The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes L\'{e}vy flights. Sometimes the infinite variance of L\'{e}vy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.

We study boundary value problems of a quasi-one-dimensional steady-state Poisson-Nernst-Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through an ion channel, focusing on effects of ion sizes and ion valences. The flow properties of interest, individual fluxes and total flow rates of the mixture, depend on multiple physical parameters such as boundary conditions (boundary concentrations and boundary potentials) and diffusion coefficients, in addition to ion sizes and ion valences. For the relatively simple setting and assumptions of the model in this paper, we are able to characterize, almost completely, the distinct effects of the nonlinear interplay between these physical parameters. The boundaries of different parameter regions are identified through a number of critical values that are explicitly expressed in terms of the physical parameters. We believe our results will provide useful insights for numerical and even experimental studies of ionic flows through membrane channels.

We consider the Benard convection in a three-dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface. The domain is horizontally periodic. The fluid dynamics are governed by the Boussinesq approximation and the effect of surface tension is neglected on the free surface. Here we develop a local well-posedness theory for the equations of general case in the framework of the nonlinear energy method.

In this work, we study the quasineutral limit of the one-dimensional Vlasov-Poisson equation for ions with massless thermalized electrons. We prove new weak-strong stability estimates in the Wasserstein metric that allow us to extend and improve previously known convergence results. In particular, we show that given a possibly unstable analytic initial profile, the formal limit holds for sequences of measure initial data converging sufficiently fast in the Wasserstein metric to this profile. This is achieved without assuming uniform analytic regularity.

In this paper we study the non-relativistic and low Mach num-ber limits of two P1 approximation model arising in radiation hydrodynamics in T^3, i.e. the barotropic model and the Navier-Stokes-Fourier model. For the barotropic model, we consider the case that the initial data is a small perturbation of stable equilbria while for the Navier-Stokes-Fourier model, we consider the case that the initial data is large. For both models, we prove the convergence to the solution of the incompressible Navier-Stokes equations with/without stationary transport equations.

We prove weak-strong uniqueness results for the compressible Navier-Stokes system with degenerate viscosity coefficients and with vacuum in one dimension. In other words, we give conditions on the weak solution constructed in [Q.S. Jiu, Z.P. Xin, Kinet. Relat. Models, 1. 313-330, 2008] so that it is unique. The novelty consists in dealing with initial density \rho_0 which contains vacuum. To do this we use the notion of relative entropy developed recently by Germain, Feireisl et al and Mellet and Vasseur combined with our new formulation of the compressible system [10, 11]) (more precisely we introduce a new effective velocity v which makes the system parabolic on the density and hyperbolic on the velocity v).

The multi-dimensional Euler-Poisson system describes the dynamic behavior of many important physical flows, yet as a hyperbolic system its solution can blow up for some initial configurations. This paper strives to advance our understanding on the critical threshold phenomena through the study of a two-dimensional weakly restricted Euler-Poisson (WREP) system. This system can be viewed as an improved model upon the restricted Euler-Poisson (REP) system introduced in [H. Liu and E. Tadmor, Comm. Math. Phys. 228 (2002), 435-466]. We identify upper-thresholds for finite time blow up of solutions for WREP equations with attractive/repulsive forcing. It is shown that the thresholds depend on the size of the initial density relative to the initial velocity gradient through both trace and a nonlinear quantity.

This work studies the stability of a class of globally hyperbolic moment systems (GHMS) with the single relaxation-time collision model in the sense of hyperbolic relaxation systems. We prove the equilibrium stability of the GHMS in both one- and multi-dimensional space. For a five-moment system in one dimension, we prove its linear instability for some quiescent nonequilibrium states and demonstrate numerically the nonlinear instability of the nonequilibrium states.

In this paper we obtain a weak solution to a quantum energy-transport model for semiconductors. The model is formally derived from the quantum hydrodynamic model in the large-time and small-velocity regime by J\"{u}ngel and Mili\v{s}i\'{c} (Nonlinear Anal.: Real World Appl., 12(2011), pp. 1033-1046). It consists of a fourth-order nonlinear parabolic equation for the electron density, an elliptic equation for the electron temperature, and the Poisson equation for the electric potential. Our solution is global in the time variable, while the N space variables ($N<6$) lie in a bounded Lipschitz domain with a mixed boundary condition. The existence proof is based upon a carefully-constructed approximation scheme which generates a sequence of positive approximate solutions. These solutions are so regular that they can be used to form a variety of test functions to produce a priori estimates. Then these estimates are shown to be enough to justify passing to the limit in the approximate problems.

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by L\'evy noise. We define a Hilbert-Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the L\'evy noise. We then prove a decomposition of the solution process into a stationary component, the law of which is identified with the unique invariant probability measure $\mu$ of the process, and a component which vanishes asymptotically for large times in the $L^p(\mu)$-sense, for all $1 \leq p < +\infty$.

The maximum entropy principle is widely used in diverse fields. We address the issue of why the second order maximum entropy model, by using only firing rates and second order correlations of neurons as constraints, can well capture the observed distribution of neuronal firing patterns in many neuronal networks, thus, conferring its great advantage in that the degree of complexity in the analysis of neuronal activity data reduces drastically from O(2^n) to O(n^2), where n is the number of neurons under consideration. We first derive an expression for the effective interactions of the n-th order maximum entropy model using all orders of correlations of neurons as constraints and show that there exists a recursive relation among the effective interactions in the model. Then, via a perturbative analysis, we explore a possible dynamical state in which this recursive relation gives rise to the strengths of higher order interactions always smaller than the lower orders. Finally, we invoke this hierarchy of effective interactions to provide a possible mechanism underlying the success of the second order maximum entropy model and to predict whether such a model can successfully capture the observed distribution of neuronal firing patterns.

The stabilized semi-implicit time-stepping method is an efficient algorithm to simulate phased field problems with fourth order dissipation. We consider the 3D Cahn-Hilliard equation and prove unconditional energy stability of the corresponding stabilized semi-implicit Fourier spectral scheme independent of the time step. We do not impose any Lipschitz-type assumption on the nonlinearity. It is shown that the size of the stabilization term depends only on the initial data and the diffusion coefficient. Unconditional Sobolev bounds of the numerical solution are obtained and the corresponding error analysis under nearly optimal regularity assumptions is established.

The chemical reaction rate from reactant to product depends on the geometry of potential energy surface (PES) as well as the temperature. We consider a design problem of how to choose the best PES from a given family of smooth potential functions in order to maximize (or minimize) the reaction rate for a given chemical reaction. By utilizing the transition-path theory, we relate reaction rate to committor functions which solves boundary-value elliptic problems, and perform the sensitivity analysis of the underlying elliptic equations via adjoint approach. We derive the derivative of the reaction rate with respect to the potential function. The shape derivative with respect to the domains defining reactant and product is also investigated. The numerical optimization method based on the gradient is applied for two simple numerical examples to demonstrate the feasibility of our approach.

Explicit energy-transport equations for the spinorial carrier transport in ferromagnetic semiconductors are calculated from a general spin energy-transport system that was derived by Ben Abdallah and El Hajj from a spinorial Boltzmann equation. The novelty of our approach are the simplifying assumptions leading to explicit models which extend both spin drift-diffusion and semiclassical energy-transport equations. The explicit models allow us to examine the interplay between the spin and charge degrees of freedom. In particular, the dissipation of the entropy (or free energy) is quantified, and the existence of weak solutions to a time-discrete version of one of the models is proved, using novel truncation arguments. Numerical experiments in one-dimensional multilayer structures using a finite-volume discretization illustrate the effect of the temperature and the polarization parameter.

The Hodge projection of a vector field is the divergence-free component of its Helmholtz decomposition. In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. In the decomposition by the Gibou-Min method, an important $L^{2}$-orthogonality holds between the gradient field and the solenoidal field, which is similar to the continuous Hodge decomposition. Using the orthogonality, we present a novel analysis which shows that the convergence order is 1.5 in the $L^2$-norm for approximating the divergence-free vector field. Numerical results are presented to validate our analyses.

This paper studies the non-autonomous globally modified Navier-Stokes equations. The authors first prove that the associated process possesses a pullback attractor. Then they establish that there exists a unique family of Borel invariant probability measures on the pullback attractor.

A time-dependent method is coupled with the method of approximate particular solutions (MAPS) and the method of approximate fundamental solutions (MAFS) of Delta-shaped basis functions to solve a nonlinear Poisson-type boundary value problem on an irregular shaped domain. The problem is first converted into a sequence of time-dependent nonhomogeneous modified Helmholtz boundary value problems through a fictitious time integration method. Then the superposition princi- ple is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. A Delta-shaped basis function is used to provide an approximation of the source function at each time step. This allows for an easy derivation of an approximate particular solution. The corresponding homogeneous boundary value problem is solved using MAFS, and also with the method of fundamental solutions (MFS) for comparison purposes. Numerical results support the accuracy and validity of this computational method.

In this paper we prove an exact controllability result for the Vlasov-Stokes system in the two-dimensional torus with small data by means of an internal control. We show that one can steer, in arbitrarily small time, any initial datum of class C^1 satisfying a smallness condition in certain weighted spaces to any final state satisfying the same conditions. The proof of the main result is achieved thanks to the return method and a Leray-Schauder fixed-point argument.

In this paper, we study the homogenization of a thermal diffusion problem in a highly heterogeneous medium formed by two constituents. The main characteristics of the medium are the discontinuity of the thermal conductivity over the domain as we go from one constituent to another and the presence of an imperfect interface between the two constituents, where both the temperature and the flux exhibit jumps. The limit problem, obtained via the periodic unfolding method, captures the influence of the jumps in the limit temperature field, in an additional source term, and in the correctors, as well.

We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context the recent implicitexplicit schemes developed in [C. Chalons, M. Girardin, and S. Kokh, SIAM J. Sci. Comput., 35(6):pp. a2874–a2902, 2013], [C. Chalons, M. Girardin, and S. Kokh, Communications in Computational Physics, 2016, to appear.] in the framework of compressible flows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases.

In this paper, we establish the existence of time periodic solutions to the two and three dimensional compressible Navier-Stokes-Poisson equations with the linear damping, under some smallness and symmetry assumptions on the time periodic external force. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Finally, the existence result in the whole space is obtained by a limiting process.

We present a high-order Lagrange-projection like method for the approximation of the compressible Euler equations with a general equation of state. We extend the method introduced in Renac (Numer. Math., 2016, DOI 10.1007/s00211-016-0807-0) in the case of the isentropic gas dynamics to the compressible Euler equations and minimize the numerical dissipation by quantifying it from a parameter evaluated locally in each element of the mesh. The method is based on a decomposition between acoustic and transport operators associated to an implicit-explicit time integration, thus relaxing the constraint of acoustic waves on the time step as proposed in Coquel et. al. (Math. Comput., 79 (2010), pp. 1493-1533) in the context of a first-order finite volume method. We derive conditions on the time step and on a local numerical dissipation parameter to keep positivity of the mean value of the discrete density and internal energy in each element of the mesh and to satisfy a discrete inequality for the physical entropy at any approximation order in space. These results are then used to design limiting procedures in order to restore these properties at nodal values within elements. Moreover, the scheme is designed to avoid over-resolution in space and time in the low Mach number regime. Numerical experiments support the conclusions of the analysis and highlight stability and robustness of the present method when applied to either discontinuous flows or vacuum. Large time steps are allowed while keeping accuracy on smooth solutions even for low Mach number flows.

Highly-accurate numerical methods that can efficiently handle problems with interfaces and/or problems in domains with complex geometry are essential for the resolution of a wide range of temporal and spatial scales in many partial differential equations based models from Biology, Materials Science and Physics. In this paper we continue our work started in 1D, and we develop high-order accurate methods based on the Difference Potentials for 2D parabolic interface/composite domain problems. Extensive numerical experiments are provided to illustrate high-order accuracy and efficiency of the developed schemes.

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a \in (0, +\infty). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, (0,1) uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance.

Multi-scale modeling and numerical simulations of the isothermal chemical vapor infiltration (CVI) process for the fabrication of carbon fiber reinforced silicon carbide (C/SiC) composites was presented in [Y. Bai, X. Yue and Q. Zeng, Commun. Comput. Phys., 7:3 (2010), 597-612]. The homogenization theory, which played a fundamental role in the multi-scale algorithm, will be rigorously established in this paper. The governing system, which is a multi-scale reaction-diffusion equation, is different in the two stages of CVI process, so we will consider the homogenization for the two stages respectively. One of the main features is that the reaction only occurs on the surface of fiber, so it behaves as a singular surface source. The other feature is that in the second stage of the process when the micro pores inside the fiber bundles are all closed, the diffusion only occurs in the macro pores between fiber bundles and we face up with a problem in a locally periodic perforated domain.

We consider a one-dimensional version of a model obtained in [Engwer, C., Hunt, A., Surulescu, IMA J. Math. Med. Bio, 2015] and describing the anisotropic spread of tumor cells in a tissue network. The model consists of a reaction-diffusion-taxis equation for the density of tumor cells coupled with an ODE for the density of tissue fibers and allows for strong degeneracy both in the diffusion and the haptotaxis terms. In this setting we prove the global existence of weak solutions to an associated no-flux initial-boundary value problem. Numerical simulations are performed in order to illustrate the model behavior.

We consider the full time-harmonic Maxwell equations in the presence of a thin, high-contrast dielectric object. As an extension of previous work by two of the authors, we continue to study limits of the electric field as the thickness of the scatterer goes to zero simultaneously as the contrast goes to infinity. We present both analytical and computational results, including simulations which demonstrate that the interior transverse component of the electric field has limit zero, and a rigorous asymptotic approximation accurate outside of the scatterer. Finally, we propose an inversion method to recover the geometry of the scatterer given its two-dimensional plane and we present numerical simulations using this method.

The logarithmic KdV (log-KdV) equation admits global solutions in an energy space and exhibits Gaussian solitary waves. Orbital stability of Gaussian solitary waves is known to be an open problem. We address properties of solutions to the linearized log-KdV equation at the Gaussian solitary waves. By using the decomposition of solutions in the energy space in terms of Hermite functions, we show that the time evolution of the linearized log-KdV equation is related to a Jacobi difference operator with a limit circle at infinity. This exact reduction allows us to characterize both spectral and linear orbital stability of Gaussian solitary waves. We also introduce a convolution representation of solutions to the linearized log-KdV equation with the Gaussian weight and show that the time evolution in such a weighted space is dissipative with the exponential rate of decay.

We extend Brenier's transport collapse scheme on the initial-boundary value problem for scalar conservation laws. It is based on averaging out the solution to the corresponding kinetic equation, and it leads to a new solution concept for the problem under consideration. We also provide numerical examples.

In this paper, we obtain global well-posedness for the 2D damped Boussinesq equations. Based on the estimate of the damped Euler equations leading to the uniform corresponding bound (do not grow in time), we can achieve this goal by using a new decomposition technic. Comparing with the previous works, we do not need any small assumptions of the initial velocity. As an application of our method, we obtain a similar result for the 2D damped MHD equations.

This paper studies the stability of smooth traveling wave solutions to a nonlinear PDE problem in reducing image noise. Specifically, we prove that the solution to the Cauchy problem approaches to the traveling wave solution if the initial data is a small perturbation of the traveling wave. We use a weighted energy method to show that if the initial perturbation decays algebraically or exponentially as |x| \to \infty, then the Cauchy problem solution approaches to the traveling wave at corresponding rates as t\to \infty.

We apply the two-scale formulation approach to propose uniformly accurate (UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic limit regime. The nonlinear Dirac equation involves two small scales \epsilon and \epsilon^2 with \epsilon \to 0 in the nonrelativistic limit regime. The small parameter causes high oscillations in time which bring severe numerical burden for classical numerical methods. We present a suitable two-scale formulation as a general strategy to tackle a class of highly oscillatory problems involving the two small scales \epsilon and \epsilon^2. A numerical scheme with uniform (with respect to \epsilon \in (0,1]) second order accuracy in time and a spectral accuracy in space are proposed. Numerical experiments are done to confirm the UA property.

In this article, we show that the recently studied compressible and incompressible models for viscoelastic fluids with infinite Weissenberg number can be well regarded as specific examples of general hyperbolicparabolic systems studied by Shizuta and Kawashima. It will be seen that two physically motivated compatibility conditions compensate the breaking of the Kawashima condition. Thus, the global existences of classical small solutions near equilibrium can be easily proved by following the general framework.

In this work, we prove the existence, uniqueness and smoothing properties of the solution to the Cauchy problem for the spatially homogeneous Boltzmann equation with Debye-Yukawa potential for probability measure initial datum.

We present a numerical method for solving the Langevin dynamics model. Rather than the trajectory-wise accuracy, we emphasize on the consistency to the equilibrium statistics at the discrete level. A discrete fluctuation-dissipation theorem is imposed to ensure that the statistical properties are preserved.

In this paper we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain. The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor.

This paper studies a Boltzmann Nordheim equation in a slab with two-dimensional velocity space and pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem with large initial data in an L^1\cap L^\infty setting. The main results are existence, uniqueness and stability of solutions conserving mass, momentum and energy that explode in L^\infty if they are only local in time. The solutions are obtained as limits of solutions to corresponding anyon equations.

A compressible Oldroyd-B type model with stress diffusion is derived from a compressible Navier-Stokes-Fokker-Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent, are idealized as pairs of massless beads connected with Hookean springs. We develop a priori bounds for the model, including a logarithmic bound, which guarantee the nonnegativity of the elastic extra stress tensor, and we prove the existence of large data global-in-time finite-energy weak solutions in two space dimensions.

This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation (\alpha, ]beta): \alpha+\beta=1. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data, as long as \alpha>2/3. The main new idea is the introduction of a new, second generation Hmidi-Keraani-Rousset type, change of variables, which further improves the linear derivative in temperature term in the vorticity equation. This approach is then complemented by new set of commutator estimates (in both negative and positive index Sobolev spaces!), which may be of independent interest.

This paper introduces the localized sparsifying preconditioner for the pseudospectral approximations of indefinite systems on periodic structures. The work is built on top of the recently proposed sparsifying preconditioner with two major modifications. First, the local potential information is utilized to improve the accuracy of the preconditioner. Second, an FFT based method to compute the local stencil is proposed to reduce the setup time of the algorithm. Numerical results show that the iteration number of this improved method grows only mildly as the problem size grows, which implies that solving pseudospectral approximation systems is computationally as efficient as solving sparse systems, up to a mildly growing factor.

In this note, we show the existence of regular solutions to the stationary version of the Navier--Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods for weak solutions to the compressible Navier--Stokes system.

Mean-Field Games are games with a continuum of players that incorporate the time dimension through a control-theoretic approach. Recently, simpler approaches relying on the Best Reply Strategy have been proposed. They assume that the agents navigate their strategies towards their goal by taking the direction of steepest descent of their cost function (i.e. the opposite of the utility function). In this paper, we explore the link between Mean-Field Games and the Best Reply Strategy approach. This is done by introducing a Model Predictive Control framework, which consists of setting the Mean-Field Game over a short time interval which recedes as time moves on. We show that the Model Predictive Control offers a compromise between a possibly unrealistic Mean-Field Game approach and the sub-optimal Best Reply Strategy.

In this paper we are concerned with the global existence of smooth solutions to two types of initial-boundary value problems to a system of equations describing one-dimensional motion of selfgravitating, radiative and chemically reactive gas whose viscosity coefficient depends on density. The main ingredient of the analysis is to derive the positive lower and upper bounds on both the specific volume and the absolute temperature.

This paper studied the Cauchy problem for the generalized sixth-order Boussinesq equation in multi-dimension (n \ge 3), which was derived in the shallow fluid layers and nonlinear atomic chains. Firstly the global classical solution for the problem are obtained by means of long wave-short wave decomposition, energy method and the Green’s function. Secondly and what’s more, the pointwise estimates of the solutions are derived by virtue of the Fourier analysis and Green’s function.

The following paper addresses the connection between two classical models of phase transition phenomena describing different stages of clusters growth. The first one, the Becker-Doring model (BD) that describes discrete-sized clusters through an infinite set of ordinary differential equations. The second one, the Lifshitz-Slyozov equation (LS) that is a transport partial differential equation on the continuous half-line $x\in (0,+\infty)$. We introduce a scaling parameter $\veps>0$, which accounts for the grid size of the state space in the BD model, and recover the LS model in the limit $\veps\to 0$. The connection has been already proven in the context of outgoing characteristic at the boundary $x=0$ for the LS model when small clusters tend to shrink. The main novelty of this work resides in a new estimate on the growth of small clusters, which behave at a fast time scale. Through a rigorous quasi steady state approximation, we derive boundary conditions for the incoming characteristic case, when small clusters tend to grow.