existence results for the flow of viscoelastic fluids with a differential constitutive law where 2 is the symmetric extra stress tensor the total stress is given by g pl i where p is the hydrodynamic pressure 4 is the rate of deformation tensor g u vu vu u being the velocity field a is the relaxation time a2 the retardation time 0 i a a1 and q is the fluid viscosity the symbol a d t denotes an objective frame indifferent derivative i 181 more precisely

the fujita kato theorem for some oldroyd b model abstract in this paper we investigate the cauchy problem associated to a system of pdes of oldroyd type the considered model describes the evolution of certain viscoelastic fluids within a corotational framework the non corotational setting is also addressed in dimension two we show that some widespread results concerning the incompressible navier stokes equations can be extended to the considered systems in particular we show the existence and uniqueness of global in time classical solutions for large data in dimension two this result is supported by suitable condition on the initial data to provide a global in time lipschitz regularity for the flow which allows to overcome specific challenging due to the lack of a decay in time of the main forcing terms secondly we address the global in time well posedness in dimension d 3 we prove the propagation of lipschitz regularities for the flow for this result we just assume the initial data to be sufficiently small in a critical lorentz space

adachi like chaotic neural networks requiring linear time computations by enforcing a tree shaped topology the adachi neural network adnn is a fascinating neural network nn which has been shown to possess chaotic properties and to also demonstrate associative memory am and pattern recognition pr characteristics variants of the adnn have also been used to obtain other pr phenomena and even blurring an unsurmountable problem associated with the adnn and the variants referred to above is that all of them require a quadratic number of computations this is essentially because the nns in each case are completely connected graphs in this paper we consider how the computations can be significantly reduced by merely using a linear number of computations to achieves this we extract from the original completely connected graph one of its spanning trees we then address the problem of computing the weights for this spanning tree this is done in such a manner that the modified tree based nn has approximately the same input output characteristics and thus the new weights are themselves calculated using a gradient based algorithm by a detailed experimental analysis we show that the new linear time adnn like network possesses chaotic and pr properties for different settings as far as we know such a tree based adnn has not been reported and the results given here are novel

global existence results for oldroyd b fluids in exterior domains the case of non small coupling parameters consider the set of equations describing oldroyd b fluids in an exterior domain it is shown that this set of equations admits a unique global solution in a certain function space provided the initial data but not necessarily the coupling constant is small enough

on a micro macro model for polymeric fluids near equilibrium in this paper we study a micro macro model for polymeric fluid the system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation we employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure for the initial data not far from the equilibrium we prove the global existence and uniqueness of classical solutions to the system 2006 wiley periodicals inc

global well posedness and decay rates for the three dimensional compressible oldroyd b model abstract in this paper we are concerned with the global well posedness and decay rates of strong solutions for the three dimensional compressible oldroyd b model we prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in h 2 framework moreover if the initial data belong to l 1 the convergence rate of the solutions in l p norm with 2 p 6 and convergence rates of their spatial derivatives in l 2 norm are obtained it is noteworthy that the smallness restriction on the coupling constant is not necessary in this paper

diffusing particles with electrostatic repulsion we study a diffusion model of an interacting particles system with general drift and diffusion coefficients and electrostatic inter particles repulsion more precisely the finite particle system is shown to be well defined thanks to recent results on multivalued stochastic differential equations see 2 and then we consider the behaviour of this system when the number of particles n goes to infinity through the empirical measure process in the particular case of affine drift and constant diffusion coefficient we prove that a limiting measure valued process exists and is the unique solution of a deterministic pde our treatment of the convergence problem as n uparrow infty is partly similar to that of t chan 3 and l c g rogers z shi 5 except we consider here a more general case allowing collisions between particles which leads to a second order limiting pde

chaos and indecomposability abstract we use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space earlier mouron proved that if x is an arc like continuum which admits a homeomorphism f with positive topological entropy then x contains an indecomposable subcontinuum barge and diamond proved that if g is a finite graph and f g g is any map with positive topological entropy then the inverse limit space lim g f contains an indecomposable continuum in this paper we show that if x is a g like continuum for some finite graph g and f x x is any map with positive topological entropy then lim x f contains an indecomposable continuum as a corollary we obtain that in the case that f is a homeomorphism x contains an indecomposable continuum moreover if f has uniformly positive upper entropy then x is an indecomposable continuum our results answer some questions raised by mouron and generalize the above mentioned work of mouron and also that of barge and diamond we also introduce a new concept called zigzag pair which attempts to capture the complexity of a dynamical systems from the continuum theoretic perspective and facilitates the proof of the main result

strong solutions of stochastic models for viscoelastic flows of oldroyd type abstract in this work we study stochastic oldroyd type models for viscoelastic fluids in r d d 2 3 we show existence and uniqueness of strong local maximal solutions when the initial data are in h s for s d 2 d 2 3 probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data

local existence for the fene dumbbell model of polymeric fluids we study the well posedness of a multi scale model of polymeric fluids the microscopic model is the kinetic theory of the finitely extensible nonlinear elastic fene dumbbell model the macroscopic model is the incompressible non newton fluids with polymer stress computed via the kramers expression the boundary condition of the fene type fokker planck equation is proved to be unnecessary by the singularity on the boundary other main results are the local existence uniqueness and regularity theorems for the fene model in certain parameter range

the l2 decay for the 2d co rotation fene dumbbell model of polymeric flows abstract in this paper we mainly study the long time behavior of solutions to the finite extensible nonlinear elastic fene dumbbell model with dimension two in the co rotation case firstly we obtain the l 2 decay rate of the velocity of the 2d co rotation fene model is 1 t 1 2 with small data then by virtue of the littlewood paley theory we can remove the small condition our obtained sharp result improves considerably the recent results in 8 14

from the boltzmann equations to the equations of incompressible fluid mechanics i we consider here the problem of deriving rigorously from boltzmann s equation globally in time and for general initial conditions fluid mechanics equations such as the navier stokes or euler equations

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

transport equation extension of classical results for div b bmo we investigate the transport equation ut b u 0 our result improves the criteria on uniqueness of weak solutions replacing the classical condition divb l by divb bmo

global existence results for oldroyd b fluids in exterior domains abstract in this paper we consider the set of equations describing oldroyd b fluids in exterior domains it is shown that these equations admit a unique global solution defined in a certain function space provided the initial data and the coupling constant are small enough

strong solution for a stochastic model of two dimensional second grade fluids existence uniqueness and asymptotic behavior abstract we investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of r 2 global existence and uniqueness of strong probabilistic solution is established in contrast to previous results on this model we show that the sequence of galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem we also give two results on the long time behavior of the solution mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time independent second grade fluids equations if the deterministic part of the external force does not depend on time if the deterministic forcing term explicitly depends on time then the strong probabilistic solution decays exponentially in mean square

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

synchronization in an array of linearly stochastically coupled networks with time delays abstract in this paper the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays the stochastic coupling term which can reflect a more realistic dynamical behavior of coupled systems in practice is introduced to model a coupled system and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly based on a simple adaptive feedback control scheme and some stochastic analysis techniques several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays finally an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results

the topology and dynamics of the hyperspaces of normal fuzzy sets and their inverse limit spaces given a compact and connected metric space continuum x we study topological and dynamical properties of the hyperspace of normal fuzzy sets f1 x equipped with the hausdorff endograph or sendograph metric among the many results we show that it is contractible path connected locally contractible locally path connected locally simply connected and locally connected for the endograph metric the hyperspace f1 x is a continuum and then for a topological graph x we show how using the inverse limit approach of barge and martin the inverse limit of a fuzzy dynamical system on x can be realized as an attractor of a fuzzy dynamical system on a manifold

global well posedness of the d d oldroyd b type models with fractional laplacian dissipation abstract this paper focuses on the cauchy problem of the d dimensional incompressible oldroyd b type models for viscoelastic flow with fractional laplacian dissipation namely with 1 u and 2 for 1 1 2 d 4 2 0 and 1 2 1 d 2 we obtain the global regularity of strong solutions when the initial data u 0 0 are sufficiently smooth

adaptive low rank methods problems on sobolev spaces this paper is concerned with the development and analysis of an iterative solver for high dimensional second order elliptic problems based on subspace based low rank tensor formats both the subspaces giving rise to low rank approximations and the corresponding sparse approximations of lower dimensional tensor components are determined adaptively a principal obstruction to a simultaneous control of rank growth and accuracy turns out to be the fact that the underlying elliptic operator is an isomorphism only between spaces that are not endowed with cross norms therefore as the central part of this scheme we devise a method for preconditioning low rank tensor representations of operators under standard assumptions on the data we establish convergence to the solution of the continuous problem with a guaranteed error reduction moreover for the case that the solution exhibits a certain low rank structure and representation sparsity we derive bounds on the computational complexity including in partic

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

mathematical existence results for the doi edwards polymer model in this paper we present some mathematical results on the doi edwards model describing the dynamics of flexible polymers in melts and concentrated solutions this model developed in the late 1970s has been used and extensively tested in modeling and simulation of polymer flows from a mathematical point of view the doi edwards model consists in a strong coupling between the navier stokes equations and a highly nonlinear constitutive law the aim of this article is to provide a rigorous proof of the well posedness of the doi edwards model namely that it has a unique regular solution we also prove which is generally much more difficult for flows of viscoelastic type that the solution is global in time in the two dimensional case without any restriction on the smallness of the data

regularity of the solutions for micro macro models near equilibrium abstract in this paper we study the regularity of solutions to two microscopic macroscopic models hookean spring model and fene dumbbell model by making use of the structure of fokker planck operator and establishing the general hardy littlewood s inequality we show that the classical solutions near the equilibrium obtained in f lin c liu ping zhang on a micro macro model for polymeric fluids near equilibrium comm pure appl math 60 6 2007 838 866 and f lin ping zhang the fene dumbbell model near equilibrium acta math sinica chin ser 24 2008 529 538 become smooth with respect to the end to end vector variable q for any positive time

an eulerian description of fluids containing visco elastic particles equations governing the flow of fluid containing visco hyperelastic particles are developed in an eulerian framework the novel feature introduced here is to write an evolution equation for the strain it is envisioned that this will simplify numerical codes which typically compute the strain on lagrangian meshes moving through eulerian meshes existence results for the flow of linear visco hyperelastic particles in a newtonian fluid are established using a galerkin scheme

incompressible limit of oldroyd b fluids in the whole space abstract this paper is dedicated to the study of compressible oldroyd b model in the framework of critical spaces we first prove the local well posedness for the compressible oldroyd b model with initial data 0 1 u 0 0 b 2 1 d 2 b 2 1 d 2 1 d b 2 1 d 2 d d the global well posedness is also established provided the initial data and coupling constant are sufficiently small on the basis of this we also consider the incompressible limit problem for ill prepared initial data we prove that as the mach number tends to zero the global solution to the compressible oldroyd b fluids converges to the solution to the corresponding incompressible model in some function spaces in the meanwhile the converge rates are obtained in some sense to fulfill our proof we give a new commutator estimate

the dynamics of continuous maps of finite graphs through inverse limits suppose that f g g is a continuous piecewise monotone function on a finite graph g then the following are equivalent i f has positive topological entropy ii there are disjoint intervals i1 and i2 and a positive integer n with i1 u i2 c fn ji n fn i2 iii the inverse limit space constructed by using f on g as a single bonding map contains an indecomposable subcontinuum this result generalizes known results for the interval and circle

instability of the rest state of fluids of arbitrary grade greater than one i am going to prove that the rest state of fluids of grade n any n t 1 is unstable in the spectral sense of linearized theory when the ratio of the coefficients of a and a 1 in the constitutive equation is negative negative ratios and only negative ratios are implied by integral expansions of the stress moreover in the only case n 2 which has been checked in experiments with polymeric liquids the ratio is negative to explain my result i have to specify further what i mean by linearized theory and what i mean by the ratio of coefficients i am following the analysis of my book on stability 1976 and carrying it further i take the stress for an incompressible simple fluid in the form t pl s where instead of normalizing the extra stress s in the usual way with trs 0 i normalize so that s 0 on rigid motions and of course s 0 when there is no motion thus s need not be traceless and in fact the usual approximations set down for s are not traceless there are two types of approximations which are used to express the stress on motions perturbing the rest state i will follow coleman noll 1960 1961 in setting my work in the frame of simple fluids with fading memory i will assume that the reader is familiar with the common notations of continuum mechanics and i recall that the stress is given by a functional of the history of the cauchy strain o

global wellposedness to the generalized oldroyd type models in r3 abstract it remains unknown whether smooth solutions of the 3d oldroyd models exist globally in viscoelastic fluids using the trick of transferring dissipation we get the global strong solutions of the generalized oldroyd models in three dimensions under the assumption that initial data is sufficiently small it improves the result in two dimensions obtained by elgndi and rousset 2015 4

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

boundary conditions for the microscopic fene models we consider the microscopic equation of finite extensible nonlinear elasticity fene models for polymeric fluids under a steady flow field it is shown that for the underlying fokker planck type of equations any preassigned distribution on the boundary will become redundant once the nondimensional number text it li frac hb k bt geq 2 where h is the elasticity constant sqrt b is the maximum dumbbell extension t is the temperature and k b is the usual boltzmann constant moreover if the probability density function is regular enough for its trace to be defined on the sphere m sqrt b then the trace is necessarily zero when text it li 2 these results are consistent with our numerical simulations as well as some recent well posedness results by preassuming a zero boundary distribution

existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead spring chain models for dilute polymers the two dimensional case abstract we prove the existence of global in time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids where the polymer molecules are idealized as bead spring chains with finitely extensible nonlinear elastic fene type spring potentials the class of models under consideration involves the unsteady compressible isentropic isothermal navier stokes system in a bounded domain in r d d 2 for the density the velocity u and the pressure p of the fluid with an equation of state of the form p c p where c p is a positive constant and 1 the right hand side of the navier stokes momentum equation includes an elastic extra stress tensor which is the classical kramers expression the elastic extra stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a fokker planck type parabolic equation a crucial feature of which is the presence of a centre of mass diffusion term this extends the result in our paper j w barrett and e suli 2016 9 which established the existence of global in time weak solutions to the system for d 2 3 and 3 2 but the elastic extra stress tensor required there the addition of a quadratic interaction term to the classical kramers expression to complete the compactness argument on which the proof was based we show here that in the case of d 2 and 1 the existence of global in time weak solutions can be proved in the absence of the quadratic interaction term our results require no structural assumptions on the drag term in the fokker planck equation in particular the drag term need not be corotational with a nonnegative initial density 0 l for the continuity equation a square integrable initial velocity datum u 0 for the navier stokes momentum equation and a nonnegative initial probability density function 0 for the fokker planck equation which has finite relative entropy with respect to the maxwellian m associated with the spring potential in the model we prove via a limiting procedure on a pressure regularization parameter the existence of a global in time bounded energy weak solution t t u t t to the coupled navier stokes fokker planck system satisfying the initial condition 0 u 0 0 0 u 0 0

well posedness for the fene dumbbell model of polymeric flows we prove local and global well posedness for the fene dumbbell model for a very general class of potentials indeed in prior local or global well posedness results conditions on the strength of the singularity or on the parameter b were made here we give a proof in the general case we also prove global existence results if the data is small or if we restrict to the co rotational model in dimension 2 2008 wiley periodicals inc

dissipative weak solutions to compressible navier stokes fokker planck systems with variable viscosity coefficients abstract motivated by a recent paper by barrett and suli 2016 6 we consider the compressible navier stokes system coupled with a fokker planck type equation describing the motion of polymer molecules in a viscous compressible fluid occupying a bounded spatial domain with polymer number density dependent viscosity coefficients the model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids where the polymer molecules are idealized as bead spring chains with finitely extensible nonlinear elastic fene type spring potentials the motion of the solvent is governed by the unsteady compressible barotropic navier stokes system where the viscosity coefficients in the newtonian stress tensor depend on the polymer number density our goal is to show that the existence theory developed in the case of constant viscosity coefficients can be extended to the case of polymer number density dependent viscosities provided that certain technical restrictions are imposed relating the behavior of the viscosity coefficients and the pressure for large values of the solvent density as a first step in this direction we prove here the weak sequential stability of the family of dissipative finite energy weak solutions to the system

global existence and uniqueness result for the diffusive peterlin viscoelastic model abstract the aim of this paper is to present the existence and uniqueness result for the diffusive peterlin viscoelastic model describing the unsteady behaviour of some incompressible polymeric fluids the polymers are treated as two beads connected by a nonlinear spring the peterlin approximation of the spring force is used to derive the equation for the conformation tensor the latter is the time evolution equation with spatial diffusion of the conformation tensor using the energy estimates we prove global in time existence of a weak solution in two space dimensions we are also able to show the regularity and consequently the uniqueness of the weak solution

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

ℱ mixing property and ℱ1 ℱ2 everywhere chaos of inverse limit dynamical systems abstract this paper investigates some chaotic properties via furstenberg families generated by inverse limit dynamical systems it is proved that the inverse limit dynamical system lim x f f of a dynamical system x f is transitive resp mixing 1 2 everywhere chaotic if and only if the system n 0 f n x f n 0 f n x is transitive resp mixing 1 2 everywhere chaotic where 1 and 2 are furstenberg families

global existence and exponential stability of small solutions to nonlinear viscoelasticity the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn dimensional euclidean space with smooth boundary dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to is described time periodic solutions are also studied as an application of our main theorem nonlinear viscoelasticity strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated

long time asymptotics of a multiscale model for polymeric fluid flows in this paper we investigate the long time behavior of some micro macro models for polymeric fluids hookean model and fene model in various settings shear flow general bounded domain with homogeneous dirichlet boundary conditions on the velocity general bounded domain with non homogeneous dirichlet boundary conditions on the velocity we use both probabilistic approaches coupling methods and analytic approaches entropy methods

blow up criterion for the incompressible viscoelastic flows abstract in this paper we are concerned with the incompressible viscoelastic flows in the periodic domain we establish a serrin type blow up criterion for 3 d periodic initial boundary problem which states a strong solution exists globally provided that the velocity satisfies serrin s condition and the l t l x norm of the deformation gradient are bounded we also establish blow up criterion in terms of the upper bound of the deformation gradient for 2 d periodic initial boundary problem the main ingredient of the proof is a priori estimate for an important quantity under the assumption that the deformation gradient is upper bounded whose divergence can be viewed as the effective viscous flux

global small solutions to an mhd type system the three dimensional case in this paper we consider the global well posedness of a three dimensional incompressible mhd type system with smooth initial data that is close to some nontrivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of in the coupled equations and only the horizontal derivatives of is dissipative with respect to time to achieve this we first employ anisotropic littlewood paley analysis to establish the key l1 lip 3 estimate to the third component of the velocity field then we prove the global well posedness to this system by the energy method which depends crucially on the divergence free condition of the velocity field 2014 wiley periodicals inc

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

soret and dufour effects on three dimensional oldroyd b fluid abstract in this article a realistic three dimensional magnetohydrodynamic flow of an oldroyd b fluid is studied after developing a convergent analytic scheme soret and dufour effects with mixed convection are taken into account the governing highly non linear partial differential equations are transformed into the system of ordinary differential equations using similarity transformations the resulting problems are computed by homotopy analysis method ham profiles of dimensionless velocities temperature and concentration are plotted and discussed for various emerging physical parameters numerical values of physical quantities of interest such as local nusselt number and local sherwood number are tabulated a comparative study with existing literature are found in an excellent agreement

stability results for fluids of oldroyd b type on exterior domains consider the set of equations describing fluids of oldroyd b type on an exterior domain it is shown that the solution of the linearized equation is governed by a bounded analytic semigroup on lp w1 p 1 p which is strongly stable moreover it is shown that the trivial solution of the full system is asymptotically stable in the sense that any solution starting in small ball around 0 0 converges towards 0 as t

finite element approximation of soluble surfactant spreading on a thin film we consider a fully practical finite element approximation of the following system of nonlinear degenerate parabolic equations begin align frac partial u partial t frac12 nabla cdot u 2 nabla sigma v frac13 nabla cdot u 3 nabla w 0 w c delta u delta u nu a u 3 hspace 0 1cm frac partial v partial t nabla cdot u v nabla sigma v frac12 nabla cdot u 2 v nabla w rho s delta v k left psi v right 0 frac partial psi partial t frac12 u nabla sigma v cdot nabla psi frac13 u 2 nabla w cdot nabla psi rho b u 1 nabla cdot left u nabla psi right beta k u 1 psi v 0 hspace 2cm nonumber end align the above equations model a marangoni driven thin film laden with a soluble surfactant in which the bulk surfactant concentration has been vertically averaged the model accounts for the presence of both attractive a geq 0 and repulsive delta 0 with nu 3 van der waals forces here u is the height of the film v is the concentration of the interfacial surfactant species psi is the concentration of the surfactant species within the bulk phase and sigma v 1 v is the typical surface tension moreover rhos geq 0 rhob 0 and c 0 are the inverses of the surface peclet number the bulk peclet number and the modified capillary number respectively finally beta 0 and k 0 are parameters that characterize the solubility and the rate of interfacial adsorption in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenerate parabolic system i in one space dimension when rhos 0 and moreover ii in two space dimensions if in addition nu geq 7 furthermore iterative schemes for solving the resulting nonlinear discrete system are discussed finally some numerical experiments are presented

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

existence of solution for a micro macro model of polymeric fluid the fene model abstract we analyse a non linear micro macro model of polymeric fluids in the case of a shear flow more precisely we consider the fene dumbbell model which models polymers by nonlinear springs accounting for the finite extensibility of the polymer chain we prove the existence of a unique solution to the stochastic differential equation which rules the evolution of a representative polymer in the flow and next deduce a local in time existence and uniqueness result on the system coupling the stochastic differential equation and the momentum equation on the fluid

global existence of weak solutions to the fene dumbbell model of polymeric flows systems coupling fluids and polymers are of great interest in many branches of sciences one of the most classical models to describe them is the fene finite extensible nonlinear elastic dumbbell model we prove global existence of weak solutions to the fene dumbbell model of polymeric flows the main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties the proof uses many weak convergence techniques in particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure in addition this quantity controls a rescaled defect measure of the gradient of the velocity

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

global solutions of two dimensional incompressible viscoelastic flows with discontinuous initial data the global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem and it is studied in this paper we show global existence if the initial deformation gradient is close to the identity matrix in l2 l and the initial velocity is small in l2 and bounded in lp for some p 2 while the assumption on the initial deformation gradient is automatically satisfied for the classical oldroyd b model the additional assumption on the initial velocity being bounded in lp for some p 2 may due to techniques we employed the smallness assumption on the l2 norm of the initial velocity is however natural for global well posedness one of the key observations in the paper is that the velocity and the effective viscous flux g are sufficiently regular for positive time the regularity of g leads to a new approach for the pointwise estimate for the deformation gradient without using l bounds on the velocity gradients in spatial variables 2015 wiley periodicals inc

continuity of velocity gradients in suspensions of rod like molecules we investigate the doi model for suspensions of rod like molecules in the dilute regime for certain parameter values the velocity gradient vs stress relation defined by the stationary and homogeneous flow is not rank one monotone we then consider the evolution of possibly large perturbations of stationary flows we prove that even in the absence of a microscopic cut off discontinuities in the velocity gradient cannot occur in finite time the proof relies on a novel type of estimate for the smoluchowski equation

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

global solutions to repulsive hookean elastodynamics the global existence of classical solutions to the three dimensional repulsive hookean elastodynamics around an equilibrium is considered by linearization and hodge s decomposition the compressible part of the velocity the density and the compressible part of the transpose of the deformation gradient satisfy klein gordon equations with speed sqrt 2 while the incompressible parts of the velocity and of the transpose of the deformation gradient satisfy wave equations with speed one the space time resonance method combined with the vector field method is used in a novel way to obtain the decay of the solution and hence global existence

global analysis of the flows of fluids with pressure dependent viscosities to describe the flows of fluids over a wide range of pressures it is necessary to take into account the fact that the viscosity of the fluid depends on the pressure that the viscosity depends on the pressure has been verified by numerous careful experiments while the existence of solutions local in time to the equations governing the flows of such fluids are available for small special data and rather unrealistic dependence of the viscosity on the pressure no global existence results are in place our interest here is to establish the existence of weak solutions for spatially periodic three dimensional flows that are global in time for a large class of physically meaningful viscosity pressure relationships

initial boundary value problems for the compressible viscoelastic fluid abstract this paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of oldroyd type the global in time solution is proved to exist uniquely near the equilibrium state in h 2

remarks on the blowup criteria for oldroyd models abstract we provide a new method to prove and improve the chemin masmoudi criterion for viscoelastic systems of oldroyd type in j y chemin n masmoudi about lifespan of regular solutions of equations related to viscoelastic fluids siam j math anal 33 1 2001 84 112 in two space dimensions our method is much easier than the one based on the well known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

global existence of weak solutions to two dimensional compressible viscoelastic flows abstract the global existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions is studied in this paper we show the global existence if the initial velocity u 0 is small in h with an arbitrary 0 1 2 and the perturbation of 0 f 0 around the constant state 1 i are small in l 2 b p 1 2 p with p 1 1 16 2 4 one of the main ingredients is that the velocity and the effective viscous flux g i are sufficiently regular for positive time the regularity of g i helps to obtain the l estimate of density and deformation gradient and hence eliminates the possible concentration and oscillation issues

on the boltzmann equation for long range interactions we study the boltzmann equation without grad s angular cutoff assumption we introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables angular singularities occur as soon as one is interested in long range interactions while singularities in the relative velocity variable occur in the study of soft potentials in particular coulomb interaction together with several new estimates this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by lions appearance of strong compactness under general fully realistic assumptions 2001 john wiley sons inc

on hydrodynamics of viscoelastic fluids in this paper we study a hydrodynamic system describing fluids with viscoelastic properties after a brief examination of the relations between several models we shall concentrate on a few analytical issues concerning them in particular we establish local existence and global existence with small initial data of classical solutions for an oldroyd system without an artificially postulated damping mechanism

regularity results for the 2d critical oldroyd b model in the corotational case this paper studies the regularity results of classical solutions to the two dimensional critical oldroyd b model in the corotational case the critical case refers to the full laplacian dissipation in the velocity or the full laplacian dissipation in the non newtonian part of the stress tensor whether or not their classical solutions develop finite time singularities is a difficult problem and remains open the object of this paper is two fold firstly we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non newtonian part of the stress tensor secondly when the critical case occurs in the non newtonian part of the stress tensor we first present many interesting global a priori bounds then establish a conditional global regularity in terms of the non newtonian part of the stress tensor this criterion comes naturally from our approach to obtain a global l bound for the vorticity

on the initial boundary value problem of the incompressible viscoelastic fluid system in this paper we shall establish the local well posedness of the initial boundary value problem of the viscoelastic fluid system of the oldroyd model we shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state c 2007 wiley periodicals inc

estimates and regularity results for the diperna lions flow in this paper we derive new simple estimates for ordinary differential equations with sobolev coefficients these estimates not only allow to recover some old and recent results in a simple direct way but they also have some new interesting corollaries

well posedness for the dumbbell model of polymeric fluids the dumbbell model is a coupled hydrodynamic kinetic model for polymeric fluids in which the configurations of the dumbbells are described by stochastic differential equations we prove well posedness of this model by deriving directly a priori estimates on the stochastic model our results can be used to analyze stochastic simulation methods such as the ones that are based on brownian configuration fields

global solutions for micro macro models of polymeric fluids abstract we provide a new proof for the global well posedness of systems coupling fluids and polymers in two space dimensions compared to the well known existing method based on the losing a priori estimates our method is more direct and much simpler the co rotational fene dumbbell model and the coupling smoluchowski and navier stokes equations are studied as examples to illustrate our main ideas

existence of global weak solutions to finitely extensible nonlinear bead spring chain models for dilute polymers with variable density and viscosity abstract we prove the existence of global in time weak solutions to a general class of coupled bead spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains with finitely extensible nonlinear elastic fene spring potentials the class of models under consideration involves the unsteady incompressible navier stokes equations with variable density and density dependent dynamic viscosity in a bounded domain in r d d 2 or 3 for the density the velocity and the pressure of the fluid with an elastic extra stress tensor appearing on the right hand side in the momentum equation the extra stress tensor stems from the random movement of the polymer chains and is defined by the kramers expression through the associated probability density function that satisfies a fokker planck type parabolic equation a crucial feature of which is the presence of a centre of mass diffusion term and a nonlinear density dependent drag coefficient we require no structural assumptions on the drag term in the fokker planck equation in particular the drag term need not be corotational with initial density 0 min max for the continuity equation where min 0 a square integrable and divergence free initial velocity datum u 0 for the navier stokes equation and a nonnegative initial probability density function 0 for the fokker planck equation which has finite relative entropy with respect to the maxwellian m associated with the spring potential in the model we prove via a limiting procedure on certain regularization parameters the existence of a global in time weak solution t t u t t to the coupled navier stokes fokker planck system satisfying the initial condition 0 u 0 0 0 u 0 0 such that t t min max t u t belongs to the classical leray space and t t has bounded relative entropy with respect to m and t t m has integrable fisher information w r t the gibbs measure d m q d q d x over any time interval 0 t t 0

global well posedness for 2d polymeric fluid models and growth estimate motivated by p constantin n masmoudi global well posedness for a smoluchowski equation coupled with navier stokes equations in 2d comm math phys 278 2008 179 191 f lin ping zhang zhifei zhang on the global existence of smooth solution to the 2 d fene dumbbell model comm math phys 277 2008 531 553 we prove the global existence of smooth solutions to a coupled microscopic macroscopic model for polymeric fluid in 2d under the co rotational assumption furthermore we provide an estimate on the time growth of these solutions our method is general and can be applied to the co rotational fene model and to the rod like model without the co rotational assumption c 2008 elsevier b v all rights reserved

global well posedness for a smoluchowski equation coupled with navier stokes equations in 2d we prove global existence for a nonlinear smoluchowski equation a nonlinear fokker planck equation coupled with navier stokes equations in 2d the proof uses a deteriorating regularity estimate in the spirit of 5 see also 1

constitutive relations for rivlin ericksen fluids based on generalized rational approximation a well known constitutive expression for the stress in an incompressible non newtonian fluid is provided by the representation of the extra stress as a function of the rivlin ericksen tensors mathbf a 1 mathbf a 2 ldots if this function is ordered in terms of the number of space plus time derivatives and appropriately scaled one obtains mathbf f mathbf a 1 mathbf a 2 ldots mu mathbf a 1 mu 2 alpha 1 mathbf a 2 alpha 2 mathbf a 2 1 cdots truncation at first order yields the usual newtonian viscous stress while truncation at second order provides the second order rivlin ericksen fluid many rheologists believe that alpha 1 0 in polymeric fluids however the requirement alpha 1 0 causes the rest state of the second order fluid to be unstable this paper shows how the approximation of mathbf f via generalized rational functions eliminates the instability paradox

global well posedness for the microscopic fene model with a sharp boundary condition abstract we prove global well posedness for the microscopic fene model under a sharp boundary requirement the well posedness of the fene model that consists of the incompressible navier stokes equation and the fokker planck equation has been studied intensively mostly with the zero flux boundary condition in this article we show that for the well posedness of the microscopic fene model b 2 the least boundary requirement is that the distribution near boundary needs to approach zero faster than the distance function under this condition it is shown that there exists a unique weak solution in a weighted sobolev space moreover such a condition still ensures that the distribution is a probability density the sharpness of this boundary requirement is shown by a construction of infinitely many solutions when the distribution approaches zero no faster than the distance function

global solution to the incompressible oldroyd b model in the critical l p framework the case of the non small coupling parameter this paper is dedicated to the global well posedness issue of the incompressible oldroyd b model in the whole space mathbb r d with d geqq 2 it is shown that this set of equations admits a unique global solution in a certain critical lp type besov space provided that the initial data but not necessarily the coupling parameter is small enough as a consequence even through the coupling effect between the equations of velocity u and the symmetric tensor of constrains is not small one may construct the unique global solution to the oldroyd b model for a class of large highly oscillating initial velocity the proof relies on the estimates of the linearized systems of u and u mathbb p rm div tau which may be of interest for future works this result extends the work by chemin and masmoudi siam j math anal 33 84 112 2001 to the non small coupling parameter case

viscoelastic fluid models derived from kinetic equations for polymers differential relationships between velocity and stress are derived from the kinetic theory of polymeric liquids the fokker planck equation corresponding to the so called dumbbell theory for dilute solutions of polymers is considered achapman enskog expansion for time dependent nonhomogeneous flows is proposed when the elastic character of the fluid is small and the results included in the book of r bird c curtiss r armstrong and o hassager dynamics of polymeric liquids vol kinetic theory john wiley sons new york 1987 are recovered a comparison with oldroyd b fene and fene p fluids is presented in the frame of the plane couette flow

synchronization based approach for parameters identification in delayed chaotic neural networks abstract in this paper an adaptive procedure to the problem of synchronization and parameters identification for chaotic neural networks with time varying delay is introduced by combining the adaptive control and linear feedback with appropriate update law based on the invariance principle of functional differential equations all the connection weight matrices can be efficiently estimated according to a simple rigorous and systematic technique this approach is also able to track the changes in the operating parameters of the experimental neural networks rapidly the speed of synchronization and parameters estimation can be adjusted under the adaptive gain properly chosen in addition the method is simple to implement in practice and it is quite robust against the effect of slight noise in the given time series and the estimated value of a parameter fluctuates around the correct value

global well posedness for the 2d micro macro models in the bounded domain in this paper we establish new a priori estimates for the coupled 2d navier stokes equations and fokker planck equation as its applications we prove the global existence of smooth solutions for the coupled 2d micro macro models for polymeric fluids in the bounded domain

the adaptive lagrangian particle method for macroscopic and micro macro computations of time dependent viscoelastic flows we propose a new numerical technique referred to as the adaptive lagrangian particle method alpm for computing time dependent viscoelastic flows using either a differential constitutive equation macroscopic approach or a kinetic theory model micro macro approach in alpm the eulerian finite element solution of the conservation equations is decoupled from the lagrangian computation of the extra stress at a number of discrete particles convected by the how in the macroscopic approach the extra stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories in the micro macro approach the extra stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model at each time step alpm automatically enforces that all elements of the mesh have a number of lagrangian particles ranging within a user specified interval results are given for the start up flow between highly eccentric rotating cylinders using the fene and fene p dumbbell models for dilute polymer solutions c 1999 elsevier science s a all rights reserved