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.

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

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.

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.

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.

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.

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.

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.

We consider the motion of a compressible, viscous, and heat conducting fluid in the regime of small viscosity and heat conductivity. It is shown that weak solutions of the associated Navier-Stokes-Fourier system converge to a (strong) solution of the Euler system on its life span. The problem is studied in a bounded domain \Omega \in R^3, on the boundary of which the velocity field satisfies the complete slip boundary conditions.

This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that may be degenerate, which also includes an advection term appearing in many phase-field models for multi-component fluid flows. A class of maximum principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility. For time discretization, the standard semi-implicit scheme as well as the stabilized semi-implicit scheme will be adopted, while for space discretization, the central finite difference is used for approximating the diffusion term and the upwind scheme is employed for the advection term. We establish the maximum principle for both semi-discrete (in time) and fully discretized schemes. We also provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are carried out to verify our theoretical results.

The asymmetric stem-cell division of Drosophila SOP precursor cells is driven by
the asymmetric localisation of the key protein Lgl (Lethal giant larvae) during mitosis, when Lgl is
phosphorylated by the kinase aPKC on a subpart of the cortex and subsequently released into the
cytoplasm.

In this paper, we present a volume-surface reaction-diffusion system, which models the localisation
of Lgl within the cell cytoplasm and on the cell cortex. We prove well-posedness of global solutions as
well as regularity of the solutions. Moreover, we rigorously perform the fast reaction limit to a reduced
quasi-steady-state approximation system, when phosphorylated Lgl is instantaneously expelled from the
cortex. Finally, we apply a suitable first order finite element scheme to simulate and discuss interesting
numerical examples, which illustrate i) the influence of the presence/absence of surface-diffusion to the
behaviour of the system and the complex balance steady state and ii) the dependency on the release
rate of phosphorylated cortical Lgl.

The paper is concerned with a direct proof of the uniqueness of global conservative solutions to the two-component Camassa-Holm system, based on characteristics. Given a conservative solution u=u(t,x) and \rho=\rho(t,x), an equation is introduced to single out a unique charcteristic curve through each initial point. It is proved that the Cauchy problem with general initial data u_0\in H^1(R), \rho_0\in L^2(R) has a unique global conservative solution.

We consider a system of coupled nonlinear Schrodinger equations in one space dimension. First, we prove the existence of multi-speed solitary waves, i.e solutions to the system with each component behaving at large times as a solitary wave. Then, we investigate numerically the interaction of two solitary waves supported each on one component. Among the possible outcomes, we find elastic and inelastic interactions, collision with mass extraction and reflexion.

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space {0, 1, ..., N} and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. While the mean-field model is a well known approximation, this lower bound is new, and unlike an asymptotic result, these bounds can be used for finite N. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.

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.

Modelling extreme events is a central issue in climate science and engineering. The capacity of imperfect models to capture intermittent behavior with fat-tailed probability distributions of a passive scalar field advected by turbulent flow systems is investigated here. We consider the effects with complicated flow systems including strong nonlinear and non-Gaussian interactions, and construct much simpler and cheaper imperfect models with model error to capture the crucial statistical features in the stationary tracer field. The Lorenz 96 (L-96) system is utilized as a test model to generate the turbulent advection flow field. Tracer statistics under this L-96 flow field are analyzed both theoretically and numerically, and strong intermittent fat tails can be observed in different dynamical regimes of the flow system with distinct statistical features. The complexity and large computational expense in resolving the true advection flow require the introduction of simpler and more tractable imperfect models which still maintain the ability to capture the key intermittent features in the tracer field. The simplest linear stochastic models containing no positive Lyapunov exponents are proposed here to approximate the tracer advected by the original L-96 system with large degrees of internal instabilities. It is demonstrated that the prediction skill of this imperfect linear model can be greatly improved through fitting the autocorrelation functions by empirical information theory. A systematic framework of measuring the autocorrelation function under spectral representation with the help of empirical information theory is developed, and the optimal model parameters under this unbiased information measure can be achieved easily in a training phase before running the predictions. This imperfect model using optimal parameters achieved through the information-theoretic framework is tested in a variety of dynamical regimes of the L-96 system. Uniformly high skill of the optimal model is displayed in accurately capturing the crucial tracer statistical features in a stationary statistical steady state, especially in getting accurate intermittent fat tails in tracer density distributions. This information framework for tuning autocorrelation functions can be further generalized to more complicated turbulent models and should have many applications.

In the current work we demonstrate the principal possibility of prediction of the response of the largest Lyapunov exponent of a chaotic dynamical system to a small constant forcing perturbation via a linearized relation, which is computed entirely from the unperturbed dynamics. We derive the formal representation of the corresponding linear response operator, which involves the (computationally infeasible) infinite time limit. We then compute suitable finite-time approximations of the corresponding linear response operator, and compare its response predictions with actual, directly perturbed and measured, responses of the largest Lyapunov exponent. The test dynamical system is a 20-variable Lorenz 96 model, run in weakly, moderately, and strongly chaotic regimes. We observe that the linearized response prediction is a good approximation for the moderately and strongly chaotic regimes, and less so in the weakly chaotic regime due to intrinsic nonlinearity in the response of the Lyapunov exponent, which the linearized approximation is incapable of following.

The enthalpy regularization is a preliminary step in many numerical methods for the simulation of phase change problems. It consists in smoothing the discontinuity (on the enthalpy) caused by the latent heat of fusion and yields a thickening of the free boundary. The phase change occurs in a curved strip, i.e. the mushy zone, where solid and liquid phases are present simultaneously. The width \epsilon of this (mushy) region is most often considered as the parameter to control the regularization effect. The purpose we have in mind is a rigorous study of the effect of the process of enthalpy smoothing. The melting Stefan problem we consider is set in a semi-infinite slab, heated at the extreme-point. After proving the existence of an auto-similar temperature, solution of the regularized problem, we focus on the convergence issue as \epsilon \to 0. Estimates found in the literature predict an accuracy like \sqrt{\epsilon}. We show that the thermal energy trapped in the mushy zone decays exactly like \sqrt{\epsilon}, which indicates that the global convergence rate of \sqrt{\epsilon} cannot be improved. However, outside the mushy region, we derive a bound for the gap between the smoothed and exact temperature fields that decreases like \epsilon. We also present some numerical computations to validate our results.

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions. Several numerical experiments are performed to verify the well-balanced property and ability of the proposed scheme to accurately capture small perturbations of such steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.

Marketing on random networks displays similarities to epidemiological models in the sense that ``word-of-mouth'' information passes between individuals and may ``infect'' susceptible buyers such that they end up buying the product. The difference to epidemics is that there are usually many competing products (rather than just one disease), and in addition to word-of-mouth transmission, products are also advertised by the producers, which can be thought of as external nodes connected to the network. In this paper we develop a model in which these various transmission pathways compete, and, in addition, where product fatigue and product switching are possible. This is a genuine and realistic extension of the model developed in [Li, M. and Edwards, R. and Illner, R. and Ma, J., Commun. Math. Sci., 13, 497--509, 2015], where a customer would never abandon a product after purchase. The model presented here is similar to and was inspired by SIS epidemiological models. We discuss the homogeneous limit for a fully connected graph, present some analytical properties of the models and conduct a number of numerical experiments, including an investigation of a modelling assumption we call ``edge chaos''. The validity of this assumption turns out to depend on the type of the underlying random network.

In this paper, we examine structured tensors which have sum-of-squares (SOS) tensor decomposition, and study the SOS-rank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, B0-tensors, double B-tensors, quasi-double B0-tensors, MB0-tensors, H-tensors, absolute tensors of positive semi-definite Z-tensors and extended Z-tensors. We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS decomposition and the SOS-width for general SOS tensor cone using the known results in the literature of polynomial theory. Then, we provide an explicit sharper estimate for the SOS-rank of SOS tensor decomposition with bounded exponent and identify the SOS-width for the tensor cone consisting of all tensors with bounded exponent that have SOS decompositions. Finally, as applications, we show how the SOS tensor decomposition can be used to compute the minimum H-eigenvalue of an even order symmetric extended Z-tensor and test the positive definiteness of an associated multivariate form. Numerical experiments are also provided to show the ef- ficiency of the proposed numerical methods ranging from small size to large size numerical examples.

An initial-boundary value problem for a chemical system with unknown velocity related to gas chromatography is considered. The system is hyperbolic and existence of entropy solutions is achieved in fractional BV spaces: BV^*, s\ge 1/3, with less regularity than usual. We prove that BV^{1/3} is the critical space for this problem. A Lagrangian formulation of the system for the initial value problem provides a smoothing effect in BV and uniqueness when the first gas is more active than the second one.

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.

We introduce a model of liquidity risk through a stochastic supply curve for price taking traders. The supply curve gives the actual execution cost investors face in trading assets. We use the solutions to the modified Black-Scholes type PDE and obtain the delta-hedging strategies. We then show the replicating portfolio including liquidity costs converges to the payoff of the option. We demonstrate the replication error of discrete- time trading strategy decreases with inhomogeneous rebalancing times, and investigate an optimal positioning of them.

Observed avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models.

The notion of symmetry classically defined for hyperbolic systems of conservation law is extended to the case of evolution equations of conservative form for which the flux function can be an operator. We explain how such a symmetrization can work from a general point of view using an extension of the classical Godunov structure. We then apply it to the Green-Naghdi type equations which are a dispersive extension of the hyperbolic shallow-water equations. In fact, in the case of these equations, the general Godunov structure of the system is obtained from its Hamiltonian structure.

The classical models for irreversible diffusion-controlled reactions can be derived by introducing absorbing boundary conditions to over-damped continuous Brownian motion (BM) theory. As there is a clear corresponding stochastic process, we can describe them by the duality between the Kolmogorov forward equation for the dynamics of the probability distribution function and the specific stochastic trajectory of one particle. This duality is a fundamental characteristic of stochastic processes and allows simple particle based simulations to accurately match the expected statistical behavior. However, in the traditional theory using boundary conditions to model reversible reactions with geminate recombinations, several subtleties arise: It is unclear what the underlying stochastic process is, which causes complications in producing accurate simulations; and it is non-trivial how to perform an appropriate discretization for numerical computations. In this work, we derive a discrete stochastic model for reversible reactions that recovers the classical models and their boundary conditions in the continuous limit. Furthermore, all the complications encountered in the continuous models become trivial. Our analysis again confront the question: With computations in mind, what model should be considered more fundamental?

The incompressible Boussinesq equations serve as an important model in geophysics as well as in the study of Rayleigh-Benard convection. One generalization is to replace the standard Laplacian operator by a fractional Laplacian operator, namely (-\Delta)^{\alpha/2} in the velocity equation and (-\Delta)^{\beta/2} in the temperature equation. This paper is concerned with the two-dimensional (2D) incompressible Boussinesq equations with critical dissipation (\alpha+\beta=1) or supercritical dissipation (\alpha+\beta<1). We prove two main results. This first one establishes the global-in-time existence of classical solutions 10 to the critical Boussinesq equations with \alpha+\beta=1 and 0.7692 \approx 10/13<\alpha<1. The second one proves the eventual regularity of Leray-Hopf type weak solutions to the Boussinesq \alpha+\beta<1 and 0.7692 \approx 10/13<\alpha<1.