multilevel quadratic spline quasi interpolation abstract in this paper we present new approximation schemes based on both quasi interpolation and multilevel methods by using bivariate quadratic b spline functions defined on simple and multiple knot type 2 triangulations improving the classical quasi interpolating spline results we also prove polynomial reproduction optimal approximation order and propose some numerical results and applications

strong solutions for two dimensional nonlocal cahn hilliard navier stokes systems abstract a well known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the navier stokes system coupled with a convective cahn hilliard equation in some recent contributions the standard cahn hilliard equation has been replaced by its nonlocal version the corresponding system is physically more relevant and mathematically more challenging indeed the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness in fact even in the two dimensional case uniqueness of weak solutions is still an open problem here we take a step forward in the case of regular potentials first we prove the existence of a unique strong solution in two dimensions then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data this result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions we also demonstrate that each trajectory converges to a single equilibrium provided that the potential is real analytic and the external forces vanish

an efficient finite element method for simulation of droplet spreading on a topologically rough surface abstract we study numerically the dynamics of a three dimensional droplet spreading on a rough solid surface using a phase field model consisting of the coupled cahn hilliard and navier stokes equations with a generalized navier boundary condition gnbc an efficient finite element method on unstructured meshes is introduced to cope with the complex geometry of the solid surfaces we extend the gnbc to surfaces with complex geometry by including its weak form along different normal and tangential directions in the finite element formulation the semi implicit time discretization scheme results in a decoupled system for the phase function the velocity and the pressure in addition a mass compensation algorithm is introduced to preserve the mass of the droplet to efficiently solve the decoupled systems we present a highly parallel solution strategy based on domain decomposition techniques we validate the newly developed solution method through extensive numerical experiments particularly for those phenomena that can not be achieved by two dimensional simulations on a surface with circular posts we study how wettability of the rough surface depends on the geometry of the posts the contact line motion for a droplet spreading over some periodic rough surfaces are also efficiently computed moreover we study the spreading process of an impacting droplet on a microstructured surface a qualitative agreement is achieved between the numerical and experimental results the parallel performance suggests that the proposed solution algorithm is scalable with over 4 000 processors cores with tens of millions of unknowns

second order convex splitting schemes for periodic nonlocal cahn hilliard and allen cahn equations we devise second order accurate unconditionally uniquely solvable and unconditionally energy stable schemes for the nonlocal cahn hilliard nch and nonlocal allen cahn nac equations for a large class of interaction kernels we present numerical evidence that both schemes are convergent we solve the nonlinear equations resulting from discretization using an efficient nonlinear multigrid method and demonstrate the performance of our algorithms by simulating nucleation and crystal growth for several different choices of interaction kernels

spectral characteristics of network redundancy many real world complex networks contain a significant amount of structural redundancy in which multiple vertices play identical topological roles such redundancy arises naturally from the simple growth processes which form and shape many real world systems since structurally redundant elements may be permuted without altering network structure redundancy may be formally investigated by examining network automorphism symmetry groups here we use a group theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks in particular we describe how a network s automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs

on a special case of model matching a special case of model matching a generalized version of the feedback simulation problem is considered and a necessary condition of its solvability is established this condition is also a new necessary and sufficient condition of solvability for the feedback simulation problem

nonlocal cahn hilliard navier stokes systems with shear dependent viscosity abstract we consider a diffuse interface model for the phase separation of an incompressible and isothermal non newtonian binary fluid mixture in three dimensions the averaged velocity u is governed by a navier stokes system with a shear dependent viscosity controlled by a power p 2 this system is nonlinearly coupled through the korteweg force with a convective nonlocal cahn hilliard equation for the order parameter that is the relative concentration difference of the two components the resulting equations are endowed with the no slip boundary condition for u and the no flux boundary condition for the chemical potential the latter variable is the functional derivative of a nonlocal and nonconvex ginzburg landau type functional which accounts for the presence of two phases we first prove the existence of a weak solution in the case p 11 5 then we extend some previous results on time regularity and uniqueness if p 11 5

modelling merging and fragmentation in multiphase flows with surfer we introduce a new numerical method called surfer for the simulation of two and three dimensional flows with several fluid phases and free interfaces between them we consider incompressible fluids obeying the navier stokes equation with newtonian viscosity in the bulk of each phase capillary forces are taken into account even when interfaces merge or break up fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm thus allowing for full symmetry between fluid phases the navier stokes equation is solved using staggered finite differences on a mac grid and a split explicit time differencing scheme while incompressibility is enforced using an iterative multigrid poisson solver capillary effects are represented as a stress tensor computed from gradients of the volume fraction function this formulation is completely independent of the topology of interfaces and relatively easy to implement in 3d it also allows exact momentum conservation in the discretized algorithm numerical spurious effects or parasite currents are noticed and compared to similar effects in boltzmann lattice gas methods for immiscible fluids simulations of droplets pairs colliding in 2d and in 3d are shown interface reconnection is performed easily despite the large value of capillary forces during reconnection

on an application of tikhonov s fixed point theorem to a nonlocal cahn hilliard type system modeling phase separation abstract this paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by p podio guidugli 2006 36 the model consists of an initial boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter and the chemical potential singular contributions to the local free energy in the form of logarithmic or double obstacle potentials are admitted in contrast to the local model which was studied by p podio guidugli and the present authors in a series of recent publications in the nonlocal case the equation governing the evolution of the order parameter contains in place of the laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long range interactions between the atoms it is shown that just as in the local case the model equations are well posed where the technique of proving existence is entirely different it is based on an application of tikhonov s fixed point theorem in a rather unusual separable and reflexive banach space

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

spectral comparison principles for the cahn hilliard and phase field equations and time scales for coarsening we initiate a study of the instability properties of steady solutions of the cahn hilliard ch equation and the phase field pf system in terms of the coarseness of those solutions other spectral results are obtained as well steady state solutions for the ch equation with no flux or periodic boundary conditions are also stationary states for the bistable reaction diffusion equation and the pf system with constant temperature thus it is meaningful to linearize each of these equations about a common stationary solution and discuss the stability of this state in each dynamical process converting the linearized operators to self adjoint form and expressing eigenvalues in terms of modified rayleigh quotients enables us to relate the spectra of the three operators in particular we relate the indices of instability with respect to the three processes we also discuss the change in the spectrum of the linearized ch operator in one dimension when we change the stationary solution at which it is linearized we conjecture that the eigenvalues increase with the number of oscillations of the steady state for a given mass and prove that this is so in certain cases this lends credence to the idea that motion of these dynamical systems near coarse solutions takes place on longer time scales

the dynamics of nucleation for the cahn hilliard equation when a constant metastable solution of the cahn hilliard equation is subjected to a spatially localized large amplitude perturbation a transition process may be triggered leading to a globally stable stationary solution in one space dimension the existence and instability of a third stationary solution with the same mass is proved a spike like solution called a canonical nucleus within the class of solutions which are even with respect to the center of the spike it has a one dimensional unstable manifold in addition the process of nucleation by formal arguments using two space scales and two timescales is described the last stage in the process can be approximated by a nonlinear stefan free boundary problem

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

a dynamic mesh algorithm for curvature dependent evolving interfaces a new finite element method is discussed for approximating evolving interfaces in2nwhose normal velocity equals mean curvature plus a forcing function the method is insensitive to singularity formation and retains the local structure of the limit problem and thus exhibits a computational complexity typical of3n 1without having the drawbacks of front tracking strategies a graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh time stepping is explicit but stability constraints force small time steps only when singularities develop whereas relatively large time steps are allowed before or past singularities when the evolution is smooth the explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step thereby making mesh updating simple and thus practical performance and potentials are fully documented via a number of numerical simulations in 2d 3d 4d and 8d with axial symmetries they include tori and cones for the mean curvature flow minimal and prescribed mean curvature surfaces with given boundary fattening for smooth driving force and volume constraint

isogeometric analysis of the navier stokes cahn hilliard equations with application to incompressible two phase flows abstract in this work we provide a unified and comparative description of the most prominent phase field based two phase flow models and present our numerical results of the application of galerkin based isogeometric analysis iga to incompressible navier stokes cahn hilliard nsch equations in velocity pressure phase field chemical potential formulation for the approximation of the velocity and pressure fields lbb compatible non uniform rational b spline spaces are used which can be regarded as smooth generalizations of taylor hood pairs of finite element spaces the one step scheme is used for the discretization in time the static and rising bubble in addition to the nonlinear rayleigh taylor instability flow problems are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

nonlinear aspects of the cahn hilliard equation abstract this paper treats phase separation within the context of the phenomenological cahn hilliard equation c t m c f c k 2 c where c is the concentration m c is the mobility and f c is homogeneou s free energy which is assumed here to be a fourth degree polynomial natural boundary conditions are introduced the full set of equilibrium solutions is specified a comparison theorem for stability criteria which was postulated by langer is proved here within the framework of the natural boundary conditions energy methods are used to define and estimate the limit of monotonic global stability it is pointed out that within the parameter region where the uniform homogeneous state is the only equilibrium solution there may still exist some internal excitable region in which the homogeneous solution possesses growing fluctuations furthermore a periodic instability is shown to exist in the metastable region in addition to the well known nucleation instability

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

solution to morgan s problem a necessary and sufficient condition is presented for the solution of the row by row decoupling problem known as morgan s problem in the general case that is without any restrictive assumption added to the system to the feedback law u fx gy g may be noninvertible this is a structural condition in terms of invariant lists of integers which are easily computable from a given state realization of the system these integers are the infinite zero orders morse s list i sub 4 and the essential orders of the system which only depend on the input output behavior and morse s list i sub 2 of the system which depends on the choice of a particular state realization

on a nonlocal phase separation model abstract an alternative to the cahn hilliard model of phase separation for two phase systems in a simplified isothermal case is given the model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations using the free energy as lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle

on the solvability of models for two phase flows of viscous incompressible fluid with shear dependent viscosity in this paper we consider a model describing the motion of a mixture of two incompressible and partially immiscible fluids in d dimension email protected 4 space domain whose viscosities have a p structure depending on the shear rate or called with shear dependent viscosity the existence of weak solutions is established for the exponent p p email protected d 3d 2 4dd 2 under the no slip boundary condition to velocity and non flux boundary condition to relative density this improves the known results proved for p 3d 2d 2 in kim et al 2006

constant solutions of polynomial equations abstract a necessary and sufficient condition is given for the equation ax by c in polynomial matrices to have a constant solution pair x y and also for x to be non singular a sufficient condition is then established under which the equation with a and b fixed has a constant solution for each c from a given class applications to the construction of static state feedback or static output injection in linear systems are discussed

reduced form of the interactor matrix some structural results about the interactor of a given right invertible system are presented an original form for this operator which directly exhibits structural information such as infinite zero structure or essential orders is provided moreover this form allows the linkage of the infinite and essential structures of a system

synchronization of networks of chaotic units with time delayed couplings a network of chaotic units is investigated where the units are coupled by signals with a transmission delay any arbitrary finite network is considered where the chaotic trajectories of the uncoupled units are a solution of the dynamic equations of the network it is shown that chaotic trajectories cannot be synchronized if the transmission delay is larger than the time scales of the individual units for several models the master stability function is calculated which determines the maximal delay time for which synchronization is possible

fundamental theorem of state feedback for singular systems abstract the limits of state feedback in altering the dynamics of singular systems are studied a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system the condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials the development is based on the properties of polynomial matrices a procedure is given for the calculation of a feedback gain which achieves the desired dynamics

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

existence and energy estimates of weak solutions for nonlocal cahn hilliard equations on an unbounded domain abstract this paper considers the initial boundary value problem for the nonlocal cahn hilliard equation t 1 a j g 0 in 0 t in an unbounded domain r n with smooth bounded boundary where n n t 0 and a j g are given functions in the case that is a bounded domain and 1 is replaced with this problem has been studied by using a faedo galerkin approximation scheme considering the compactness of the neumann operator 1 cf 8 16 however the compactness of the neumann operator 1 breaks down when is an unbounded domain the present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain

global existence of weak solutions to a nonlocal cahn hilliard navier stokes system abstract a well known diffuse interface model consists of the navier stokes equations nonlinearly coupled with a convective cahn hilliard type equation this system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors here we consider a variant of this model where the standard cahn hilliard equation is replaced by its nonlocal version more precisely the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter while the potential f may have any polynomial growth therefore the coupling with the navier stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of we establish the global existence of a weak solution in the two dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate provided that f fulfills a suitable coercivity condition

on a diffuse interface model for two phase flows of viscous incompressible fluids with matched densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids of the same density in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account this leads to a coupled navier stokes cahn hilliard system which is capable of describing the evolution of droplet formation and collision during the flow we prove the existence of weak solutions of the non stationary system in two and three space dimensions for a class of physical relevant and singular free energy densities which ensures in contrast to the usual case of a smooth free energy density that the concentration stays in the physical reasonable interval furthermore we find that unique strong solutions exist in two dimensions globally in time and in three dimensions locally in time moreover we show that for any weak solution the concentration is uniformly continuous in space and time because of this regularity we are able to show that any weak solution becomes regular for large times and converges as t to a solution of the stationary system these results are based on a regularity theory for the cahn hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on maximal regularity of a stokes system with variable viscosity and forces in l2 0 hs s in 0 frac12 which are new themselves

existence of weak solutions for a diffuse interface model for viscous incompressible fluids with general densities we study a diffuse interface model for the flow of two viscous incompressible newtonian fluids in a bounded domain the fluids are assumed to be macroscopically immiscible but a partial mixing in a small interfacial region is assumed in the model moreover diffusion of both components is taken into account in contrast to previous works we study the general case that the fluids have different densities this leads to an inhomogeneous navier stokes system coupled to a cahn hilliard system where the density of the mixture depends on the concentration the velocity field is no longer divergence free and the pressure enters the equation for the chemical potential we prove existence of weak solutions for the non stationary system in two and three space dimensions

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

a semi canonical form for a class of right invertible linear systems we present a semi canonical form for a class of right invertible systems under the action of the transforma tion group t f g p where p is a permutation matrix acting on the outputs under certain additional conditions this form is canonical in particular when the system is nonsingular and controllable

partial synchronization and partial amplitude death in mesoscale network motifs we study the interplay between network topology and complex space time patterns and introduce a concept to analytically predict complex patterns in networks of stuart landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology these patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes the underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks in addition it directly links the stability of complete in phase synchronization to only a small subset of topological eigenvalues of a network

eigenstructure assignment in linear descriptor systems given a linear controllable system ex spl dot fx gu we study the limits of linear proportional feedback u kx spl upsi in assigning a desired finite and infinite eigenstructure of the closed loop system ex spl dot f gk x g spl upsi the closed loop system is not restricted to be regular its rank makes part of the specifications necessary and sufficient conditions are established for the existence of a feedback gain k which assigns the desired eigenstructure and a procedure is outlined to calculate one such gain

network quotients structural skeletons of complex systems a defining feature of many large empirical networks is their intrinsic complexity however many networks also contain a large degree of structural repetition an immediate question then arises can we characterize essential network complexity while excluding structural redundancy in this article we utilize inherent network symmetry to collapse all redundant information from a network resulting in a coarse graining which we show to carry the essential structural information of the parent network in the context of algebraic combinatorics this coarse graining is known as the quotient we systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of real world quotients with respect to those of their parent networks in particular we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity heterogeneity and hub vertices and communication diameter and mean geodesic distance suggesting that quotients constitute the essential structural skeletons of their parent networks we summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms

a nonlocal stochastic cahn hilliard equation abstract we consider a stochastic extension of the nonlocal convective cahn hilliard equation containing an additive wiener process noise we first introduce a suitable analytical setting and make some mathematical and physical assumptions we then establish in a variational context the existence of a weak statistical solution for this problem finally we prove existence and uniqueness of a strong solution

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

controlling synchrony by delay coupling in networks from in phase to splay and cluster states we study synchronization in delay coupled oscillator networks using a master stability function approach within a generic model of stuart landau oscillators normal form of supercritical or subcritical hopf bifurcation we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states we propose the coupling phase as a crucial control parameter to switch between in phase synchronization or desynchronization for general network topologies or between in phase cluster or splay states in unidirectional rings our results are robust even for slightly nonidentical elements of the network

a cascaded phase field lattice boltzmann model for the simulation of incompressible immiscible fluids with high density contrast abstract in this work a conservative phase field model for the simulation of immiscible multiphase flows is developed using an incompressible velocity based cascaded lattice boltzmann method clbm extensions are made to the lattice boltzmann lb equations for interface tracking and incompressible hydrodynamics proposed by fakhari et al 1 by performing relaxation operations in central moment space this was motivated by the work of fei et al 2 3 where promising results from such a transformation were observed the relaxation of central moments is defined in a reference frame moving with the fluid while the existing multiple relaxation time 4 5 scheme performs collision in a fixed frame of reference moreover the derivations make use of continuous maxwellian distribution functions as a result the clbm enhances the galilean invariance and stability of the method when high lattice mach numbers are evident the cascaded scheme has been previously used in the literature to simulate multiphase flows based on the pseudo potential model where it allowed for high density and viscosity contrasts to be captured 6 7 here the clbm is implemented within the phase field framework and is verified through the analysis of a layered poiseuille flow the performance of the clbm is then investigated in terms of spurious currents galilean invariance and computational efficiency finally the work of fakhari et al 1 is extended by validating the model s ability to capture the relation between surface tension and the rise velocity of a planar taylor bubble in both stagnant and flowing fluids new counter current results indicate that the rise velocity model of ha ngoc and fabre 8 also applies in this regime

interlacing and majorization in invariant factor assignment problems this paper shows some connections between pole assignment in control theory and assignment of invariant factors on matrices with some prescribed submatrices the majorization in the hardy littlewood polya sense and the interlacing inequalities for invariant factors play a fundamental role in the solution of some problems common to both theories

optimal distributed control of a nonlocal convective cahn hilliard equation by the velocity in three dimensions in this paper we study a distributed optimal control problem for a nonlocal convective cahn hilliard equation with degenerate mobility and singular potential in three dimensions of space while the cost functional is of standard tracking type the control problem under investigation cannot easily be treated via standard techniques for two reasons the state system is a highly nonlinear system of pdes containing singular and degenerating terms and the control variable which is given by the velocity of the motion occurring in the convective term is nonlinearly coupled to the state variable the latter fact makes it necessary to state rather special regularity assumptions for the admissible controls which while looking a bit nonstandard are however quite natural in the corresponding analytical framework in fact they are indispensable prerequisites to guarantee the well posedness of the associated state system in this paper we employ recently proved existence uniqueness and regularity results f

the dirichlet boundary problem for a nonlocal cahn hilliard equation abstract we study the existence uniqueness and continuous dependence on initial data of the solution for a nonlocal cahn hilliard equation with dirichlet boundary condition on a bounded domain under a nondegeneracy assumption the solutions are classical but when this is relaxed the equation is satisfied in a weak sense also we prove that there exists a global attractor in some metric space

thermodynamically consistent spatial discretization of the one dimensional regularized system of the navier stokes cahn hilliard equations abstract this paper is devoted to the construction of a thermodynamically consistent with non increasing total energy on the discrete level semi discrete approximation continuous in time and discrete in space for the one dimensional regularized system of the navier stokes cahn hilliard equations that describes a compressible isothermal flow of two phase binary fluid with surface effects thermodynamical consistency is achieved using special form of momentum equation with surface force term written in so called potential form results of numerical study are presented

the neumann boundary problem for a nonlocal cahn hilliard equation we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal cahn hilliard equation on a bounded domain the equation generates a gradient flow for a free energy functional with nonlocal interaction also we apply a nonlinear poincare inequality to show the existence of an absorbing set in each constant mass affine space

amplitude death in an array of limit cycle oscillators we analyze a large system of limit cycle oscillators with mean field coupling and randomly distributed natural frequencies we prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance the system undergoes amplitude death the oscillators pull each other off their limit cycles and into the origin which in this case is astable equilibrium point for the coupled system we determine the region in couplingvariance space for which amplitude death is stable and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system

a novel linear unconditional energy stable scheme for the incompressible cahn hilliard navier stokes phase field model abstract in this paper we propose a linear unconditional energy stable time discretization scheme for cahn hilliard navier stokes model which is a phase field model for two phase incompressible flow based on a lagrange multiplier approach our proposed scheme is linearized by using implicit explicit treatments and the computation of the velocity field u the pressure p the phase function are decoupled using the mathematical induction the bound n l is obtained rigorous error analysis for the variable u p and are carried out in semi discrete form finally some numerical experiments are done to demonstrate the effectiveness for the proposed method

total and partial amplitude death in networks of diffusively coupled oscillators networks of weakly nonlinear oscillators are considered with diffusive and time delayed coupling averaging theory is used to determine parameter ranges for which the network experiences amplitude death whereby oscillations are quenched and the equilibrium solution has a large domain of attraction the amplitude death is shown to be a common phenomenon which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions in addition when the network consists of dissimilar oscillators there exist parameter values for which only parts of the network are suppressed sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities coupling coefficients and connection delays

state feedback in linear control theory abstract the role of system invariants in solutions of classical control problems when regular state feedback is used is reviewed the structural modifications that arise when these problems are extended to the case of non regular state feedback are presented an interpretation of state feedback problems in terms of matrix pencils completion is also discussed

cluster and group synchronization in delay coupled networks we investigate the stability of synchronized states in delay coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics using a master stability approach we find that the master stability function shows a discrete rotational symmetry depending on the number of groups the coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum which helps to simplify the evaluation of the master stability function our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times multiple coupling functions but also with multiple kinds of local dynamics in the networks nodes we illustrate our results by calculating stability in the example of delay coupled semiconductor lasers and in a model for neuronal spiking dynamics

a semi analytical fourier spectral method for the allen cahn equation in recent years fourier spectral methods have been widely used as a powerful tool for solving phase field equations to improve its effectiveness many researchers have employed stabilized semi implicit fourier spectral sifs methods which allow a much larger time step than a usual explicit scheme our mathematical analysis and numerical experiments however suggest that an effective time step is smaller than a time step specified in the sifs schemes in consequence the sifs scheme is inaccurate for a considerably large time step and may lead to incorrect morphologies in phase separation processes in order to remove the time step constraint and guarantee the accuracy in time for a sufficiently large time step we present a first and a second order semi analytical fourier spectral safs methods for solving the allen cahn equation the core idea of the methods is to decompose the original equation into linear and nonlinear subequations which have closed form solutions in the fourier and physical spaces respectively both the first and the second order methods are unconditionally stable and numerical experiments demonstrate that our proposed methods are more accurate than the stabilized semi implicit fourier spectral method

the row by row decoupling via state feedback a polynomial approach necessary and sufficient conditions for the row by row and integrator decoupling of a linear system x ax bu y cx via nonregular static state feedback are established a procedure is outlined to calculate one such gain

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

new results about morgan s problem some aspects of the nonsquare morgan s problem are considered in particular it is shown that the condition given by j descusse et al 1986 to solve this problem is not sufficient it is also shown that the essential orders are not necessarily preserved in the static decoupling problem a necessary and sufficient condition for static decoupling of a system without modifying the essential orders is given for the case k p 1 where p is the number of outputs and k is the rank at infinity of the proper part of the interactor

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

network synchronization of groups in this paper we study synchronized motions in complex networks in which there are distinct groups of nodes where the dynamical systems on each node within a group are the same but are different for nodes in different groups both continuous time and discrete time systems are considered we initially focus on the case where two groups are present and the network has bipartite topology i e links exist between nodes in different groups but not between nodes in the same group we also show that group synchronous motions are compatible with more general network topologies where there are also connections within the groups

on the modification of a structure at infinity by a return of static state le bouclage d un systeme lineaire par un retour d etat statique non regulier c est a dire du type u fx gv avec g non inversible modifie en general sa structure et en particulier sa structure a l infini qui joue un role cle daps les problemes de commande l objet de cet article est d etudier ces modifications nous donnons une condition structurelle necessaire et sufsante pour qu une lists donnee soft la lists des ordres des zeros infinis du systeme boucle par un retour d etat statique nous calculons alors explicitement un retour d etat f g qui produit cette modification de structure a l infini

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

instability of two phase flows a lower bound on the dimension of the global attractor of the cahn hilliard navier stokes system abstract we consider a model for the flow of a mixture of two viscous and incompressible fluids in a two or three dimensional channel like domain the model consists of the navier stokes equations governing the fluid velocity coupled with a convective cahn hilliard equation for the relative density of atoms of one of the fluids we prove the instability of certain stationary solutions for such a system endowed with periodic boundary conditions on elongated domains 0 2 0 0 2 or 0 2 0 0 2 0 2 0 for a special class of periodic body forces provided that 0 and 0 are small enough as a consequence we deduce a lower bound for the hausdorff dimension of the global attractor

the nonlocal cahn hilliard equation with singular potential well posedness regularity and strict separation property abstract we consider the nonlocal cahn hilliard equation with singular potential and constant mobility well posedness and regularity of weak solutions are studied then we establish the validity of the strict separation property in dimension two further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided finally regularity results and the strict separation property are also proven for the two dimensional nonlocal cahn hilliard navier stokes system with singular potential

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

new decoupling invariants the essential orders the aim of this paper is to introduce a new list of integers which is invariant under the action of biproper compensation and which plays a key role in the solution of the row by row decoupling problem several equivalent characterizations are provided within both the geometric and the transfer matrix approaches

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a convergent convex splitting scheme for the periodic nonlocal cahn hilliard equation in this paper we devise a first order in time second order in space convex splitting scheme for the periodic nonlocal cahn hilliard equation the unconditional unique solvability energy stability and ell infty 0 t ell 4 l 0 t l 4 stability of the scheme are established using the a priori stabilities we prove error estimates for our scheme in both the ell infty 0 t ell 2 l 0 t l 2 and ell infty 0 t ell infty l 0 t l norms the proofs of these estimates are notable for the fact that they do not require point wise boundedness of the numerical solution nor a global lipschitz assumption or cut off for the nonlinear term the ell 2 l 2 convergence proof requires no refinement path constraint while the one involving the ell infty l norm requires only a mild linear refinement constraint s le c h s c h the key estimates for the error analyses take full advantage of the unconditional ell infty 0 t ell 4 l 0 t l 4 stability of the numerical solution and an interpolation estimate of the form left phi right 4 le c left phi right 2 alpha left nabla h phi right 2 1 alpha alpha frac 4 d 4 d 1 2 3 4 c 2 h 2 1 4 d 4 d 1 2 3 which we establish for finite difference functions we conclude the paper with some numerical tests that confirm our theoretical predictions

analysis of a cahn hilliard ladyzhenskaya system with singular potential abstract the evolution of a mixture of two incompressible and partially immiscible fluids is here described by ladyzhenskaya navier stokes type equations for the average fluid velocity coupled with a convective cahn hilliard equation with a singular e g logarithmic potential the former is endowed with no slip boundary conditions while the latter is subject to no flux boundary conditions so that the total mass is conserved here we first prove the existence of a weak solution in three dimensions and some regularity properties then we establish the existence of a weak trajectory attractor for a sufficiently general time dependent external force finally taking advantage of the validity of the energy identity we show that the trajectory attractor actually attracts with respect to the strong topology

a nonlocal quasilinear multi phase system with nonconstant specific heat and heat conductivity in this paper we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi phase transitions under nonhomogeneous third type boundary conditions the system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro differential inclusion governing the vectorial phase parameter dynamics the specific heat and the heat conductivity k are allowed to depend both on the order parameter and on the absolute temperature of the system and the convex component of the free energy may or may not be singular uniqueness and continuous data dependence are also proved under additional assumptions

stabilized linear semi implicit schemes for the nonlocal cahn hilliard equation abstract comparing with the well known classic cahn hilliard equation the nonlocal cahn hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials on the other hand it evidently brings more computational costs in numerical simulations thus efficient and accurate time integration schemes are highly desired in this paper we propose two energy stable linear semi implicit methods with first and second order temporal accuracies respectively for solving the nonlocal cahn hilliard equation the temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly while the spatial discretization is carried out by the fourier collocation method with fft based fast implementations the energy stabilities are rigorously established for both methods in the fully discrete sense numerical experiments are conducted for a typical case involving gaussian kernels we test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one in addition long time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed

asymptotic behavior for nonlocal dispersal equations abstract this paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation by means of super subsolution method and monotone iteration we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation then we apply these results to our equation and show that the nonnegative solution is unique and the behavior of this solution depends on parameter in equation for 1 the solution decays to zero as t while for 1 the solution converges to the unique positive stationary solution as t in addition we show that the solution blows up under some conditions

boundary fluxes for nonlocal diffusion abstract we study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary this problem may be seen as a generalization of the usual neumann problem for the heat equation first we prove existence uniqueness and a comparison principle next we study the behavior of solutions for some prescribed boundary data including blowing up ones finally we look at a nonlinear flux boundary condition

on a nonlocal phase field system abstract we study the existence uniqueness and continuous dependence on initial data of the solution to a nonlocal phase field system on a bounded domain the system is a gradient flow for a free energy functional with nonlocal interaction also we study the asymptotic behavior of the solution and show the existence of an absorbing set in some metric space

assigning the kronecker invariants of a matrix pencil by row or column completions abstract the challenge consists in describing the relationships between the kronecker invariants of a matrix pencil and one of its subpencils for a given subpencil an algorithm for constructing a matrix pencil with prescribed kronecker invariants should also be proposed