We present a BGK approximation of a kinetic Boltzmann model for a mixture of polyatomic gases, in which non-translational degrees of freedom of each gas are represented by means of a set of discrete internal energy levels. We deal also with situations in which even chemical reactions implying transfer of mass may occur. The consistency of the proposed BGK model is proved in both inert and reactive frames, and numerical simulations in space homogeneous settings are presented.

In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coeffcients, (ii) is ergodic or (iii) is a class of well-known results for fluid flows driven by elliptic diffusion processes.

It is believed that social preference, economic disparity, and heterogeneous environments are mechanisms for segregation. However, it is difficult to unravel the exact role of each mechanism in such a complex system. We introduce a versatile, simple and intuitive particle-interaction model that allows to easily examine the effect of each of these factors. It is amenable to numerical simulations, and allows for the derivation of the macroscopic equations. As the population size and number of groups with different economic status approach infinity, we derive various local and non-local system of PDEs for the population density. Through the analysis of the continuous limiting equations, we conclude that social preference is a necessary but not always sufficient mechanism for segregation. On the other hand, when combined with the environment and economic disparity (which on the their own also do not cause segregation), social preference does enhance segregation.

In this paper, we first present the derivation of the anisotropic Lagrangian averaged gyrowaterbag continuum (LAGWB-alpha) equations. The gyrowaterbag (nickname for gyrokinetic-waterbag) continuum can be viewed as a special class of exact weak solution of the gyrokinetic-Vlasov equation, allowing to reduce this latter into an infinite dimensional set of hydrodynamic equations while keeping its kinetic features such as Landau damping. In order to obtain the LAGWBC-alpha equations from the gyrowaterbag continuum we use an Eulerian variational principle and Lagrangian averaging techniques introduced by Holm, Marsden and Ratiu, Marsden and Shkoller, for the mean motion of ideal incompressible flows, extended to barotropic compressible flows by Bhat et. al. and some supplementary approximations for the electrical potential uctuations. Regarding to the original gyrowaterbag continuum, the LAGWBC-alpha equations show some additional properties and several advantages from the mathematical and physical viewpoints, which make this model a good candidate for describing accurately gyrokinetic turbulence in magnetically confined plasma. In the second part of this paper we prove local-in-time well-posedness of an approximated version of the anisotropic LAGWBC-alpha equations, that we call the anisotropic" LAGWBC-alpha equations, by using quasilinear PDE type methods and elliptic regularity estimates for several operators.

A new class of high-order accuracy numerical methods for the BGK model of the Boltzmann equation is presented. The schemes are based on a semi-Lagrangian formulation of the BGK equation; time integration is dealt with DIRK (Diagonally Implicit Runge Kutta) and BDF methods; the latter turn out to be accurate and computationally less expensive than the former. Numerical results and examples show that the schemes are reliable and efficient for the investigation of both rarefied and fluid regimes in gasdynamics.

In this paper, we investigate the effects of environment fuctuations on the disease's
dynamics through studying the stochastic dynamics of an SIS model incorporating media coverage.
The value of this study lies in two aspects: Mathematically, we show that the disease dynamics the
SDE model can be governed by its related basic reproduction number R_0^S:
if R_0^S \le 1, the disease will die out stochastically, while R_0^S>1,
the disease will break out with probability one. Epidemiologically, we partially provide the effects of the environment fuctuations affect the disease spreading
incorporating media coverage. First, noise can suppress the disease outbreak. Notice that R_0^S

We construct small-amplitude steady periodic gravity water waves arising as the free surface of water flows that contain stagnation points and possess a discontinuous distribution of vorticity in the sense that the flows consists of two layers of constant but different vorticities. We also describe the streamline pattern in the moving frame for the constructed flows.

For multispecies ions, we study boundary layer solutions of charge conserving Poisson-Boltzmann (CCPB) equations (with a small parameter \epsilon) over a finite one-dimensional (1D) spatial domain, subjected to Robin type boundary conditions with variable coefficients. Hereafter, 1D boundary layer solutions mean that as \epsilon approaches zero, the profiles of solutions form boundary layers near boundary points and become flat in the interior domain. These solutions are related to electric double layers with many applications in biology and physics. We rigorously prove the asymptotic behaviors of 1D boundary layer solutions at interior and boundary points. The asymptotic limits of the solution values (electric potentials) at interior and boundary points with a potential gap (related to zeta potential) are uniquely determined by explicit nonlinear formulas (cannot be found in classical Poisson-Boltzmann equations) which are solvable by numerical computations.

For the general $2\times 2$ hyperbolic conservation laws with relaxation, the convergence to the rarefaction wave of the equilibrium equation as the relaxation parameter tends to zero is proved, and the convergence rate is given.

In this paper, we study the attractor of quantum Zakharov system on unbounded domain $\mathbb{R}^d$ ($d=1,2,3$). We first prove the existence and uniqueness of solution by standard energy method. Then, by making use of the particular characters of quantum Zakharov system and the special decomposition of the solution operator, we obtain the existence of attractor for this system.

Experimental studies of vehicular traffic provide data on quantities like density, flux, and average speed of the vehicles. However, the diagrams that relate these variables can have different interpretations. In this paper, resting on the kinetic theory for vehicular traffic models, we introduce a new framework which takes into account the heterogeneous nature of the flow of vehicles. In more detail, we extend the model presented in Fermo and Tosin to the case of two populations of vehicles (such as e.g., cars and trucks), each with its own distribution function. Thus we consider traffic as a mixture of vehicles with different features, in particular different length and maximum speed. With this approach we can explain some interesting features of experimental diagrams. In fact, mathematical models for vehicular traffic typically yield fundamental diagrams that are single-valued functions of the density; in contrast, actual measurements show scattered data in the phase of congested traffic, which are naturally reproduced by our 2-population model as a result of the heterogeneous composition of the mixture of vehicles.

Multiscale partial differential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic, and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macro model. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating by the coupling between the different scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well.

We study the long-time behavior an extended Navier-Stokes system in R^2 where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.

We establish in the present paper that under long-wavelength, small amplitude approximation, the solution to the gas dynamics system converges globally in time to the solution of the Burgers equation for well prepared initil data.

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.

In this paper we present an unconditionally solvable and energy stable second order numerical scheme for the three-dimensional (3-D) Cahn-Hilliard (CH) equation. The scheme is a two-step method based on a second order convex splitting of the physical energy, combined with a centered difference in space. The equation at the implicit time level is nonlinear but represents the gradients of a strictly convex function and is thus uniquely solvable, regardless of time step-size. The nonlinear equation is solved using an efficient nonlinear multigrid method. In addition, a global in time H_h^2 bound for the numerical solution is derived at the discrete level, and this bound is independent on the final time. As a consequence, an unconditional convergence (for the time steps in terms of the spatial grid size h is established, in a discrete L_s^\infty(0,T; H_h^2) norm, for the proposed second order scheme. The results of numerical experiments are presented and conrm the efficiency and accuracnorm, for the proposed second order scheme. The results of numerical experiments are presented and confirm the efficiency of the scheme.

In this paper, a generalized two-component Camassa-Holm model, closely connected to the shallow water theory, is discussed. This two-component Camassa-Holm system is investigated on the local well-posedness and blow-up phenomena. The present work is mainly concerned with the detailed blow-up criteria where some special classes of initial data are involved. Moreover, as a by-product, the blow-up rate is established.

Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index r such that the entries of the generating vector of a Hankel tensor are circulant with module r. In the special case when r=n, where n is the dimension of the Hankel tensor, the generalized anticirculant tensor reduces to the anti-circulant tensor. Hence, generalized anti-circulant tensors are still special Hankel tensors. For the cases that GCD(m,r)-1, GCD(m,r)-2 and some other cases, including the matrix case that m=2, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are sum of squares tensors. This shows that, in these cases, there are no PNS (positive semidefinite tensors which are not sum of squares) Hankel tensors.

We are interested in the development of a numerical method for solving optimal control problems governed by hyperbolic systems of conservation laws. The main difficulty of computing the derivative in the case of shock waves is resolved in the presented scheme. Our approach is based on a combination of a relaxation approach in combination with a numerical schemes to resolve the evolution of the tangent vectors. Numerical results for optimal control problems are presented.

We present a Cucker-Smale type flocking model for interacting multi-agents(or particles) moving with constant speed in arbitrary dimensions, and derive a sufficient condition for the asymptotic flocking in terms of spatial and velocity diameters, coupling strength and a communication weight. In literature, several Vicsek type models with a unit speed constraint have been proposed in the modeling of self-organization and planar models were extensively studied via the dynamics of the heading angle. Our proposed model has a velocity coupling that is orthogonal to the velocity of the test agent to ensure the consistancy of speed of the test agent along the dynamic process. For a flocking estimate, we derive a system of dissipative differential inequalities for spatial and velocity diameters, and we also employ a robust Lyapunov functional approach.

The paper is concerned with time periodic solutions to the three dimensional compressible fluid models of Korteweg type under some smallness and structure conditions on a time periodic force. The proof is based on a regularized approximation scheme and the topological degree theory for time periodic solutions in a bounded domain. Furthermore, via a limiting process, the existence results can be obtained in the whole space.

We consider, through PDE methods, branching Brownian motion with drift and absorption. It is well known that there exists a critical drift which separates those processes which die out almost surely and those which survive with positive probability. In this work, we consider lower order corrections to the critical drift which ensures a non-negative, bounded expected number of particles and convergence of this expectation to a limiting non-negative number, which is positive for some initial data. In particular, we show that the average number of particles stabilizes at the convergence rate $O(\log(t)/t)$ if and only if the multiplicative factor of the $O(t^{-1/2})$ correction term is $3\sqrt{\pi} t^{-1/2}$. Otherwise, the convergence rate is $O(1/\sqrt{t})$. We point out some connections between this work and recent work investigating the expansion of the front location for the initial value problem in Fisher-KPP.

This paper is concerned with the pure-state N-representability problem for systems under a magnetic field. Necessary and sufficient conditions are given for a spin-density 2\times 2 matrix R to be representable by a Slater determinant. We also provide sufficient conditions on the parametric current j for the pair (R,j) to be Slater-representable in the case where the number of electrons N is greater than 12. The case N<12 is left open.

We compare three types of mathematical models of growth factor reaction and diffusion in angiogenesis: one describes the reaction on the blood capillary surface, one in the capillary volume, and one on the capillary centerline. Firstly, we explore the analytical properties of these models including solution regularity and positivity. We prove that the surface-reaction models have smooth and positive solutions, and the volume-reaction models have continuous and positive solutions. The line-reaction models utilize distributions on the capillary centerline to represent the reaction line source. The line-reaction model-I employs the Dirac delta function and the mean value of the growth factor around the centerline, which gives a valid model. The line-reaction model-II and III use the local value of the growth factor, which either create singulaity of decouple the reaction from diffusion, thus invalid. Secondly, we compare the programming complexity and computational cost of these models in numerical implementations: the surface-reaction model is the most complicated and suitable for small domains, while the volume-reaction and linear-reaction models are simpler and suitable for large domains with a large number of blood capillaries. Finally, we qauantitatively compare these models in the prediction of the growth factor dynamics. It turns out the volume-reaction and line-reaction model-I agree well with the surface-reaction model for most parameters used in literature, but may differ significantly when the diffusion constant is small.

A new model to describe biological invasion influenced by a line with fast diffusion has been introduced by H. Berestycki, J.-M. Roquejoffre and L. Rossi in 2012. The purpose of this article is to present a related model where the line of fast diffusion has a nontrivial range of influence, i.e. the exchanes between the line and the surrounding space has a nontrivial support. We show the existence of a spreading velocity depending on the diffusion on the line. Two intermediate model are also discussed. Then, we try to understand the influence of different exchange terms on this spreading speed. We show that various behaviour may happen, depending on the considered exchange distributions.

The compressible Navier-Stokes-Maxwell system with the linear damping is inves- tigated in R^3 and the global existence and large time behavior of solutions are established in the present paper. We rst construct the global unique solution under the assumptions that the H^3 norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If further the initial data belongs to H^{-s} (s \in [0, 3/2)) or B_{2, \infty}^{-s}, s\in (0, 3/2) , by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the L^p-L^2 (1 \le p \le 2) type of the decay rates follow without requiring that the L^p norm of initial data is small.

Routing strategies in unreliable production networks are an essential tool to meet given demands and to avoid high inventory levels. Therefore we are interested in studying state-independent and state-dependent control policies to maximize the total throughput of the production network. Different to M/M/1 queuing theory the underlying model is based on partial and ordinary differential equations with random breakdowns capturing the time varying behavior of the system. The key idea is to numerically compare suitable routing strategies with results computed by nonlinear optimization. We comment on the efficiency of the proposed methods and their qualitative behavior as well.

The Madden-Julian Oscillation (MJO) is the dominant component of intraseasonal (30-90 days) variability in the tropical atmosphere. Here, traveling wave solutions are presented for the MJO skeleton model of Majda and Stechmann. The model is a system of nonlinear partial differential equations that describe the evolution of the tropical atmosphere on planetary (10,000-40,000 km) spatial scales. The nonlinear traveling waves come in four types, corresponding to the four types of linear wave solutions, one of which has the properties of the MJO. In the MJO traveling wave, the convective activity has a pulse-like shape, with a narrow region of enhanced convection and a wide region of suppressed convection. Furthermore, an amplitude-dependent dispersion relation is derived, and it shows that the nonlinear MJO has a lower frequency and slower propagation speed than the linear MJO. By taking the small-amplitude limit, an analytic formula is also derived for the dispersion relation of linear waves. To derive all of these results, a key aspect is the model's conservation of energy, which holds even in the presence of forcing. In the limit of weak forcing, it is shown that the nonlinear traveling waves have a simple sech-squared waveform.

In this paper, we study the nonlinearly coupled Schrodinger equations for atomic Bose-Einstein condensates. By using the Galerkin method and a priori estimates, the global existence of smooth solution is obtained. And under some conditions of the coefficients and p, the blow-up theorem is established.

The vortex sheet solutions are considered for the inviscid liquid-gas two-phase flow. In particular, the linear stability of rectilinear vortex sheets in two spatial dimensions is established for both constant and variable coefficients The linearized problem of vortex sheet solutions with constant coefficients is studied by means of Fourier analysis, normal mode analysis and Kreiss' symmetrizer, while the linear stability with variable coefficients is obtained by Bony-Meyer's paradierential calculus theory. The linear stability is crucial to the existence of vortex sheet solutions of the nonlinear problem. A novel symmetrization and some weighted Sobolev norms are introduced to study the hyperbolic linearized problem with characteristic boundary.

We present a fast Newton-like algorithm, within the framework of the method of evolving junctions (MEJ), to find the shortest path in a cluttered environment. We demonstrate that the new algorithm converges much faster than the existing methods via numerical examples.

Semiclassical asymptotics for Schrodinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P. L. Lions and T. Paul in 1993. If we consider more singular potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based on posterior error controal. Thus rigorous uppen bounds for the asymptotic error on concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle points admit a regularized transport solution for the WM.

We consider the solutions to a modication of the Courant's minimax characterization of the Dirichlet eigenfunctions of second order linear symmetric elliptic operators in a bounded domain \Omega in R^d. In particular, we perturb the objective functional by an arbitrary bounded penalty term. Without perturbation, it is well-known that Courant minimax principle yields the eigenfunctions, which form an orthonormal basis for L^2(\Omega). We prove that the solutions of the perturbed problem still form an orthonormal basis in the case of d=1, and d=2, provided that the perturbation is sufficiently small in the latter case. As an application, we prove completeness results for compressed plane waves and compressed modes, which are the solutions to analogous variational problems with perturbations being an L^1-regularization term. The completeness theory for these functions sets a foundation for finding a computationally efficient basis for the representation of the solution of dierential equations.

There are numerous contexts where one wishes to describe the state of a randomly evolving system. Effective solutions combine models that quantify the underlying uncertainty with available observational data to form scientically reasonable estimates for the uncertainty in the system state. Stochastic differential equations are often used to mathematically model the underlying system. The Kusuoka-Lyons-Victoir (KLV) approach is a higher order particle method for approximating the weak solution of a stochastic differential equation that uses a weighted set of scenarios to approximate the evolving probability distribution to a high order of accuracy. The algorithm can be performed by integrating along a number of carefully selected bounded variation paths and the iterated application of the KLV method has a tendency for the number of particles to increase. Together with local dynamic recombination that simplies the support of discrete measure without harming the accuracy of the approximation, the KLV method becomes eligible to solve the filtering problem for which one has to maintain an accurate description of the ever-evolving conditioned measure. Besides the alternate application of the KLV method and recombination for the entire family of particles, we make use of the smooth nature of the likelihood function to lead some of the particles immediately to the next observation time and to build an algorithm that is a form of automatic high order adaptive importance sampling. We perform numerical simulations to evaluate the efficiency and accuracy of the proposed approaches in the example of the linear stochastic differential equation driven by three dimensional Brownian motions. Our numerical simulation show that, even when the sequential Monte-Carlo method poorly performs, the KLV method and recombination can together be used to approximate higher order moments of the filtering solution in a moderate dimension with high accuracy and efficiency.

In this paper, we study the transonic shock solutions to the Euler-Poisson systems in quasi-one-dimensional nozzles. For given supersonic flow at the entrance of the nozzle, under some proper assumptions on the data and the nozzle length we first obtain a class of steady transonic shock solutions for the exit pressure lying in a suitable range. The shock position is monotonically determined by the exit pressure. More importantly, by the estimates on the coupled eelectric field and the geometry of the nozzle, we prove the dynamic stability of the transonic shock solutions under suitable physical conditions. As a consequence, there indeed exist dynamically stable transonic shock solutions for the Euler-Poisson system in convergent nozzles, which is not true for the Euler systems in [T.P. Liu, Comm. Math. Phys. 83, 243-260, 1982].

We study in this article multiplicities of tensor eigenvalues. There are two natural
multiplicities associated to an eigenvalue \lambda a tensor: algebraic multiplicity am(\lambda) of a tensor: algebraic multiplicity gm(\lambda). The former is the multiplicity of the eigenvalue as a root of the characteristic
polynomial, and the latter is the dimension of the eigenvariety (i.e., the set of eigenvectors)
corresponding to the eigenvalue.
We show that the algebraic multiplicity could change along the orbit of tensors by the orthogonal
linear group action, while the geometric multiplicity of the zero eigenvalue is invariant under this
action, which is the main diffculty to study their relationships. However, we show that for a generic
tensor, every eigenvalue has a unique (up to scaling) eigenvector, and both the algebraic multiplicity
and geometric multiplicity are one. In general, we suggest for an m-th order
n-dimensinal tensor the relationship

am(\lambda)\ge gm(\lambda)(m-1)^(gm(\lambda)-1).

We show that it is true for several cases, especially when the eigenvariety contains a linear subspace
of dimension gm(\lambda) in coordinate form. As both multiplicities are invariants under the orthogonal
linear group action in the matrix counterpart, this generalizes the classical result for a matrix: the
algebraic multiplicity is not smaller than the geometric multiplicity.

In this study we investigate how to use sample data, generated by a fully resolved multiscale model, to construct stochastic representation of unresolved scales in reduced models. We explore three methods to model these stochastic representations. They employ empirical distributions, conditional Markov chains and conditioned Ornstein-Uhlenbeck processes, respectively. The Kac-Zwanzig heat bath model is used as a prototype model to illustrate the methods. We demonstrate that all tested strategies reproduce the dynamics of the resolved model variables accurately. Furthermore, we show that the computational cost of the reduced model is several orders of magnitude lower than that of the fully resolved model.

We study the homogenization of a stationary conductivity problem in a random heterogeneous medium with highly oscillating conductivity coefficients and an ensemble of simply closed conductivity resistant membranes. This medium is randomly deformed and then rescaled from a periodic one with periodic membranes, in a manner similar to the random medium proposed by Blanc, Le Bris and Lions (2006). Across the membranes, the flux is continuous but the potential field itself undergoes a jump of Robin type. We prove that, for almost all realizations of the random deformation, as the small scale of variations of the medium goes to zero, the random conductivity problem is well approximated by that on an effective medium which has deterministic and constant coefficients and contains no membrane. The effective coefficients are explicitly represented. One of our main contributions is to provide a solution to the associated auxiliary problem that is posed on the whole domain with infinitely many interfaces, in a setting that is neither periodic nor stationary ergodic in the usual sense.

In [F. Jiang, S. Jiang, On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014)], the author and Jiang investigated the instability of pressible viscous flow driven by gravity in a bounded domain \Omega of class C^2. In particular, they proved the steady-state is nonlinearly unstable under a restrictive condition of that the derivative function of steady density possesses a positive lower bound. In this article, by exploiting a standard energy functional and more-refined analysis of error estimates in the bootstrap argument, we can show the nonlinear instability result without the restrictive condition.

This is a continuous study on E. coli chemotaxis under the framework of pathway-based mean-field theory (PBMFT) proposed in [G. Si, M. Tang and X. Yang, Multiscale Model. Simul. , 12 (2014), 907{926], following the physical studies in [G. Si, T. Wu, Q. Quyang and Y. Tu, Phys. Rev. Lett. , 109 (2012), 048101]. In this paper, we derive an augmented Keller-Segel system with macroscopic intercellular signaling pathway dynamics. It can explain the experimental observation of phase-shift between the maxima of ligand concentration and density of E. coli in fast-varying environments at the population level. This is a necessary complement to the original PBMFT where the phase-shift can only be modeled by moment systems. Formal analysis are given for the system in the cases of fast and slow adaption rates. Numerical simulations show the quantitative agreement of the augmented Keller-Segel model with the individual-based E. coli chemotaxis simulator.

We study the emergence of phase locked states to the finite-dimensional Kuramoto model from generic initial configurations which are not phase-locked states in a large coupling regime. In the literature of physics and engineering, it has often been argued that complete synchronization may occur for a generic initial configuration in a large coupling regime. Such arguments are generally based on the results of numerical simulations. Unfortunately, this plausible scenario has not been completely verified by rigorous mathematical arguments, although there are several partial results available for a restricted class of initial configurations. In this paper, we provide a sufficient framework for complete synchronization from a generic initial configuration} in a large coupling regime. Our analysis depends on the gradient flow structure of the Kuramoto model and the uniform boundedness of the phase configuration.

We develop a global and hierarchical scheme for the forward Kolmogorov (Fokker-Planck) equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several alleles at the same locus in a population. The key of our scheme is to connect the solutions before and after the loss of an allele. Whereas in an approach via stochastic processes or partial differential equations, such a loss of an allele leads to a boundary singularity, from a biological or geometric perspective, this is a natural process that can be analyzed in detail. Our method depends on evolution equations for the moments of the process and a careful analysis of the boundary flux.

We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse this fact accurately, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in terms of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space.

The Ensemble Kalman lter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of catastrophic filter divergence have indicated that stability is a genuine mathematical concern and can not be taken for granted in implementation. In this article we propose a simple modication of ensemble based methods which resolves these stability issues entirely. The method involves a new type of adaptive covariance inflation, which comes with minimal additional cost. We develop a complete nonlinear stability theory for the adaptive method, yielding Lyapunov functions and geometric ergodicity under weak assumptions. We present numerical evidence which suggests the adaptive methods have improved accuracy over standard methods and completely eliminate catastrophic filter divergence. This enhanced stability allows for the use of extremely cheap, unstable forecast integrators, which would otherwise lead to widespread filter malfunction.

In this article, we clarify the mathematical framework underlying the construction of norm-conserving semilocal pseudopotentials for Kohn-Sham models, and prove the existence of optimal pseudopotentials for a family of optimality criteria. Most of our results are proved for the Hartree (also called reduced Hartree-Fock) model, obtained by setting the exchange-correlation energy to zero in the Kohn-Sham energy functional. Extensions to the Kohn-Sham LDA (local density approximation) model are discussed.

We investigate a periodic homogenization problem involving two isotropic materials with conductivities of different signs: a classical material and a metamaterial (or negative material). Combining the T-coercivity approach and the unfolding method for homogenization, we prove well-posedness results for the initial and the homogenized problems and we obtain a convergence result. These results are obtained under the condition that the contrast between the two conductivities is large enough in modulus. The homogenized matrix, is generally anisotropic and indefinite, but it is shown to be isotropic and (positive or negative) definite for particular geometries having symmetries.

We consider a microscopic non-linear model for friction mediated by transient elastic linkages introduced in our previous works. In the present study, we prove existence and uniqueness of a solution to the coupled system under weaker hypotheses. The theory we present covers the case where the off-rate of linkages is unbounded but increasing at most linearly with respect to the mechanical load.

Nonlocal Lotka-Volterra equations have the property that solutions concentrate as Dirac masses in the limit of small diffusion. In this paper, we show how the presence of an advection term changes the location of the concentration points in the limit of small diffusion and slow drift. The mathematical interest lies in the formalism of constrai ned Hamilton-Jacobi equations. Our motivations come from previous models of evolutionary dynamics in phenotype-structured populations [R.H. Chisholm, T. Lorenzi, A. Lorz, et al., Cancer Res., 75, 930-939, 2015], where the diffusion operator models the effects of heritable variations in gene expression, while the advection term models the effect of stress-induced adaptation.

This paper is concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann relation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact wave through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system. The analysis is based on the techniques developed in [R.-J. Duan and S.Q. Liu, arXiv:1403.2520] and an elementary L^2 energy method.

We study the dynamics of a quantum heavy particle undergoing a repulsive interaction with
a light particle. The main motivation is the detailed description of the loss of coherence induced on a quantum
system (in our model, the heavy particle) by the interaction with the environment (the light particle).

The content of the paper is analytical and numerical.

Concerning the analytical contribution, we show that an approximate description of the dynamics of the
heavy particle can be carried out in two steps: first comes the interaction, then the free evolution. In particular,
all effects of the nteraction can be embodied in the action of a collision operator that acts on the initial state
of the heavy particle. With respect to previous analytical results on the same topics, we turn our focus from
the Moller wave operator to the full scattering operator, whose
analysis proves to be simpler.

concerning the numerical contribution, we exploit the previous analysis to construct an eseparately. This leads to a considerable gain in simulation time. We present and interpret some simulations
carried out on specic one-dimensional systems by using the new scheme.

According to simulations, decoherence is produced by an interference-free bump which arises from the initial
state of the heavy particle immediately after the collision. We support such a picture by numerical evidence
as well as by an approximation theorem.

In this paper, we investigate the degenerate Keller-Segel-Stokes system (K-S-S) in a bounded convex domain \Omega \in R^2 with smooth boundary. A particular feature is that the chemotactic sensitivity S is a given parameter matrix on \Omega\times [0, \infty)^2, where Frobenius norm satisfies |S(x,n,c)|\le C_s with some C_s>0. It is shown that for any porous medium diffusion m>1, the system (K-S-S) with nonnegative and smooth initial data possesses at least a global-in-time weak solution, which is uniformly bounded.

In real-world geophysical applications (such as predicting climate change), the re duced models of real-world complex multiscale dynamics are used to predict the response of the actual multiscale climate to changes in various global atmospheric and oceanic parameters. However, while a reduced model may be adjusted to match a particular dynamical regime of a multiscale process, it is unclear why it should respond to external perturbations in the same way as the underlying multiscale process itself. In the current work, the authors study the statistical behavior of a reduced model of the linearly coupled multiscale Lorenz 96 system in the vicinity of a chosen dynamical regime by perturbing the reduced model via a set of forcing parameters and observing the response of the reduced model to these external perturbations. Comparisons are made to the response of the underlying multiscale dynamics to the same set of perturbations.

We consider the time evolution of quantum states by many-body Schrodinger dynamics and study the rate of convergence of their reduced density matrices in the bosonic mean field limit. If the prepared state at initial time is of coherent or factorized type and the number of particles $n$ is large enough then it is known that $1/n$ is the correct rate of convergence at any time. We show in the simple case of bounded pair potentials that the previous rate of convergence holds in more general situations with possibly correlated prepared states. In particular, it turns out that the coherent structure at initial time is unessential and the important fact is rather the speed of convergence of all reduced density matrices of the prepared states. We illustrate our result with several numerical simulations and examples of multi-partite entangled quantum states borrowed from quantum information.

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 analyze the global convergence properties of the filtered spherical harmonic (FP_N) equations for radiation transport. The well-known spherical harmonic (P_N) equations are a spectral method (in angle) for the radiation transport equation and are known to suffer from Gibbs phenomena around discontinuities. The filtered equations include additional terms to address this issue that are derived via a spectral filtering procedure. We show explicitly how the global L^2 convergence rate (in space and angle) of the spectral method to the solution of the transport equation depends on the smoothness of the solution (in angle only) and on the order of the filter. The results are confirmed by numerical experiments. Numerical tests have been implemented in MATLAB and are available online.

In the recent article [Hairer, M., Hutzenthaler, M., & Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468-527] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that the Euler scheme converges to the solution in the strong sense but with no polynomial rate. Hairer et al.’s result naturally leads to the question whether this slow convergence phenomenon can be overcome by using a more sophisticated approximation method than the simple Euler scheme. In this article we answer this question to the negative. We prove that there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion converges in absolute mean to the solution with a polynomial rate. Even worse, we prove that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.

We consider a conservation law perturbed by a linear diffusion and non-positive dispersion u_t+f(u)_x= \epsilon u_{xx} - \delta(|u_{xx}|^n)_x. We prove the convergence of the previous solution to the entropy weak solution of the hyperbolic conservation law u_t+f(u)_x=0, in both cases n=1 and n=2.

In this paper, we discuss a numerical approach for the simulation of a model for supply chains based on both ordinary and partial differential equations. Such methodology foresees differential quadrature rules and a Picard–like recursion. In its former version, it was proposed for the solution of ordinary differential equations and is here extended to the case of partial differential equations. The outcome is a ﬁnal non–recursive scheme, which uses matrices and vectors, with consequent advantages for the determination of the local error. A test case shows that traditional methods give worst approximations with respect to the proposed formulation.