dynamics and spatial organization in two species competition the authors investigate the time evolution of a prototypical population biology reaction which involves reproduction self regulation and competitive annihilation of two distinct species in one dimension the authors use a quasistatic analysis to argue that for a system with equal initial densities of two strongly competing species an alternating pattern of domains forms whose lengths grow logarithmically with time a scaling analysis of the underlying master equation as well as numerical integration of the reaction diffusion equations support this result for unequal initial densities the concentration of the minority species undergoes a power law decay with a non universal exponent the authors generalize the model by allowing for a nonlinear self regulation term in the rate equations as a function of the exponent of this nonlinearity the typical domain size may grow either as a power law with time or saturate at a finite value this general approach also suggests that a coarsening domain mosaic occurs in arbitrary spatial dimensions in two dimensions numerical integration of the reaction diffusion equations indicates that the average domain area grows approximately as t0 84

phase ordering and persistence in a class of stochastic processes interpolating between the ising and voter models we study the dynamics of a class of two dimensional stochastic processes depending on two parameters which may be interpreted as two different temperatures respectively associated to interfacial and to bulk noise special lines in the plane of parameters correspond to the ising model voter model and majority vote model the dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing annihilating and branching random walkers we use the freedom allowed by the space of parameters to measure by numerical simulations the persistence probability of a generic model in the low temperature phase where the system coarsens this probability is found to decay at large times as a power law with a seemingly constant exponent we also discuss the connection between persistence and the nature of the interfaces between domains

derivation of the hydrodynamical equation for one dimensional ginzburg landau model the hydrodynamical behavior of one dimensional scalar ginzburg landau model with conservation law is investigated the dynamics of the system is given by solving a stochastic partial differential equation under appropriate space time scaling a deterministic limit is obtained and the limit is described by a certain nonlinear diffusion equation

derivation of a hydrodynamic equation for ginzburg landau models in an external field the lattice approximation to a time dependent ginzburg landau equation is investigated in the presence of a small external field the evolution law conserves the spin but is not reversible a nonlinear diffusion equation of divergence type is obtained in the hydrodynamic limit the proof extends to certain stochastically perturbed hamiltonian systems

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

optimal linear glauber model contrary to the actual nonlinear glauber model the linear glauber model lgm is exactly solvable although the detailed balance condition is not generally satisfied this motivates us to address the issue of writing the transition rate w j in a best possible linear form such that the mean squared error in satisfying the detailed balance condition is least the advantage of this work is that by studying the lgm analytically we will be able to anticipate how the kinetic properties of an arbitrary ising system depend on the temperature and the coupling constants the analytical expressions for the optimal values of the parameters involved in the linear w j are obtained using a simple moore penrose pseudoinverse matrix this approach is quite general in principle applicable to any system and can reproduce the exact results for one dimensional ising system in the continuum limit we get a linear time dependent ginzburg landau equation from the glauber s microscopic model of non conservative dynamics we analyze the critical and dynamic properties of the model and show that most of the important results obtained in different studies can be reproduced by our new mathematical approach we will also show in this paper that the effect of magnetic field can easily be studied within our approach in particular we show that the inverse of relaxation time changes quadratically with weak magnetic field and that the fluctuation dissipation theorem is valid for our model

fluctuations of one dimensional ginzburg landau models in nonequilibrium we study the fluctuation of one dimensional ginzburg landau models in nonequilibrium along the hydrodynamic diffusion limit the hydrodynamic limit has been proved to be a nonlinear diffusion equation by fritz guo papanicolaou varadhan etc we proved that if the potential is uniformly convex then the fluctuation process is governed by an ornstein uhlenbeck process whose drift term is obtained by formally linearizing the hydrodynamic equation

the hydrodynamic limit for a system with interactions prescribed by ginzburg landau type random hamiltonian as a microscopic model we consider a system of interacting continuum like spin field overr d its evolution law is determined by the ginzburg landau type random hamiltonian and the total spin of the system is preserved by this evolution we show that the spin field converges under the hydrodynamic space time scalling to a deterministic limit which is a solution of a certain nonlinear diffusion equation this equation describes the time evolution of the macroscopic field the hydrodynamic scaling has an effect of the homogenization on the system at the same time

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

efficient time stepping methods for miscible displacement problems in porous media efficient procedures for time stepping galerkin methods for approximating the solution of a coupled system for c c x t and p p x t with nonlinear neumann boundary conditions of the form begin gathered nabla cdot a x c nabla p y x c nabla g equiv nabla cdot u f 1 x t quad x in omega quad t in 0 t hfill nabla cdot b x c nabla p nabla c u cdot nabla c phi x frac partial c partial t f 2 x t c quad x in omega quad t in 0 t hfill u cdot v q 1 x t quad x in partial omega quad t in 0 t hfill b frac partial c partial v q x t c quad x in partial omega quad t in 0 t hfill c x 0 c 0 x quad x in omega hfill end gathered where omega subset mathbb r d 2 leqq d leqq 3 are presented and analyzed this system is a possible model system for describing the miscible displacement of one incompressible fluid by another in a porous medium when flow conditions a

derivation of the hydrodynamical equation for the zero range interaction process small volume is so large compared with the distances between molecules that it contains an infinite number of them which are in equilibrium with given characteristics density temperature of course these parameters vary with x say p x t x if we look at the system after a time t the system will be still locally in equilibrium with other local characteristics p x t t x t varying slowly in time although each particle individually moves very rapidly we want to derive the equation of evolution of these macroscopical parameters however it is surprising that this derivation is almost never based on the analysis of the movement of the particles but rather on conservation laws the limiting procedure that we describe below has been used by various authors 2 8 in efforts to give rigorous treatment of the program alluded to above consider a system of infinitely many particles in rd or xd moving according

discontinuous galerkin finite element convergence for incompressible miscible displacement problems of low regularity in this article we analyze the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous galerkin method under minimal regularity assumptions the main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem in order to deal with the nonconformity of the method and to avoid overpenalization of jumps across interelement boundaries the careful construction of a reflexive subspace of the space of bounded variation which compactly embeds into l 2 omega and of a lifting operator which is compatible with the nonlinear diffusion coefficient are required an equivalent skew symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows us to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method numerical experiments underline the robustness of the proposed algorithm

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

on the diffusive nature of entropy flow in infinite systems remarks to a paper by guo papanicolau varadhan the hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy free energy arguments the methods of 13 are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise following 14 15 and 3 5 we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended the flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems 4

qualitative properties of solutions for nonlinear schrodinger equations with nonlinear boundary conditions on the half line in this paper we study the interaction between a nonlinear focusing robin type boundary source a nonlinear defocusing interior source and a weak damping term for nonlinear schrodinger equations posed on the infinite half line we construct solutions with negative initial energy satisfying a certain set of conditions which blow up in finite time in the h1 sense we obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow up solutions in addition to the blow up property we also discuss the stabilization property and the critical exponent for this model

hydrodynamic limit for a system with finite range interactions we study a system of interacting diffusions the variables present the amount of charge at various sites of a periodic multidimensional lattice the equilibrium states of the diffusion are canonical gibbs measures of a given finite range interaction under an appropriate scaling of lattice spacing and time we derive the hydrodynamic limit for the evolution of the macroscopic charge density

local stability and hydrodynamical limit of spitzer s one dimensional lattice model an infinite system of ordinary differential equations is considered the right hand side is just the negative gradient of potential energy of a one dimensional system of unbounded spins interacting by a symmetric and convex pair potential constant configurations are stationary points and the mean spin is conserved it is shown that each of these stationary points has its own domain of attraction the initial distribution need not be translation invariant as a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit

qualitative aspects for the cubic nonlinear schrodinger equations with localized damping exponential and polynomial stabilization abstract results of exponential polynomial decay rates of the energy in l 2 level related to the cubic nonlinear schrodinger equation with localized damping posed on the whole real line will be established in this paper we use kato s theory and a priori estimates to obtain the result of global well posedness and we determine exponential polynomial stabilization combining the ideas of unique continuation due to zhang semigroup property and komornik s approach

large deviations of interacting particle systems in infinite volume we prove a large deviation theorem from the hydrodynamical limit for the empirical measure of ginzburg landau and zero range processes in infinite volume starting from deterministic initial configurations in the ginzburg landau case the main tool is the study of the evolution of the h 1 norm and in the zero range case the attractiveness which allows couplings

on the hydrodynamic limit of a ginzburg landau lattice model the law of large numbers in arbitrary dimensions a lattice system of interacting diffusion processes is investigated the evolution is attractive and time reversible the spin satisfies a conservation law it is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation

convergence of a discontinuous galerkin method for the miscible displacement equation under low regularity discontinuous galerkin time discretizations are combined with the mixed finite element and continuous finite element methods to solve the miscible displacement problem stable schemes of arbitrary order in space and time are obtained under low regularity assumptions on the data convergence of the scheme is proved by using compactness results for functions that may be discontinuous in time

analytical and variational numerical methods for unstable miscible displacement flows in porous media the miscible displacement of one fluid by another in a porous medium has received considerable attention in sub surface environmental and petroleum engineering applications when a fluid of higher mobility displaces another of lower mobility unstable patterns referred to as viscous fingering may arise their physical and mathematical study has been the object of numerous investigations over the past century the objective of this paper is to present a review of these contributions with particular emphasis on variational methods these algorithms are tailored to real field applications thanks to their advanced features handling of general complex geometries robustness in the presence of rough tensor coefficients low sensitivity to mesh orientation in advection dominated scenarios and provable convergence with fully unstructured grids this paper is dedicated to the memory of dr jim douglas jr for his seminal contributions to miscible displacement and variational numerical methods

the noisy voter model under the influence of contrarians abstract the influence of contrarians on the noisy voter model is studied at the mean field level the noisy voter model is a variant of the voter model where agents can adopt two opinions optimistic or pessimistic and can change them by means of an imitation herding and an intrinsic noise mechanisms an ensemble of noisy voters undergoes a finite size phase transition upon increasing the relative importance of the noise to the herding form a bimodal phase where most of the agents share the same opinion to a unimodal phase where almost the same fraction of agent are in opposite states by the inclusion of contrarians we allow for some voters to adopt the opposite opinion of other agents anti herding we first consider the case of only contrarians and show that the only possible steady state is the unimodal one more generally when voters and contrarians are present we show that the bimodal unimodal transition of the noisy voter model prevails only if the number of contrarians in the system is smaller than four and their characteristic rates are small enough for the number of contrarians bigger or equal to four the voters and the contrarians can be seen only in the unimodal phase moreover if the number of voters and contrarians as well as the noise and herding rates are of the same order then the probability functions of the steady state are very well approximated by the gaussian distribution

variational inequality approach to enforcing the non negative constraint for advection diffusion equations abstract predictive simulations are crucial for the success of many subsurface applications and it is highly desirable to obtain accurate non negative solutions for transport equations in these numerical simulations to this end optimization based methodologies based on quadratic programming qp have been shown to be a viable approach to ensuring discrete maximum principles and the non negative constraint for anisotropic diffusion equations in this paper we propose a computational framework based on the variational inequality vi which can also be used to enforce important mathematical properties e g maximum principles and physical constraints e g the non negative constraint we demonstrate that this framework is not only applicable to diffusion equations but also to non symmetric advection diffusion equations an attractive feature of the proposed framework is that it works with any weak formulation for the advection diffusion equations including single field formulations which are computationally attractive a particular emphasis is placed on the parallel and algorithmic performance of the vi approach across large scale and heterogeneous problems it is also shown that qp and vi are equivalent under certain conditions state of the art qp and vi solvers available from the petsc library are used on a variety of steady state 2d and 3d benchmarks and a comparative study on the scalability between the qp and vi solvers is presented we then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems our numerical experiments indicate that vis are indeed a viable approach for enforcing the maximum principles and the non negative constraint in a large scale computing environment also provided are firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework

non equilibrium behaviour of a many particle process density profile and local equilibria one considers a simple exclusion particle jump process on where the underlying one particle motion is a degenerate random walk that moves only to the right one starts with the configuration in which the left halfline is completely occupied and the right one free it is shown that the number of particles at time t between site u t and v t divided by t converges a s to int limits u nu f w dw where f might be called the density profile it is explicitely determined and shown to be an affine function secondly we prove that the distribution of the process looked at by an observer travelling at constant speed u converges weakly to the bernoulli measure with density f u as the time tends to infinity

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

splitting the voter potts model critical point recently some two dimensional models with double symmetric absorbing states were shown to share the same critical behavior that was called the voter universality class we show that for an absorbing states potts model with finite but further than nearest neighbor range of interactions the critical point is split into two critical points one of the ising type and the other of the directed percolation universality class similar splitting takes place in the three dimensional nearest neighbor model

time dependent critical fluctuations of a one dimensional local mean field model one dimensional stochastic ising systems with a local mean field interaction kac potential are investigated it is shown that near the critical temperature of the equilibrium gibbs distribution the time dependent process admits a scaling limit given by a nonlinear stochastic pde the initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way earlier results of bertini presutti rudiger saada are improved the proof is based on an energy inequality obtained by coupling the glauber dynamics to its voter type linear approximation

rescaled voter models converge to super brownian motion we show that a sequence of voter models suitably rescaled in space and time converges weakly to super brownian motion the result includes both nearest neighbor and longer range voter models and complements a limit theorem of mueller and tribe in one dimension

numerical solutions of the incompressible miscible displacement equations in heterogeneous media abstract this paper presents a numerical method based on mixed finite element discontinuous galerkin methods in space and high order runge kutta method in time for solving the miscible displacement problem no slope limiters are needed the proposed method exhibits high order of convergence in space and time when comparing with analytical solutions the simulation shows robustness of the method for heterogeneous media with highly varying permeabilities

langevin equations for l valued gaussian processes and fluctuation limits of infinite particle systems l gaussian processes of a certain class are shown to satisfy generalized langevin equations examples are fluctuation limits of several infinite particle systems in particular infinite particle branching brownian motions with immigration under various scalings and the voter model with hydrodynamic scaling

uniform decay rate estimates for schrodinger and plate equations with nonlinear locally distributed damping on a compact n dimensional riemannian manifold m g we establish uniform decay rate estimates for the linear schrodinger and plate equations subject to an internal nonlinear damping locally distributed on the manifold our approach can be also employed for other equations provided that inverse inequality for the linear model occurs in the particular case of the wave equation where the well known geometric control condition gcc is equivalent to the observability inequality our method generalizes the results due to cavalcanti et al 2010 2009 9 10 regarding the optimal choice of dissipative regions

hydrodynamics of the voter model we study the voter model on mathbb z d d geqq 3 for a sequence mu varepsilon of initial states which have a gradient in the mean magnetization of the order varepsilon varepsilon rightarrow 0 we prove that the magnetization field m varepsilon f t varepsilon d sum f varepsilon x eta x varepsilon 2 t tends to a deterministic field m f t int dqf q m q t as varepsilon rightarrow 0 m q t is the solution of the diffusion equation the fluctuations of m varepsilon f t around its mean are given by an infinite dimensional non homogeneous ornstein uhlenbeck process in the limit as varepsilon rightarrow 0 locally i e around varepsilon 1 q varepsilon 2 t the voter model is in equilibrium with parameter m q t

adaptive enriched galerkin methods for miscible displacement problems with entropy residual stabilization we present a novel approach to the simulation of miscible displacement by employing adaptive enriched galerkin finite element methods eg coupled with entropy residual stabilization for transport in particular numerical simulations of viscous fingering instabilities in heterogeneous porous media and hele shaw cells are illustrated eg is formulated by enriching the conforming continuous galerkin finite element method cg with piecewise constant functions the method provides locally and globally conservative fluxes which are crucial for coupled flow and transport problems moreover eg has fewer degrees of freedom in comparison with discontinuous galerkin dg and an efficient flow solver has been derived which allows for higher order schemes dynamic adaptive mesh refinement is applied in order to reduce computational costs for large scale three dimensional applications in addition entropy residual based stabilization for high order eg transport systems prevents spurious oscillations numerical tests are presented to show the capabilities of eg applied to flow and transport enriched galerkin approximations for coupled flow and transport system entropy residual stabilization for enriched galerkin to stabilize higher order transport scheme dynamic mesh adaptivity for viscous fingering in two and three dimensions

heat flow in an exactly solvable model a chain of one dimensional oscillators is considered they are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors the total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures the stationary measures are obtained when the chain is finite the thermodynamic limit is then considered approach to the gibbs distribution is proven and a linear temperature profile is obtained

voter dynamics on an ising ladder coarsening and persistence coarsening and persistence of ising spins on a ladder is examined under voter dynamics the density of domain walls decreases algebraically with time as t 1 2 for sequential as well as parallel dynamics the persistence probability decreases as under sequential dynamics and as under parallel dynamics where p 2 s 0 88 numerical values of the exponents are explained the results are compared with the voter model on one and two dimensional lattices as well as ising model on a ladder under zero temperature glauber dynamics

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data

competition and cooperation in one dimensional stepping stone models mutualism is a major force driving evolution and sustaining ecosystems although the importance of spatial degrees of freedom and number fluctuations is well known their effects on mutualism are not fully understood with range expansions of microbes in mind we show that even when mutualism confers a selective advantage it persists only in populations with high density and frequent migrations when these parameters are reduced mutualism is generically lost via a directed percolation dp process with a phase diagram strongly influenced by an exceptional symmetric dp dp2 transition

mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media abstract in this article we introduce a family of discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media the proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise then discretise approach for the approximation of the optimal control problem leading to nonsymmetric algebraic systems and employing minimum regularity requirements estimates for the error between a local reference solution of the infinite dimensional optimal control problem and its hybrid mixed discontinuous approximation measured in suitable norms are derived showing optimal orders of convergence

partitioned coupling of advection diffusion reaction systems and brinkman flows we present a partitioned algorithm aimed at extending the capabilities of existing solvers for the simulation of coupled advection diffusion reaction systems and incompressible viscous flow the space discretisation of the governing equations is based on mixed finite element methods defined on unstructured meshes whereas the time integration hinges on an operator splitting strategy that exploits the differences in scales between the reaction advection and diffusion processes considering the global system as a number of sequentially linked sets of partial differential and algebraic equations the flow solver presents the advantage that all unknowns in the system here vorticity velocity and pressure can be fully decoupled and thus turn the overall scheme very attractive from the computational perspective the robustness of the proposed method is illustrated with a series of numerical tests in 2d and 3d relevant in the modelling of bacterial bioconvection and boussinesq systems

one dimensional hard rod caricature of hydrodynamics we give here a rigorous deduction of the hydrodynamic equation which holds in the hydrodynamic limit for a model system of one dimensional identical hard rods interacting through elastic collisions the equation should be considered as the analog of the euler equation of real hydrodynamics owing to the degeneracy of the model it is written in terms of a functiong q v t expressing the density of particles with velocityv at the pointq at timet for this equation we prove an existence and uniqueness theorem in some natural class of functions our main result is the proof that if 0 is a class of initial states which are homogeneous on a scale much less than e 1 and if the corresponding particle densities tend ase 0 in the proper scale to the initial hydrodynamic densitygo q v then under some general assumptions on the states and ong0 the particle densities of the evolved states at timee 1t tend ase 0 to the unique solution of the hydrodynamic equation with initial conditiong0 the proof is completed by exhibiting a large class of initial families 0 which possess the required properties

a soluble kinetic model for spinodal decomposition we compare the two dimensional voter model with approximate theories for spinodal decomposition the cluster size distribution and the short time dynamics of the voter model are studied by means of a monte carlo simulation the time dependent structure factor and the long time scaling of the voter dynamics are known analytically