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.

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

We 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.

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This paper focuses on the 2D incompressible Boussinesq equations with fractional dissipation, given by \Lamda^\alphau in the velocity equation and by \Lambda^\beta \theta in the temperature equation, where \Lambda=-\sqrt{\Delta} denotes the Zygmund operator. Due to the vortex stretching and the lack of sufficient dissipation, the global regularity problem for the supercritical regime \alpha + \beta <1 remains an outstanding problem. This paper presents several regularity criteria for the supercritical Boussinesq equations. These criteria are sharp and reflect the level of difficulty of the supercritical Boussinesq problem. In addition, these criteria are important tools in understanding some crucial properties of Boussinesq solutions such as the eventual regularity.

Semi-discrete Runge-Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry-Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge-Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schrodinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wave-length \epsilon. The estimates are performed for the scalar wave equation and the Schrodinger equation. Our result demonstrates that a Gaussian beam superposition with k-th order beams converges to the exact solution as O(\epsilon^{k/2-s}) in order s Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is O(\epsilon^[k/2]) and away from the essential support of the solution, the convergence is spectral in \epsilon. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate O(\epsilon^{(k-2)/2}) in n spatial dimensions.

We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators.

The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem.
We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method.

This paper is devoted to the full system of incompressible liquid crystals, as modeled in the Q-tensor framework. The main purpose is to establish the uniqueness of weak solutions in a two dimensional setting, without imposing an extra regularity on the solutions themselves. This result only requires the initial data to fulfill the features which allow the existence of a weak solution. Thus, we also present a revisit of the global existence result in dimension two and three.

We propose a model that describes phase transition including metastable states present in the van der Waals Equation of State. From a convex optimization problem on the Helmoltz free energy of a mixture, we deduce a dynamical system that is able to depict the mass transfer between two phases, for which equilibrium states are either metastable states, stable states or a coexistent state. The dynamical system is then used as a relaxation source term in an isothermal 4 by 4 two-phase model. We use a Finite Volume scheme that treats the convective part and the source term in a fractional step way. Numerical results illustrate the ability of the model to capture phase transition and metastable states.

We show that, for the space of Borel probability measures on a Borel subset of a Polish metric space, the extreme points of the Prokhorov, Monge-Wasserstein and Kantorovich metric balls about a measure whose support has at most n points, consist of measures whose supports have at most n+2 points. Moreover, we use the Strassen and Kantorovich-Rubinstein duality theorems to develop representations of supersets of the extreme points based on linear programming, and then develop these representations towards the goal of their efficient computation.

We prove nonlinear stability of viscous shock wave of arbitrary amplitudes to a onedimensional compressible isentropic Navier-Stokes equations with density dependent viscosity. Under the assumption that the viscous coefficient is given as a power function of density, any viscous shock wave is shown to be nonlinear stable for small initial perturbations with integral zero. In contrast to previous related results [A. Matsumura, K. Nishihara, Japan J. Appl. Math., 2, 17-25, 1985; A. Matsumura, Y. Wang, Methods Appl. Anal., 17, 279-290, 2010], there is no restrictions on the power index of the viscous coefficient and the amplitudes of the viscous shock wave in our result.

This paper studies the non-autonomous micropolar fluid flows in two-dimensional bounded domains with external forces containing infinite delay effects. The authors first prove the global well-posedness of the weak solutions and then establish the existence of the pullback attractors for the associated process.

We formulate the large deviations for a class of two scale chemical kinetic processes motivated from biological applications. The result is successfully applied to treat a genetic switching model with positive feedbacks. The corresponding Hamiltonian is convex with respect to the momentum variable as a by-product of the large deviation theory. This property ensures its superiority in the rare event simulations compared with the result obtained by formal WKB asymptotics. The result is of general interest to understand the large deviations for multiscale problems.

The paper is concerned with the Burgers-Hilbert equation u_t + (u^2/2)_x = H[u], where the right hand side is a Hilbert transform. Unique entropy admissible solutions are constructed, locally in time, having a single shock. In a neighborhood of the shock curve, a detailed description of the solution is provided.

We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced from a bound on the four-dimensional velocity vector field to any two of its four components, from a bound on the gradient of the velocity vector field to the gradient of any two of its four components, from a gradient of the pressure scalar field to any two of its partial derivatives. Results are further generalized to the magnetohydrodynamics system. These results may be seen as a four-dimensional extension of many analogous results that exist in the three-dimensional case and also component reduction results of many classical results.

In liquid crystals, the existing experiments and simulations suggest that for various types of molecules, no homogeneous phase is found breaking the molecular symmetry. It has been proved for rod-like molecules. We conjecture that it holds for two types of two-fold symmetries, and prove it for some molecules with these symmetries.

We analyse kinetic and macroscopic models intended to describe pursuit-evasion dynamics. We investigate well-posedness issues and the connection between the two modeling, based on asymptotic analysis. In particular, in dimension 2, we show that the macroscopic system has some regularizing effects: bounded solutions are produced, even when starting from integrable but possibly unbounded data. Our proof is based on De Giorgi's method.

We establish the existence of axially symmetric weak solutions to steady incompressible magnetohydrodynamics with non-homogeneous boundary conditions. The key issue is the Bernoulli's law for the total head pressure to a special class of solutions to the inviscid, non-resistive MHD system, where the magnetic field only contains the swirl component.

Full waveform inversion is a successful procedure for determining properties of the earth from surface measurements in seismology. This inverse problem is solved by PDE constrained optimization where unknown coefficients in a computed wavefield are adjusted to minimize the mismatch with the measured data. We propose using the Wasserstein metric, which is related to optimal transport, for measuring this mismatch. Several advantageous properties are proved with regards to convexity of the objective function and robustness with respect to noise. The Wasserstein metric is computed by solving a Monge-Ampere equation. We describe an algorithm for computing its Frechet gradient for use in the optimization. Numerical examples are given.

In this paper, the initial-boundary value problem of a 1-D bipolar quantum semiconductor hydrodynamic model is investigated under a non-linear boundary condition which means the quantum effect vanishes on the boundary. First of all, the existence and uniqueness of the corresponding stationary solution are established. Then the exponentially asymptotic stability of the stationary solution and the semi-classical limits are further studied. The adopted approach is the elementary energy method but with some new developments.

The alternating minimization is an efficient method for solving the convex minimization whose objective function is a sum of differentiable function and a separable nonsmooth one. Variants and extensions of the alternating minimization method have been developed in recent years. In this paper, we consider the convergence rate of several existing alternating minimization schemes. We improve the proved big-O convergence rate of these algorithms to little-o under an error bound condition which is actually quite common in applications. We also investigate the convergence of a variant of alternating minimization proposed in this paper.

This paper deals with a Boltzmann-type kinetic model describing the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow, including the distribution of the driving risk along a road, are ultimately generated by one-to-one interactions among drivers, the model links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, it is finally possible to recover macroscopic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles.

In this paper, we propose numerical methods for computing the boundary local time of reflecting Brownian motion (RBM) for a bounded domain in R^3 and its use in the probabilistic representation of the solution of the Laplace equation with the Neumann boundary condition. Approximations of RBM based on walk-on-spheres (WOS) and random walk on lattices are discussed and tested for sampling RBM paths and their applicability in finding accurate approximation of the local time and discretization of the probabilistic formula. Numerical tests for several domains (a cube, a sphere, an ellipsoid, and a non-convex nonsmooth domain) have shown the convergence of the numerical methods as the time length of RBM paths and number of paths sampled increase.

We prove existence and uniqueness of solutions to a transport equation modelling vehicular traffic in which the velocity field depends non-locally on the downstream traffic density via a discontinuous anisotropic kernel. The result is obtained recasting the problem in the space of probability measures equipped with the $\infty$-Wasserstein distance. We also show convergence of solutions of a finite dimensional system, which provide a particle method to approximate the solutions to the original problem.

In this note, we propose a discrete model to study one-dimensional transport equations with non-local drift and supercritical dissipation. The inspiration for our model is the equation $$ \theta_t + (H\theta) \theta_x +(-\Delta)^\alpha \theta =0 $$ where $H$ is the Hilbert transform. For our discrete model, we present blow-up results that are analogous to the known results for the above equation. In addition, we will prove regularity for our discrete model which suggests supercritical regularity in the range $1/4<\alpha<1/2$ in the continuous setting.

We consider the inverse problem of determining the highly oscillatory coefficient a in partial differential equations of the form -nabla \cdot (a \nabla u_ + bu = f from given measurements of the solutions. Here, indicates the smallest characteristic wavelength in the problem (0<\epsilon <<1). In addition to the general difficulty of finding an inverse is the challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed, and one common approach is to reduce the dimension by seeking effective parameters. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, b=0, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, b>0, and exploration seismology, b<0.

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

In this paper, a numerical scheme for a generalized planar Ginzburg-Landau energy in a circular geometry is studied. A spectral-Galerkin method is utilized, and a stability analysis and an error estimate for the scheme are presented. It is shown that the scheme is unconditionally stable. We present numerical simulation results that have been obtained by using the scheme with various sets of boundary data, including those the form u(\theta)=exp(id\theta), where the integer d denotes the topological degree of the solution. These numerical results are in good agreement with the experimental and analytical results. Results include the computation of bifurcations from pure bend or splay patterns to spiral patterns for d=1, energy decay curves for d=1, spectral accuracy plots for d=2 and computations of metastable or unstable higher-energy solutions as well as the lowest energy ground state solutions for values of d ranging from two to five.

We analyze the weak-coupling limit of the random Schrodinger equation with low frequency initial data and a slowly decorrelating random potential. For the probing signal with a sufficiently long wavelength, we prove a homogenization result, that is, the properly compensated wave field admits a deterministic limit in the "very low" frequency regime. The limit is "anomalous" in the sense that the solution behaves as exp(-Dt^s) with s>1 rather than the "usual" exp(-Dt) homogenized behavior when the random potential is rapidly decorrelating. Unlike in rapidly decorrelating potentials, as we decrease the wavelength of the probing signal, stochasticity appears in the asymptotic limit-- there exists a critical scale depending on the random potential which separates the deterministic and stochastic regimes.

In this work we extend a recent kinetic traffic model [G. Puppo, M. Semplice, A. Tosin and G. Visconti, submitted, 2015] to the case of more than one class of vehicles, each of which is characterized by few different microscopic features. We consider a Boltzmann-like framework with only binary interactions, which take place among vehicles belonging to the various classes. Our approach differs from the multi-population kinetic model proposed in [G. Puppo, M. Semplice, A. Tosin and G. Visconti, Comm. Math. Sci., 2015] because here we assume continuous velocity spaces and we introduce a parameter describing the physical velocity jump performed by a vehicle that increases its speed after an interaction. The model is discretized in order to investigate numerically the structure of the resulting fundamental diagrams and the system of equations is analyzed by studying well posedness. Moreover, we compute the equilibria of the discretized model and we show that the exact asymptotic kinetic distributions can be obtained with a small number of velocities in the grid. Finally, we introduce a new probability law in order to attenuate the sharp capacity drop occurring in the diagrams of traffic.

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

Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those arising in turbulence, in which effective low-dimensional representation of the desired probability distribution is challenging. Nonetheless recent advances have shown considerable success in filtering based on certain carefully chosen simplifications of the underlying system, which allow closed form filters. This leads to filtering algorithms with significant, but judiciously chosen, model error. The purpose of this article is to analyze the effectiveness of these simplified filters, and to suggest modifications of them which lead to improved filtering in certain time-scale regimes. We employ a Markov switching process for the true signal underlying the data, rather than working with a fully resolved DNS PDE model. Such Markov switching models haven been demonstrated to provide an excellent surrogate test-bed for the turbulent bursting phenomena which make filtering of complex physical models, such as those arising in atmospheric sciences, so challenging.

Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrodinger system with a power-type nonlinearity. We show the local well-posedness in H^(R^3) \times H^{3/2}(R^3) and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.

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

The transformed l_1 penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates l_0 and l_1 similar to l_p norm, where p is in (0,1). In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the lp thresholding functions (p is in [0,1]) are in closed form only for p=0,1,1/2, 2/3, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds (TL1IT-s1 for short), consistently performs the best in sparse signal recovery with and without measurement noise.