilibrium behaves as H=O(N^(d))(or entropy per volume is O(1))as N→∞.GPV method rather relies on the fact that the entropy production I,which is the time derivative of H,behaves as O(N^(d−2))so that I per volume is o(1),and this characterizes the limit measures.On the other hand,Yau’s method shows H=o(Nd)for H relative to local equilibria so that the entropy per volume is o(1)and this proves the hydrodynamic limit.In Sect.4,we considerKawasaki dynamics perturbed by relatively largeGlauber effect,which allows creation and annihilation of particles.This leads to the reaction–diffusion equation in the hydrodynamic limit.We discuss especially the equation with reaction term of bistable type and the problem related to the fast reaction limit or the sharp interface limit leading to the motion by mean curvature.We apply the estimate on the relative entropy due to Jara and Menezes(Non-equilibrium fluctuations of interacting particle systems,2017;Symmetric exclusion as a random environment:invariance principle,2018),which is actually obtained as a combination of GPV and Yau’s estimates.This makes possible to study the hydrodynamic limit for microscopic systems with another diverging factors apart from that caused by the space–time scaling.

ASYMPTOTIC LIMITS OF ONE-DIMENSIONAL HYDRODYNAMIC MODELS FOR PLASMAS AND SEMICONDUCTORS

This paper studies the zero-electron-mass limit, the quasi-neutral limit and the zero-relaxation-time limit in one-dimensional hydrodynamic models of Euler-Poisson system for plasmas and semiconductors. For each limit in the steady-state models, the author proves the strong convergence of the sequence of solutions and gives the corresponding convergence rate. In the time-dependent models, the author shows some useful estimates for the quasi-neutral limit and the zero-electron-mass limit. This study completes the analysis made in [11,12,13,14,19].

Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.

Recovering Navier–Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation

This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.Specifically the hydrodynamics limit is performed by employing different time and space scalings.The paper shows that,depending on the magnitude of the parameters which define the scaling,the macroscopic quantities(number density,mean velocity and local temperature)are solutions of the acoustic equation,the linear incompressible Euler equation and the incompressible Navier–Stokes equation.The derivation is formally tackled by the recent moment method proposed by[C.Bardos,et al.,J.Stat.Phys.63(1991)323]and the results generalize the analysis performed in[C.Bianca,et al.,Commun.Nonlinear Sci.Numer.Simulat.29(2015)240].

DIFFUSIVE LIMITS OF THE BOLTZMANN EQUATION IN BOUNDED DOMAIN

The goal of this paper is to study the important diffusive expansion via an alternative mathematical approach other than that in [21].

Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

This paper generalizes the exponential Runge-Kutta asymptotic preserving(AP)method developed in[G.Dimarco and L.Pareschi,SIAM Numer.Anal.,49(2011),pp.2057–2077]to compute the multi-species Boltzmann equation.Compared to the single species Boltzmann equation that the method was originally applied on,this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species.Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property.The method we propose does not contain any nonlinear nonlocal implicit solver,and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number.We prove the positivity and strong AP properties of the scheme,which are verified by two numerical examples.