A hydrodynamic model of open vesicles in solution is presented to study the enlargement and shrinkage of a pore in biological lipid membrane. The vesicle is modeled by diffusive interfaces. Transport equations permitting consistent treatment of the pore and pore rim are intro- duced. Dynamic simulations implemented by the finite difference method show the evolution of a pore in stretched vesicles. Simulation results include direct visualization of the membrane shape, water motion, and dissipation of energy. Comparison is made with data obtained from microscopy experiments.

Recently, liquid-gas flows involved in droplets, bubbles, and thin films on solid surfaces with thermal and wettability gradients have attracted widespread attention because of the many physical processes involved and their promising potential for applications in biology, chemistry, and industry. Various new physical effects have been discovered at fluid-solid interfaces by experiments and molecular dynamics simulations, e.g., fluid velocity slip, temperature slip (Kapitza resistance), mechanical-thermal cross coupling, etc. There have been various models and theories proposed to explain the experimental and numerical observations. However, to the best of our knowledge, a continuum hydrodynamic model capable of predicting the temperature and velocity profiles of liquid-gas flows on non-isothermal, heterogeneous solid substrates is still absent. The purpose of this work is to construct a continuum model for simulating the liquid-gas flows on solid surfaces that are flat and rigid, and may involve wettability gradient and thermal gradient. This model is able to describe fluid velocity slip, temperature slip, and mechanical- thermal coupling that may occur at fluid-solid interfaces. For this purpose, we first employ the diffuse interface modeling to formulate the hydrodynamic equations for one-component liquid- gas flows in the bulk region. This reproduces the dynamic van der Waals theory of Onuki [Phys. Rev. Lett., 94: 054501, 2005]. We then extend s method [Z. Naturforsch. A, 22: 1269-1280, 1967] to formulate the boundary conditions at the fluid-solid interface that match the hydrodynamic equations in the bulk. The effects of the solid surface curvature are also briefly discussed in the appendix. The guiding principles of our model derivation are the conservation laws and the positive definiteness of entropy production together with the Onsager reciprocal relation. The derived model is self-consistent in the sense that the boundary conditions are mathematically demanded by the bulk equations. A finite difference scheme is presented for numerically solving the model system. We show that some widely used boundary conditions can actually be recovered by taking appropriate limits. We also point out that the framework presented here for modeling two-phase flows on solid surfaces, from bulk equations to boundary conditions, is in a form that can be readily generalized to model other fluid-solid interfacial phenomena.

The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H^{-1}. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated, in the one-dimensional case, for the piecewise constant finite element approximation of the porous medium equation with the \delta-distribution as initial value. As a byproduct, L^p-stability of the H^{-1}-orthogonal projection onto the space of piecewise constant functions is shown for the one-dimensional case.

In this paper, we compose explicit two-stage Runge-Kutta schemes of strong order one for solutions of stochastic dierential equations driven by jump-diusion processes. By using rooted trees, we obtain the convergence rate. Our numerical tests verify our theoretical results.

We derive a continuum model for the dynamics of a dislocation array that consists of dislocations in different slip planes. In the continuum model, the dislocation array is represented by a continuous surface, of which there are many dislocations in a unit area at the scale of the continuum model. The continuum model is derived rigorously from the discrete model of the dynamics of the constituent dislocations in the array using asymptotic analysis. The obtained continuum model contains an integral over the dislocation array surface representing the long-range interaction of dislocations, and a local term that comes from the line tension effect of dislocations. The size-dependent effect due to dislocation line tension is accurately incorporated in the continuum model. Well-posedness of the continuum model is examined. Generalization to dislocation arrays in an elastically anisotropic medium is discussed.

We consider the nucleation of one-dimensional stochastic Cahn-Hilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic Cahn-Hilliard dynamics through nite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios.

This article studies the effect of discretisation error on the stationary distribution of stochastic partial differential equations (SPDEs). We restrict the analysis to the effect of space discretisation, performed by finite element schemes. The main result is that under appropriate assumptions the stationary distribution of the finite element discretisation converges in total variation norm to the stationary distribution of the full SPDE.

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving unsteady second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit for- mulation for the parabolic term. The implicit procedure uses a fast iterative algorithm with reduced evaluation cost introduced in [Renac, Marmignon and Coquel, SIAM J. Sci. Comput., 34 (2012), pp. 370-394]. The method is here extended to convection dominated flow problems. A second-order discretization in time is achieved by decomposing the integrations of convective and diffusive terms with a splitting method. Numerical examples are presented for the linear convection-diffusion equa- tion in one and two space dimensions. The performance of the present method is seen to be improved in terms of CPU time when compared to a full implicit discretization of the parabolic terms in a wide range of Peclet numbers.

We consider herein the instability properties of the periodic travelling wave solutions of a general nonlinear Boussinesq system related with a dispersive model for the 1D propagation of nonlinear long water waves with small amplitude, via an adaptation of the result of M. Grillakis, J. Shatah and W. Strauss for systems with a special Hamiltonian structure. In a particular case of this general system, we use Jacobian elliptic functions to build a curve of L-periodic travelling wave solutions having property the mean zero in [0;L] and also verify the validity of the criteria used to establish instability, in a specific range of the wave speed. Furthermore, we provide numerical evidence on a type of instability arising when perturbing with small amplitude disturbances by using a highly-accurate spectral numerical scheme.

We consider the long-time behavior and optimal decay rates of global strong solutions to the isentropic compressible Navier-Stokes-Korteweg system in R^3 in the present paper.

Following the seminal work of F. Bouchut on zero pressure gas dynamics, extensively used for gas particle-flows, the present contribution investigates quadrature-based velocity moments models for kinetic equations in the framework of the infinite Knudsen number limit, that is, for dilute clouds of small particles where the collision or coalescence probability asymptotically approaches zero. Such models define a hierarchy based on the number of moments and associated quadrature nodes, the first level of which leads to pressureless gas dynamics. We focus in particular on the four moment model where the flux closure is provided by a two-node quadrature in the velocity phase space and provides the right framework for studying both smooth and singular solutions. The link with both the kinetic underlying equation as well as with zero pressure gas dynamics, i.e. the dynamics at the frontier of the moment space of order four, is provided. We define the notion of measure solutions and characterize the mathematical structure of the resulting system of four PDEs. We exhibit a family of entropies and entropy fluxes and define the notion of entropic solution. We study the Riemann problem and provide entropic solutions in particular cases. This leads to a rigorous link with the possibility of the system of macroscopic PDEs to allow particle trajectory crossing (PTC) in the framework of smooth solutions. Generalized -shock solutions resulting from Riemann problem are also investigated. Finally, using a kinetic scheme proposed in the literature in several areas, we validate such a numerical approach and propose a dedicated extension at the frontier of the moment space in the framework of both regular and singular solutions. This is a key issue for application fields where such an approach is extensively used.

In this paper we deal with a continuous model for supply chains, introduced in [S. Gottlich, M. Herty and A. Klar, Comm. Math. Sci. 3, 545-559, 2005], consisting of a PDE for the density of processed parts and an ODE for the queue buffer occupancy. We discuss the optimal control problem stated as the minimization of the queues and the quadratic difference between the effective outflow and a desired one. Here the input flow is the control and is assumed to have uniformly bounded variation. Introducing generalized tangent vectors to piecewise constant controls, representing shifts of discontinuities, we analyse the dependence of the solution on the control function. Then existence of an optimal control for the original problem is obtained. Finally we study the sensitivity of the cost functional J as function of controlled inflow, providing an estimate of themethod. derivative of J w.r.t. switching times.

The aim of this paper is to generalize the Eulerian Gaussian beam method developed in [S. Jin, H. Wu, X. Yang, Commun. Math. Sci. 6 (2008) 995-1020] to compute high frequency wave propagation in heterogeneous media with discontinuity in one direction. At the interface where the wave speed is discontinuous, we propose proper interface conditions to capture the reflection and transmission behavior of Gaussian beams. The interface conditions rise naturally from an elegant combination of Hamiltonian preserving idea proposed in [S. Jin, X. Wen, Commun. Math. Sci. 3 (2005) 285-315] into the Eulerian Gaussian beam formulation. Numerical examples are given in one dimension to verify the accuracy of this method.

The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime $\epsilon << 1$, even a spatially spectrally accurate time splitting method requires the mesh size to be $O(\epsilon)$, which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian beams can be independently evolved along each eigenspace and summed to construct an approximate solution of the Dirac equation. Moreover, the proposed Eulerian Gaussian beam keeps the advantages of constructing the Hessian matrices by simply using level set functions's dereivative. Finally, several numerical examples show the efficiency and accuracy of the method.

We continue the study of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficency in the measure caused by displacement of mass to infinity in finite time. We introduce and study in its own right-- a regularization scheme based on a dissipative mechanism which naturally degrades mass according to distance traveled (in phase space). Our principal results are obtained based on some dynamical considerations in the form of a condition which forbids mass to return from infinity.

In this paper, we show that the coe1cients of the E-characteristic polynomial of a tensor are orthonormal invariants of that tensor. When the dimension is 2, some simpli/ed formulas of the E-characteristic polynomial are presented. A re- sultant formula for the constant term of the E-characteristic polynomial is given. We prove that both the set of tensors with in/nitely many eigenpairs, and the set of irregular tensors, have codimension 2 as subvarieties in the projective space of tensors. This makes our perturbation method workable. By using the perturbation method and exploring the di.erence between E-eigenvalues and eigenpair equiva- lence classes, we present a simple formula for the coe1cient of the leading term of the E-characteristic polynomial, when the dimension is 2.

Here, we examine the suitability of truncated Polynomial Chaos Expansions (PCE) and truncated Gram-Charlier Expansions (GrChE) as possible methods for uncertainty quantication (UQ) in nonlinear systems with intermittency and positive Lyapunov exponents. These two methods rely on truncated Galerkin projections of either the system variables in a fixed polynomial basis spanning the `uncertain' subspace (PCE) or a suitable eigenfunction expansion of the joint probability distribution associated with the uncertain evolution of the system (GrChE). Based on a simple, statistically exactly solvable non-linear and non-Gaussian test model, we show in detail that methods exploiting truncated spectral expansions, be it PCE or GrChE, have significant limitations for uncertainty quantication in systems with intermittent instabilities or parametric uncertainties in the damping. Intermittency and fat-tailed probability densities are hallmark features of the inertial and dissipation ranges of turbulence and we show that in such important dynamical regimes PCE performs, at best, similarly to the vastly simpler Gaussian moment closure technique utilized earlier by the authors in a different context for UQ within a framework of Empirical Information Theory. Moreover, we show that the non-realizability of the GrChE approximations is linked to the onset of intermittency in the dynamics and it is frequently accompanied by an erroneous blow-up of the second-order statistics at short times. These limitations of the two types of truncated spectral expansions arise from the following: (i) Non-uniform convergence in time of PCE and GrChE resulting in a rapidly increasing number of terms necessary for a good approximation of the random process as time evolves, (ii) Fundamental problems with capturing the constant flux of randomness due to white Gaussian noise forcing via finite truncations of the spectral representation of the associated Wiener process, (iii) Slow decay of PCE and GrChE coefficients in the presence of intermittency, hampering implementation of sparse truncation methods which have been widely used in nearly elliptic problems or in low Reynolds number flows. Rigorous justication of these limitations is richly illustrated by straightforward tests exploiting a simple nonlinear and non-Gaussian but statistically exactly solvable test model which is proposed here as a challenging benchmark for algorithms for UQ in systems with intermittency.

We develop a novel approach, named global geometrical optics method, for the numerical solution to wave equations in the high-frequency regime. The initial Cauchy data is assumed in the WKB form. We first study the Schrodinger equation, and then extend relevant results to the general scalar wave equations. The basic idea of this approach is to reformulate the governing equation in a moving frame, and to derive a WKB-type function merely defined on the Lagrangian manifold induced by the Hamiltonian flow. From this WKB-type function, the wave solution can be retrieved by a coherent state integral within first order accuracy. The merit of the proposed approach is manyfold. Firstly, compared with the thawed Gaussian beam approaches, it presents an approximate wave solution with first order asymptotic accuracy point-wisely, even around caustics. Secondly, compared with the canonical operator method, this approach does not require any a priori knowledge about the structure of Lagrangian manifold. Thirdly, compared with the frozen Gaussian beam approaches such as Herman-Kluk semi-classical propagator method, the proposed approach involves an integral on a manifold of much lower dimension. We report numerical tests on both Schrodinger and Helmholtz equations.

This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast- slow stochastic evolutionary equations. Under suitable conditions, it is proved that an exponentially tracking random invariant manifold exists, eliminating the fast motion for this coupled system. It is further shown that if the scaling parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential- ordinary differential equations, and coupled hyperbolic-hyperbolic partial differential equations.

This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of metamorphosis. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be dieomorphisms, but penalizing them if they are not. The possibilities of this approach are just beginning to be developed. In particular, metamorphosis and its related variants in the geometric approach to control and optimization can be expected to produce many exciting opportunities for new applications and analysis in geometric dynamics.

We give a detailed analysis of long range cumulative scattering eects from rough boundaries in waveguides. We assume small random uctuations of the boundaries and obtain a quantitative statistical description of the wave eld. The method of solution is based on coordinate changes that straighten the boundaries. The resulting problem is similar from the mathematical point of view to that of wave propagation in random waveguides with interior inhomogeneities. We quantify the net eect of scattering at the random boundaries and show how it diers from that of scattering by internal inhomogeneities.

We address the local well-posedness of the Prandtl boundary layer equations. Using a new change of variables we allow for more general data than previously considered, that is, we require the matching at the top of the boundary layer to be at a polynomial rather than exponential rate. The proof is direct, via analytic energy estimates in the tangential variables.

We study the scattered field from a thin high contrast dielectric volume of finite extent. The waves are modeled by the full three dimensional time-harmonic Maxwell equations while accounting for material boundaries. We derive a formulation of Lippmann-Schwinger type for a dielectric scatterer; this formulation has an additional surface term to account for the material discontinuities. The layer potential operator resulting from this surface term is shown to converge in a weak sense to an explicitly computable limit as the thickness of the domain approaches zero. By properly accounting for the boundary effects, we show two results about the thin high contrast limit: First, the normal component of the electric field's interior trace on the lateral boundary approaches zero. Second, the third component of the electric field (which corresponds to the direction perpendicular to the slab) goes to zero inside the slab. We propose a new two-dimensional limiting equation as a first-order computational technique.

The main objective of this article is to study the three-dimensional Rayleigh-Benard convection in a rectangular domain from a pattern formation perspective. It is well known that as the Rayleigh number crosses a critical threshold, the system undergoes a Type-I transition, characterized by an attractor bifurcation. The bifurcated attractor is an (m-1)-dimensional homological sphere where m is the multiplicity of the first critical eigenvalue. When m = 1, the structure of this attractor is trivial. When m = 2, it is known that the bifurcated attractor consists of steady states and their connecting heteroclinic orbits. The main focus of this article is then on the pattern selection mechanism and stability of rolls, rectangles and mixed modes (including hexagons) for the case where m = 2. We derive in particular a complete classification of all transition scenarios, determining the patterns of the bifurcated steady states, their stabilities and the basin of attraction of the stable ones. The theoretical results lead to interesting physical conclusions, which are in agreement with known experimental results. For example, it is shown in this article that only the pure modes are stable whereas the mixed modes are unstable.