# Forthcoming Papers

• Chunxiong Zheng
Global geometrical optics approximation to the high frequency Helmholtz equation with discontinuous media

Global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed by the author in [Commun. Math. Sci., 11(1), 105-140, 2013]. In this paper, we rederive it in a more concise way. It is shown that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, other than the WKB function used in the classical geometrical optics approximation. A new and main contribution of this paper is an interface analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident rays include caustic points.

• Denise Aregba
Godunov scheme for Maxwell equations with kerr nonlinearity

We study the Godunov scheme for a nonlinear Maxwell model arising in nonlinear optics, the Kerr model. This is a hyperbolic system of conservation laws with some eigenvalues of variable multiplicity, neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector 6\times 6 system is constrcuted and proved to exist for all data. This solution is comprated to the one of the reduced Transverse Magnetic model. The scheme is implemented in one and two space dimensions. The results are very close to the ones obtained with a Kerr-Debye relaxation approximation.

• Nengsheng Fang, Xinfeng Ruan and Caiciu Liao
Real option model upon dynamic growth process with construction

A real option model is built upon a stochastic process for some real investment decision making in incomplete markets. Typically, optimal consumption level is obtained under logarithm utility constraint, and a partial integro-dierential equation (PIDE) of the real option is deduced by martingale methods. Analytical formulation of the PIDE is solved by Fourier transformation. Two types of decision making strategies, i.e.: option price and IRP (inner risk primium) comparisons, are provided. Monte Carlo simulation and numerical computation are provided at last to verify the conclusion.

• Ewelina Zatorska, Piotr Bogus law Mucha
Multicomponent Mixture Model. The Issue of Existence via Time Discretization

We prove the existence of global-in-time weak solutions to a model of chemically reacting mixture. We consider a coupling between the compressible Navier-Stokes system and the reaction diusion equations for chemical species when the thermal eects are neglected. We rst prove the existence of weak solutions to the semi-discretization in time. Based of this, the existence of solutions to the evolutionary system is proved.

• Christophe Gomez
Wave Propagation in Underwater Acoustic Waveguides with Rough Boundaries

In underwater acoustic waveguides a pressure field can be decomposed over three kinds of modes: the propagating modes, the radiating modes and the evanescent modes. In this paper, we analyze the effects produced by a randomly perturbed free surface and an uneven bottom topography on the coupling mechanism between these three kinds of modes. Using an asymptotic analysis based on a separation of scales technique we derive the asymptotic form of the distribution of the forward mode amplitudes. We show that the surface and bottom fluctuations affect the propagating-mode amplitudes mainly in the same way. We observe an effective amplitude attenuation which is mainly due to the coupling between the propagating modes themselves. However, for the highest propagating modes this mechanism is stronger and due to an efficient coupling with the radiating modes.

• Chao Chen
Subsonic potential flows in general smooth bounded domains

In this paper, we study the existence and uniqueness of subsonic potential flows in general smooth bounded domains when the normal component of the momentum on the boundary is prescribed. It is showed that if the Bernoulli constant is given larger than a critical number, there exists a unique subsonic potential flow. Moreover, as the Bernoulli constants decrease to the critical number, the subsonic flows converge to a subsonic-sonic flow.

• Qiang Tao, Jincheng Gao and Zheng-an Yao
Global strong solutions of the compressible nematic liquid crystal flow with the cylinder symmetry

In this paper, we consider the well-posedness of the compressible nematic liquid crystal flow with the cylinder symmetry in R^n. By establishing a uniform point-wise positive lower and upper bounds of the density, we derive the global existence and uniqueness of strong solution and show the long time behavior of the global solution. Our results do not need the smallness of the initial data. Furthermore, a regularity result of global strong solution is given as well.

• Rongjie Lai, Jianfeng Lu and Stanley Osher
Density matrix minimization with l_1 regularization

We propose a convex variational principle to find sparse representation of low-lying eigenspace of symmetric metrices in the context of electronic structure calculation, this corresponds to a sparse density matrix minimization algorithm with l_1 regularization. The minimization problem can be efficiently solved by a split Bregman iteration type algorithm. We further prove that from any initial condition, the algorithm converges to a minimizer of the variational principle.

• Russel E. Caflisch, Stanley J. Osher, Hayden Schaeffer and Giang Tran
PDEs with compressed solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an L^1 norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradientfl ow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an L^1 form, such as the divisible sandlile problem and signum-Gordon. Addition of an L^1 term in the variational principle leads to a modied PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving L^1 based problems.

• Bojan Popov and Vladimir Tomov
Central schemes for Mean Field Games

Mean field type models have been recently introduced and analyzed by Lasry and Lions. They describe a limiting behavior of stochastic dierential games as the number of players tends to infinity. Numerical methods for the approximation of such models have been developed by Achdou, Camilli, Capuzzo-Dolcetta, Gueant, and others. Efficient algorithms for such problems require special efforts and so far all methods introduced have been first order accurate. In this manuscript we design a second order accurate numerical method for time dependent Mean Field Games. The discretization is based on central schemes which are widely used in hyperbolic conservation laws.

• Celment Cances, Frederic Coquel, Edwige Godlewski, Helene Mathis, and Niclas Seguin
Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

In order to validate theoretically a dynamic model adaptation method, we propose to consider a simple case where the model error can be thoroughly analyzed. The dynamic model adaptation consists in detecting at each time step the region where a given fine model can be replaced by a corresponding coarse model in an automatic way, without deteriorating the accuracy of the result, and to couple the two models, each being computed on its respective region. Our fine model is 2 \times 2 system which involves a small time scale and setting this time scale to 0 leads to a classical conservation law, the coarse model, with a flux which depends on the unknown and on space and time. The adaptation method provides an intermediate adapted solution which results from the coupling of both models at each time step. In order to obtain sharp and rigorous error estimates for the model adaptation procedure, a simple fine model is investigated and smooth transitions between fine and coarse models are considered. We refine existing stability results for conservation laws with respect to the flux function which enables us to know how to balance the time step, the threshold for the domain decomposition and the size of the transition zone. Numerical results are presented at the end and show that our estimate is optimal.

• Christophe Berthon, Bruno Debroca and Afeintou Sangam
An entropy preserving relaxation scheme for the ten-moments equations with source terms

The present paper concerns the derivation of finite volume methods to approximate weak solutions of Ten-Moments equations with source terms. These equations model compressible anisotropic flows. A relaxation type scheme is proposed to approximate such flows. Both robustness and stability conditions of the suggested finite volume methods are established. To prove discrete entropy inequalities, we derive a new strategy based on local minimum entropy principle and never use some approximate PDE's auxiliary model as usually recommended. Moreover, numerical simulations in 1D and in 2D illustrate our approach.

• Alberto Bressan and Tao Huang
Representation of Dissipative Solutions to a Nonlinear Variational Wave Equation

The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time.

• Lyle Noakes and Tudor S. Ratiu
Bi-Jacobi fields and Riemannian cubics for left-invariant SO(3)

Bi-Jacobi fields are generalized Jacobi fields, and are used to efficiently compute approximations to Riemannian cubic splines in a Riemannian manifold M. Calculating bi-Jacobi fields is straightforward when M is a symmetric space such as bi-invariant SO(3), but not for Lie groups whose Riemannian metric is only left-invariant. Because left-invariant Riemannian metrics occur naturally in applications, there is also a need to calculate bi-Jacobi fields in such cases. The present paper investigates bi-Jacobi fields for left-invariant Riemannian metrics on SO(3), reducing calculations to quadratures of Jacobi fields. Then left Lie-reductions are used to give an easily implemented numerical method for calculating bi-Jacobi fields along geodesics in SO(3), and an example is given of a nearly geodesic approximate Riemannan cubic.

• Huimin Yu
Large Time Behavior of Entropy Solutions to Unipolar Hydrodynamic Model of Semiconductors

In this paper, we study the large time behavior of entropy solutions to the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Poisson equations. First of all, a large time behavior framework for the time-increasing entropy solutions is given. In this framework, the global entropy solutions (which increase with time not very fast) are proved to decay exponentially fast to the corresponding stationary solutions. Then, for an application purpose, the existence and time-increasing-rate of the global entropy solutions with large initial data is considered by using a modified fractional step Lax-Friedrichs scheme and the theory of compensated compactness. By using the large time behavior framework, the global entropy solutions are proved to decay exponentially fast to the stationary solutions when the adiabatic index $\G>3$, without any assumption on smallness or regularity for the initial data.

• Houman Owhadi and Clint Scovel
Brittleness of Bayesian inference and new Selberg formulas

The incorporation of priors [H. Owhadi, C. Scovel and T.J. Sullivan, arXiv:1304.6772, 2013] in the Optimal Uncertainty Quantication (OUQ) framework [H. Owhardi, C. Scovel, T.J. Sullican, M. McKerns, and M. Ortiz, SIAM Review 2013] reveals brittleness in Bayesian inference; a model may share an arbitrarily large number of finite-dimensional marginals with, or be arbitrarily close (in Prokhorov or total variation metrics) to, the data-generating distribution and still make the largest possible prediction error after conditioning on an arbitrarily large number of samples. The initial purpose of this paper is to unwrap this brittleness mechanism by providing (i) a quantitative version of the Brittleness Theorem of [H. Owhadi, C. Scovel and T.J. Sullivan, arXiv:1304.6772, 2013] and (ii) a detailed and comprehensive analysis of its application to the revealing example of estimating the mean of a random variable on the unit interval [0,1] using priors that exactly capture the distribution of an arbitrarily large number of Hausdor moments.
However, in doing so, we discovered that the free parameter associated with Markov and Krein's canonical representations of truncated Hausdor moments generates reproducing kernel identities corresponding to reproducing kernel Hilbert spaces of polynomials. Furthermore, these reproducing identities lead to biorthogonal systems of Selberg integral formulas.
This process of discovery appears to be generic: whereas Karlin and Shapley used Selberg's integral formula to first compute the volume of the Hausdor moment space (the polytope defined by the first n moments of a probability measure on the interval [0, 1], we observe that the computation of that volume along with higher order moments of the uniform measure on the moment space, using different finite-dimensional representations of subsets of the infinite-dimensional set of probability measures on [0,1] representing the first n moments, leads to families of equalities corresponding to classical and new Selberg identities.

• Florent Berthelin, D. Chiron and Magali Ribot
Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure

In this article, we study a one-dimensional hyperbolic quasilinear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line R and on a bounded interval with no- ux boundary conditions. In the case of the whole line R, we nd only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an innite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations.

• Radjesvarane Alexandre, Jie Liao and Chunjin Lin
Convergence to the equilibrium state for Bose-Einstein 1-D Kac grazing limit model

The convergence to the equilibrium of the solution of a quantum Kac grazing limit model for Bose-Einstein identical particles is studied. Using the relative en-tropy method and a detailed analysis of the entropy production, the exponential decay rate is obtained under suitable assumptions on the mass and energy of the initial data. These theoretical results are further illustrated by numerical simulations.

• Stephen Pankarvich and Jack Schaeffer
Global classical solutions of the "one and one-half" dimensional Vlasov-Maxwell-Fokker-Planck system

We study the "one and one-half" dimensional Vlasov-Maxwell-Fokker-Planck system and obtain the first results concerning well-posedness of solutions. Specifically, we prove the global-in-time existence and uniqueness in the large of classical solutions to the Cauchy problem and a gain in regularity of the distribution function in its momentum argument.

• Yeping Li and Wen-an Yong
Zero Mach number limit of the compressible Navier-Stokes-Korteweg equations

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Navier-Stokes-Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Navier-Stokes-Korteweg equations, we establish a convergence-stability principle. Then we show that, when the Mach number is sufficiently small, the initial value problem of the compressible Navier-Stokes-Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Navier-Stokes equations have a smooth solution. It is important to remark that when the incompressible Navier-Stokes equations have a global smooth solution, the existence time of the solution for the compressible Navier-Stokes-Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Navier-Stokes-Korteweg equations towards those for the incompressible Navier-Stokes equations with a convergence rate. As we know, it is the first result about zero Mach number limit of the compressible Navier-Stokes-Korteweg equation.

• Bo Li, Jiayi Wen, and Shenggao Zhou
Mean-Field Theory and Computation of Electrostatics with Ionic Concentration Dependent Dielectrics

We construct a mean-field variational model to study how the dependence of dielectric coeectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and the ionic ideal-gas entropy. The electrostatic potential is determined by Poisson's equation in which the dielectric coefficient depends on the sum of concentrations of individual ionic species. This dependence is assumed to be qualitatively the same as that on the salt concentration for which experimental data are available and analytical forms can be obtained by the data fitting. We derive the first and second variations of the free-energy functional, obtain the generalized Boltzmann distributions, and show that the free-energy functional is in general nonconvex. To validate our mathematical analysis, we numerically minimize our electrostatic free-energy functional for a radially symmetric charged system. Our extensive computations reveal several features that are signicantly different from a system modeled with a dielectric coefficient independent of ionic concentration. These include the non-monotonicity of ionic concentrations, the ionic depletion near a charged surface that has been previously predicted by a one-dimensional model, and the enhancement of such depletion due to the increase of surface charges or bulk ionic concentrations.

• M. Bisi, M.J. Caceres
A BGK relaxation model for polyatomic gas mixtures

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.

• Hong Lu, Peter W. Bates, Shujuan Lv, and Mingji Zhang
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain

We study a stochastic fractional complex Ginzburg-Landau equation with multiplicative noise in three spatial dimensions with particular interest in the asymptotic behavior of its solutions. We first transform our equation into a random equation whose solutions generate a random dynamical system. A priori estimates are derived when the nonlinearity satisfies certain growth conditions. Applying the estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. Furthermore, the existence of a random attractorin H^1(R^3) of the random dynamical system is established.

• Guoding Pang and Nikola Sandric
Ergodicity and fluctuations of a fluid particle driven by diffusions with jumps

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.

• Nancy Rodriguez and Lenya Ryzhik
Exploring the effects of social preference, economic disparity, and heterogeneous environments on segregation

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.

• Nicolas Besse
Lagrangian averaged gyrokinetic-waterbag continuum

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.

• M. Groppi, G.Russo and G. Stracquadanio
High order semi-lagrangian methods for the BGK equation

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.

• Yongli Cai, Yun Kang, Malay Banerjee and Weiming Wang
A stochastic epidemic model incorporating media coverage

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^S1, it has endemic stationary distribution, which leads to the stochastically persistence of the disease. In order to understand the role of media coverage towards the disease dynamics, we present some numerical simulations to validate the analytical findings. It is interesting to note that although some parameters have no role to determine R_0^S, however the strength of noise to the susceptible population and the parameters characterizing media effect play crucial role to determine the long term dynamics of the system.

• Calin Iulian Martin and Bogdan Matioc
Gravity water flows with discontinuous vorticity and stagnation points

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.

• Chiun-Chang Lee, Hijin Lee, YunKyong Hyon, Tai-Chia Lin and Chun Liu
Boundary layer solutions of Charge Conserving Poisson-Boltzmann equations: one-dimensional case

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.

• Baoying Yang and Huihui Zeng
Zero Relaxation Limit to Rarefaction Waves for General $2\times 2$ Hyperbolic Systems With Relaxation

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.

• Yanfeng Guo, Jingjun Zhang and Chunxiao Guo
Attractor of the quantum Zakharov system on unbounded domain

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.

• Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti
Fundamental diagrams in traffic flow: the case of heterogeneous kinetic models

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.

• Doghonay Arjmand and Christian Stohrer
A finite element heterogeneous multiscale methods with improved control over the modeling error

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.

• Gung-Min Gie, Christopher Henderson, Gautam Iyer, Landon Kavlie and Jared P. Whitehead
Stability of vortex solutions to an extended Navier-Stokes system

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.

• Lian Yang and Xueke Pu
Derivation of the Burgers equation from the gas dynamics

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.

• Yanyan Yu, Weihua Deng and Yujiang Wu
Fourth order quasi-compact difference schemes for (tempered) space fractional diffusion equations

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.

• Jing Guo, Cheng Wang, Steven Wise, Xingye Yue
An H^2 Convergence of A Second-Order Convex-Splitting, Finite Difference Scheme for the Three-Dimensional Cahn-Hilliard Equation

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.

• Zhengguang Guo, Weiming Wang and Chongbin Xu
On a variation of the Camassa-Holm type system

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.

• Guoyin Li, Liqun Qi and Qun Wang
Positive Semi-Definiteness of Generalized Anti-Circular Tensors

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.

• Michael Herty and Benedetto Piccoli
Numerical method for the computation of tengent vectors to 2\times 2 hyperbolic systems of conservation laws

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.

• Sun-Ho Choi and Seung-Yeal Ha
Emergence of flocking for a holonormic multi-agent system moving with constant speed

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.

• Zhong Tan and Qiuju Xu
Time periodic solutions to the 3D compressible fluid models of Korteweg type

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.

• Christopher Henderson
Population Stabilization in Branching Brownian Motion With Drift and Absorption

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.

• David Gontier
Pure-state N-representability in current-spin-density functional theory

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.

• Fang Li and Xiaoming Zheng
Comparison of several reaction and diffusion models of growth factors in angiogenesis

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.

• Antoine Pauthier
The influence of a line with fast diffusion and nonlocal exchange terms on Fisher-KPP propagation

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.

• Zhong Tan and Leilei Tong
Decay estimates of solutions to the compressible Navier-Stokes-Maxwell system in R^3

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.

• Simone Goettlich and Sebastian Kuhn
Routing strategies in production networks with random breakdowns

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.

• Shengqian Chen and Samuel N. Stechmann
Nonlinear traveling waves for the skeleton of the Madden-Julian oscillation

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.

• Boling Guo, Qiaoxin Li and Xinglong Wu
Global Smooth Solution and Blow-up Phenomenon for a Coupled Nonlinear Schr?odinger Equations in Atomic Bose-Einstein Condensates

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.

• Lizhi Ruan, Dehua Wang, Shangkun Weng, and Changjiang Zhu
Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: linear stability

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.

• Shui-Nee Chow, Wuchen Li and Haomin Zhou
A Newton-like algorithm for the shortest path based on the method of evolving junctions

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.

• Agis Athanassoulis, Theodore Katsaounis and Irene Kyza
Regularized semiclassical limits: linear flows with infinite Lyapunov exponents

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.

• Omer Faruk Tekin, Ke Yin, Farzin Barekat
Spectral Results for Perturbed Variational Eigenvalue Problems and Their Applications to Compressed PDEs

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.

• Wonjung Lee and Terry Lyons
The adaptive patched particle filter and its implementation

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.

• Ben Duan, Zhen Luo and Jingjing Xiao
Transonic Shock Solutions to the Euler-Poisson System in Quasi-one-dimensional Nozzles

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].

• Ke Ye and Shenglong Hu
Multiplicities of eigenvalues of tensors

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.

• Nick Verheul and Daan Crommelin
Data-driven stochastic representations of unresolved features in multiscale models

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.

• Wenjia Jing
Homogenization of Randomly Deformed Conductivity Resistant Membranes

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.

• Fei Jiang
An improved result on Rayleigh-Taylor instability of nonhomogeneous incompressible viscous flows

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.

• Tong Li, Min Tang and Xu Yang
An augmented Keller-Segal model for E. coli chemotaxis in fast-varying environments

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.