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.

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

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

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.

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

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

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

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

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

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

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

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

The content of the paper is analytical and numerical.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.