This paper deals with the derivation of macroscopic equations from the underlying mesoscopic description that is suitable to capture the main features of pedestrian crowd dynamics. The interactions are modeled by means of theoretical tools of game theory, while the macroscopic equations are derived from asymptotic limits.

The present paper is devoted to the study of the Cauchy problem for the magnetic-curvature-driven electromagnetic fluid equation with random effects in a bounded domain of R^3. We first obtain a crucial property of the solution to O.U. process, thanks to the lemma, the local well-posedness of the equation with the initial and boundary value is established by the contraction mapping argument. Finally, by virtue of a priori estimates, the existence and uniqueness of global solution to the stochastic plasma equation is proved.

We consider the Cauchy problem for the equations of one-dimensional motion of a compressible inviscid gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients and the data, we prove that the problem admits a weak solution. More precisely, we show that a sequence of approximate solutions constructed by a generalized Glimn's scheme admits a subsequence converging to an entropic solution of the problem.

We study the large time behavior of solutions near a constant equilibrium state to the compressible Euler-Maxwell system in R^3. We first refine the global existence of solutions by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to H^{-s} (0\le s \le 3/2) or B^{-s}_{2, \infty} (0\le s \le 3/2), then we obtain the various time 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 follows without requiring the smallness for L^p norm of the initial data. So far, our decay results are most comprehensive ones for the bipolar Euler-Maxwell system in R^3.

We study a general Ericksen-Leslie system with non-constant density, which describes the flow of nematic liquid crystal. In particular the model investigated here is associated with Parodi's relation. We prove that: in two dimension, the solutions are globally regular with general data; in three dimension, the solutions are globally regular with small initial data, or for short time with large data. Moreover, a weak-strong type of uniqueness result is obtained.

In recent work, Li et al. (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $\epsilon$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $\epsilon$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms for selected test problems.

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.

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.

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.

Kohn-Sham density functional theory is one of the most widely used electronic structure theories. The recently developed adaptive local basis functions form an accurate and systematically improvable basis set for solving Kohn-Sham density functional theory using discontinuous Galerkin methods, requiring a small number of basis functions per atom. In this paper we develop residual-based a posteriori error estimates for the adaptive local basis approach, which can be used to guide non-uniform basis refinement for highly inhomogeneous systems such as surfaces and large molecules. The adaptive local basis functions are non-polynomial basis functions, and standard a posteriori error estimates for hp-refinement using polynomial basis functions do not directly apply. We generalize the error estimates for hp-refinement to non-polynomial basis functions. We demonstrate the practical use of the a posteriori error estimator in performing three-dimensional Kohn-Sham density functional theory calculations for quasi-2D aluminum surfaces and a single-layer graphene oxide system in water.

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.

We study the Kuramoto model for coupled oscillators. For the case of identical natural frequencies, we give a new proof of the complete frequency synchronization for all initial data; extending this result to the continuous version of the model, we manage to prove the complete phase synchronization for any non-atomic measure-valued initial datum. We also discuss the relation between the boundedness of the entropy and the convergence to an incoherent state, for the case of non identical natural frequencies.

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.

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.

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.

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.

Here we develop a model of smectic-C liquid crystals by forming their hydrostatic and hydrodynamic theories, which are motivated by the work of W. E [Arch. Rational Mech. Anal., 137(1997)]. A simplied model is also presented. In order to prove the rationality of the model, we establish the energy dissipative relation of the new model. Meanwhile, we verify that the system can also be obtained using asymptotic analysis when both the fluid and layers are incompressible.

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.

We characterize interior transmission eigenvalues of penetrable anisotropic acoustic scattering objects by a technique known as inside-outside duality. This method has recently been identified to be able to link interior eigenvalues of the penetrable scatterer with the behavior of the eigenvalues of the far field operator for the corresponding exterior time-harmonic scattering problem. A basic ingredient for the resulting connection is a suitable self-adjoint factorization of the far field operator based on wave number-dependent function spaces. Under certain conditions on the anisotropic material coeffi cients of the scatterer, the inside-outside duality allows to rigorously characterize interior transmission eigenvalues from multi-frequency far field data. This theoretical characterization moreover allows to derive a simple numerical algorithm for the approximation of interior transmission eigenvalues. Since it is merely based on far field data, the resulting eigenvalue solver does not require knowledge on the scatterer or its material coefficient; several numerical examples show its feasibility and accuracy for noisy data.

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.

We consider two-phase Navier-Stokes flow with a Boussinesq-Scriven surface fluid. In such a fluid the rheological behaviour at the interface includes surface viscosity effects, in addition to the classical surface tension effects. We introduce and analyze parametric finite element approximations, and show, in particular, stability results for semidiscrete versions of the methods, by demonstrating that a free energy inequality also holds on the discrete level. We perform several numerical simulations for various scenarios in two and three dimensions, which illustrate the effects of the surface viscosity.

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.

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.

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.

We study a model of magnetization switching driven by a spin current: the magnetization reversal can be induced without applying an external magnetic field. We first write our one dimensional model in an adimensionalized form, using a small parameter $\epsilon$. We then explain the various time and space scales involved in the studied phenomena. Taking into account these scales, we first construct an appropriate numerical scheme, that allows us to recover numerically various results of physical experiments. We then perform a formal asymptotic study as $\epsilon$ tends to 0, using a multiscale approach and asymptotic expansions. We thus obtain approximate limit models that we compare with the original model via numerical simulation.

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.

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.

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.

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.

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.

In this note we provide new non-uniqueness examples for the continuity equation by constructing infinitely many weak solutions with prescribed energy.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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