lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

pullback attractors of non autonomous micropolar fluid flows abstract pullback attractors of the two dimensional non autonomous micropolar fluid motion model in a bounded domain are investigated it is shown that a compact pullback attractor in h 1 3 exists when its external driven function is translation bounded with respect to l 2 3

optimal rigidity estimates for nearly umbilical surfaces let r 3 be a smooth compact connected surface without boundary and denote by a its second fundamental form we prove the existence of a universal constant c such that

h2 boundedness of the pullback attractor for the non autonomous sir equations with diffusion abstract we prove some regularity results for the pullback attractor of a non autonomous sir model with diffusion in a bounded domain of r d where d 1 we show a regularity result for the unique solution of the problem we establish a general result about h 2 3 boundedness of invariant sets for the associate evolution process then as a consequence we deduce that the pullback attractor of the non autonomous system of sir equations with diffusion is bounded in h 2 3

dynamic scaling of the width distribution in edwards wilkinson type models of interface dynamics edwards wilkinson type models are studied in 1 1 dimensions and the time dependent distribution mathit p mathit l mathit w 2 t of the square of the width of an interface mathit w 2 is calculated for systems of size l we find that using a flat interface as an initial condition mathit p mathit l mathit w 2 t can be calculated exactly and it obeys scaling in the form mathit w 2 mathrm mathrm ensuremath infty mathit p mathit l mathit w 2 t ensuremath phi mathit w 2 mathit w 2 mathrm mathrm ensuremath infty t mathrm l 2 where mathit w 2 mathrm mathrm ensuremath infty is the stationary value of mathit w 2 for more complicated initial states scaling is observed only in the large time limit and the scaling function depends on the initial amplitude of the longest wavelength mode the short time limit is also interesting since mathit p mathit l mathit w 2 t is found to closely approximate the log normal distribution these results are confirmed by monte carlo simulations on a single step solid on solid type model roof top model of surface evolution textcopyright 1996 the american physical society

universal configurational structure in two dimensional scalar models monte carlo methods are used to explore the universal configurational structure of two dimensional spin 1 2 spin 1 and border phi 4 models comparison of spin 1 2 and spin 1 data provides evidence that the magnetisation distribution effectively the helmholtz free energy function and its coupling derivative effectively the internal energy function constitute readily accessible signatures of a universality class it is shown that when allowance is made for relatively large corrections to scaling effects the behaviour of the border phi 4 model may be satisfactorily matched to that of the other two models substantiating the view that the border model does indeed belong to the ising universality class

embedded surfaces of arbitrary genus minimizing the willmore energy under isoperimetric constraint the isoperimetric ratio of an embedded surface in mathbb r 3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume the aim of the present work is to study the minimization of the willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus g geqq 0 the corresponding problem in the case of spherical surfaces that is g 0 was recently solved by schygulla see schygulla arch ration mech anal 203 901 941 2012 with different methods

uniform attractors for the closed process and applications to the reaction diffusion equation with dynamical boundary condition abstract in this paper we first introduce the concept of a closed process in a banach space and we obtain the structure of a uniform attractor of the closed process by constructing a skew product flow on the extended phase space then the properties of the kernel section of closed process are investigated moreover we prove the existence and structure of the uniform attractor for the reaction diffusion equation with a dynamical boundary condition in l p without any restriction on the growth order of the nonlinear term

pullback attractors for nonautonomous reaction diffusion equations in unbounded domains abstract in this paper the existence of l 2 r n l 2 r n pullback attractors and l 2 r n h 1 r n pullback attractors are proved for reaction diffusion equation in unbounded domains

an energy gap phenomenon for the whitney sphere in this paper we study lagrangian surfaces satisfying nabla t 0 where t 2 nabla check a lrcorner omega and check a is the lagrangian trace free second fundamental form we obtain a gap lemma for such a lagrangian surface

global attractors for multivalued semiflows with weak continuity properties abstract a method is proposed to prove the global attractor existence for multivalued semiflows with weak continuity properties an application to the reaction diffusion problems with nonmonotone multivalued semilinear boundary condition and nonmonotone multivalued semilinear source term is presented

ising model on nonorientable surfaces exact solution for the mobius strip and the klein bottle closed form expressions are obtained for the partition function of the ising model on an mxn simple quartic lattice embedded on a mobius strip and a klein bottle the solutions all lead to the same bulk free energy but for finite m and n the expressions are different depending on whether the strip width m is odd or even finite size corrections at criticality are analyzed and compared with those under cylindrical and toroidal boundary conditions our results are consistent with the conformal field prediction of a central charge c 1 2 provided that the twisted mobius boundary condition is regarded as a free or fixed boundary

pullback attractors for non autonomous reaction diffusion equations in lp abstract in this paper by using a new priori estimate method called asymptotic a priori estimate we prove the existence of a pullback attractor for a non autonomous reaction diffusion equation in l p without restriction on the growth order of the nonlinear term and on the exponential growth of the external force

pullback attractors of nonautonomous reaction diffusion equations abstract in this paper firstly we introduce the concept of norm to weak continuous cocycle in banach space and give a technical method to verify this kind of continuity then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle using the measure of noncompactness as an application we prove the existence of pullback attractors in h 0 1 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction diffusion equations the attractor pullback attracts all bounded subsets of h 0 1 in the norm of h 0 1

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

width distribution for random walk interfaces roughening of a one dimensional interface is studied under the assumption that the interface configurations are continuous periodic random walks the distribution of the square of the width of interface mathit w 2 is found to scale as p mathit w 2 mathit w 2 mathrm mathrm ensuremath 1 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 we calculate the scaling function ensuremath phi x exactly and compare it both to exact enumerations for a discrete slope surface evolution model and to ensuremath phi s obtained in monte carlo simulations of equilibrium and driven interfaces of chemically reacting systems

the gradient flow of the l 2 curvature energy near the round sphere abstract we investigate the low energy behavior of the gradient flow of the l 2 norm of the riemannian curvature on four manifolds we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive yamabe constant and small initial energy

dynamics for a class of non autonomous degenerate p laplacian equations abstract in this paper we investigate a class of non autonomous degenerate p laplacian equations t u div a x u p 2 u u f u g x t in where a x is allowed to vanish on a nonempty closed subset with lebesgue measure zero g x t l l o c p r d 1 p a and an unbounded domain in r n we first establish the well posedness of these equations by constructing a compact embedding then we show the existence of the minimal pullback d attractor and prove that it indeed attracts the d class in l 2 norm for any 0 our results extend some known ones in previously published papers

non autonomous reaction diffusion model with dynamic boundary conditions abstract we investigate the dynamics of a non autonomous reaction diffusion model with dynamic boundary conditions we first show that under the same assumptions the known l 2 l 2 pullback d attractor indeed can attract in l 2 l 2 norm for any 0 then we prove the continuity of the solution in h 1 h 1 2 with respect to the initial data and finally show that such attractor can also attract in h 1 h 1 2 norm under a slightly stronger integrability condition on the time dependent external forcing term the proofs are based on a new nash moser alikakos type a priori estimate about the difference of solutions near the initial time

the geometric triharmonic heat flow of immersed surfaces near spheres abstract we consider closed immersed surfaces in r 3 evolving by the geometric triharmonic heat flow using local energy estimates we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in l 2 we further use an e regularity type result to prove a gap lemma for stationary solutions using a monotonicity argument we then prove that a blowup of the flow approaching a singular time is asymptotic to a non umbilic embedded stationary surface this allows us to conclude that any solution with initial l 2 norm of the tracefree curvature tensor smaller than an absolute positive constant converges exponentially fast to a round sphere with radius equal to 3 v 0 4 3 where v 0 denotes the signed enclosed volume of the initial data

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

probability distribution of the order parameter for the three dimensional ising model universality class a high precision monte carlo study we study the probability distribution p m of the order parameter average magnetization m for the finite size systems at the critical point the systems under consideration are the 3 dimensional ising model on a simple cubic lattice and its 3 state generalization known to have remarkably small corrections to scaling both models are studied in a cubic box with periodic boundary conditions the model with reduced corrections to scaling makes it possible to determine p m with unprecedented precision we also obtain a simple but remarkably accurate approximate formula describing the universal shape of p m

star triangle relations in the exactly solvable statistical models a regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed star triangle relations in the form of functional equations are used parametric families of transfer matrices commuting due to star triangle relations are constructed the eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations the solution of these equations for the maximal eigenvalue yields the partition function of the model the method is applied for evaluation of the partition function of the critical potts models the ising model the ashkin teller model equivalent to the eight vertex model

domain statistics in a finite ising chain we present a comprehensive study for the statistical properties of random variables that describe the domain structure of a finite ising chain with nearest neighbor exchange interactions and free boundary conditions by use of extensive combinatorics we succeed in obtaining the one variable probability functions for i the number of domain walls ii the number of up domains and iii the number of spins in an up domain the corresponding averages and variances of these probability distributions are calculated and the limiting case of an infinite chain is considered analyzing the averages and the transition time between differing chain states at low temperatures we also introduce a criterion of the ferromagnetic like behavior of a finite ising chain the results can be used to characterize magnetism in monatomic metal wires and atomic scale memory devices

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

pullback attractors for the non autonomous reaction diffusion equations in rn in this paper we prove the existence of the pullback attractors in l2 rn lp rn and h1 rn for a nonlinear reaction diffusion equation in unbounded domains in which the nonlinearity f satisfies the polynomial growth of arbitrary order p 1 p 2 and the external forcing g lloc2 r l2 rn only subjects to the weaker integrability condition te s g s 22ds in this paper we prove the existence of the pullback attractors in l2 rn lp rn and h1 rn for a nonlinear reaction diffusion equation in unbounded domains in which the nonlinearity f satisfies the polynomial growth of arbitrary order p 1 p 2 and the external forcing g lloc2 r l2 rn only subjects to the weaker integrability condition te s g s 22ds

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

bubble tree of branched conformal immersions and applications to the willmore functional we develop a bubble tree construction and prove compactness results for w 2 2 branched conformal immersions of closed riemann surfaces with varying conformal structures whose limit may degenerate in bbb r n with uniformly bounded areas and willmore energies the compactness property is applied to construct willmore type surfaces in compact riemannian manifolds this includes a existence of a willmore 2 sphere in bbb s n with at least 2 nonremovable singular points b existence of minimizers of the willmore functional with prescribed area in a compact manifold n provided i the area is small when genus is 0 and ii the area is close to that of the area minimizing surface of schoen yau and sacks uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to n and pi 2 n is trivial c existence of smooth minimizers of the willmore functional if a douglas type condition is satisfied

width distribution of curvature driven interfaces a study of universality one dimensional interfaces with curvature driven growth kinetics are investigated we calculate the steady state distribution p mathit w 2 of the square of the width of the interface mathit w 2 and show that as in the case for random walk interfaces the result can be written in a scaling form mathit w 2 p mathit w 2 ensuremath phi mathit w 2 mathit w 2 where mathit w 2 is the average of mathit w 2 the scaling function ensuremath phi x is found to be distinct from that of random walk interfaces but as our monte carlo simulations indicate this function is universal for curvature driven growth it is argued that comparison of scaling functions can be a useful method for distinguishing between universality classes of growth processes

pullback attractors for the norm to weak continuous process and application to the nonautonomous reaction diffusion equations abstract in this paper we introduce the concept of norm to weak continuous process in a banach space and obtain the existence of pullback attractors for this kind of process then we give a new method for proving the existence of the pullback attractors as an application we obtain the existence of pullback attractors for nonautonomous reaction diffusion equation in h 0 1 with exponential growth of the external force

on pullback attractors in lp for nonautonomous reaction diffusion equations abstract using the method introduced in zhong et al 2006 17 together with a new way of dealing with well known estimates of solutions introduced in ukaszewicz in press 9 we prove the existence of a unique minimal pullback attractor for the evolutionary process associated with a nonautonomous nonlinear reaction diffusion system in l p p 2 in which the right hand side satisfies only a certain integrability condition in particular we generalize a result obtained recently in li et al 2009 13 where an at most exponential growth of the right hand side has been assumed for times going to both plus and minus infinity

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

willmore surfaces with nonremovable singularities and number of critical levels in this paper we show that for every prescribed genus p in mathbb n and n in mathbb n with n ge 3 there exists a closed and orientable willmore surface in mathbb r n of genus p with at least two nonremovable point singularities furthermore we prove an energy gap theorem for smooth closed and orientable willmore surfaces in mathbb r n of prescribed genus p in mathbb n with n in mathbb n and n ge 3 moreover we show that the willmore functional as a map on the set of smooth closed and orientable surfaces in mathbb r n of genus p in mathbb n with n in 3 4 has only a finite number of critical levels strictly below a douglas condition on the willmore energy in order to exclude topological splitting

universality in the two dimensional continuous spin model monte carlo methods have been used to analyse the critical behaviour of the d 2 border phi 4 model which according to a recent series expansion study violates expectations based upon universality the results suggest that the model does in fact fall into the ising universality class at least as regards those parameters which characterise the critical point configurations of the ordering variable

a classification theorem for helfrich surfaces in this paper we study the functional mathcal w lambda 1 lambda 2 which is the sum of the willmore energy lambda 1 weighted surface area and lambda 2 weighted volume for surfaces immersed in mathbb r 3 this coincides with the helfrich functional with zero spontaneous curvature our main result is a complete classification of all smooth immersed critical points of the functional with lambda 1 ge 0 and small l 2 norm of tracefree curvature with no assumption on the growth of the curvature in l 2 at infinity this not only improves the gap lemma due to kuwert and schatzle for willmore surfaces immersed in mathbb r 3 but also implies the non existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted

pullback attractors of non autonomous reaction diffusion equations in h01 abstract using a method based on the concept of the kuratowski measure of noncompactness of a bounded set as well as some new estimates of the solutions we prove the existence of a unique minimal pullback attractor in h 0 1 for the process associated with non autonomous reaction diffusion equation without any restriction on the growing order of nonlinear terms p 2 is arbitrary in addition the external forces g t satisfy only the weaker integrability condition than in the earlier similar research of song and wu 2007 25 and li et al 2009 18

probability distribution of magnetization in the one dimensional ising model effects of boundary conditions finite size scaling functions are investigated both for the mean square magnetization fluctuations and for the probability distribution of the magnetization in the one dimensional ising model the scaling functions are evaluated in the limit of the temperature going to zero t 0 the size of the system going to infinity n while n 1 tanh j kbt is kept finite j being the nearest neighbour coupling exact calculations using various boundary conditions periodic antiperiodic free block demonstrate explicitly how the scaling functions depend on the boundary conditions we also show that the block small part of a large system magnetization distribution results are identical to those obtained for free boundary conditions

width distributions and the upper critical dimension of kardar parisi zhang interfaces simulations of restricted solid on solid growth models are used to build the width distributions of d 2 5 dimensional kardar parisi zhang kpz interfaces we find that the universal scaling function associated with the steady state width distribution changes smoothly as d is increased thus strongly suggesting that d 4 is not an upper critical dimension for the kpz equation the dimensional trends observed in the scaling functions indicate that the upper critical dimension is at infinity

foliations of asymptotically flat manifolds by surfaces of willmore type the goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the willmore functional subject to an area constraint equivalently these surfaces are critical points of the geroch hawking mass thus our result has applications in the theory of general relativity

willmore blowups are never compact we prove a ojasiewicz simon gradient inequality for the willmore functional near willmore surfaces and we apply this inequality to exclude compact blowups for the willmore flow

attractors of non autonomous reaction diffusion equations in lp abstract in this paper a new theorem which is proved in s s lu h q wu c k zhong attractors for non autonomous 2d navier stokes equations with normal external forces discrete contin dyn syst 13 3 2005 701 719 is applied to a nonlinear reaction diffusion equation with normal forces we obtain the existence of the uniform attractor in l p without any restriction on the growing order of the nonlinear term the attractor attracts all bounded subsets of l p in the norm of l p

singular perturbations of mean curvature flow we consider a regularization method for mean curvature now of a submanifold of arbitrary codimension in the euclidean space through higher order equations we prove that the regularized problems converge to the mean curvature flow for all times before the first singularity

continuity and pullback attractors for a non autonomous reaction diffusion equation in r n in this paper we study the dynamics of a non autonomous reaction diffusion equation in r n with the nonlinearity f satisfying the polynomial growth of arbitrary order p 1 p 2 firstly we prove the existence of the pullback d attractor in l 2 r n secondly we use a scheme to establish some new estimates higher order integrability of the difference of the solutions near the initial time instead of using the usual estimates about higher regularities and higher order integrability of solutions thirdly we verify that the usual l 2 r n l 2 r n pullback d attractor indeed can pullback attract the d class in l 2 norm for any 0 and h 1 norm and the solutions of the equation in h 1 r n are continuous with respect to initial data finally we obtain the pullback d attractor in l p r n and h 1 r n as an application of the higher order integrability and the continuity respectively

a gap theorem for willmore tori and an application to the willmore flow abstract in 1965 willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 and attains this minimal value if and only if the torus is a mobius transform of the clifford torus this was recently proved by marques and neves 2012 in this paper we show for tori that there is a gap to the next critical point of the willmore energy and we discuss an application to the willmore flow we also prove an energy gap from the clifford torus to surfaces of higher genus

phase field approximations of the willmore functional and flow we discuss in this paper phase field approximations of the willmore functional and the associated mathrm l 2 l2 flow after recollecting known results on the approximation of the willmore energy and its mathrm l 1 l1 relaxation we derive the expression of the flows associated with various approximations and we show their behavior by formal arguments based on matched asymptotic expansions we introduce an accurate numerical scheme whose local convergence is proved to describe with more details the behavior of two flows the classical and the flow associated with an approximation model due to mugnai we propose a series of numerical simulations in 2d and 3d to illustrate their behavior in both smooth and singular situations

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

critical weak immersed surfaces within sub manifolds of the teichmuller space abstract we prove that the critical points of various energies such as the area the willmore energy the frame energy for tori etc among possibly branched immersions constrained to evolve within a smooth sub manifold of the teichmuller space satisfy the corresponding constrained euler lagrange equation we deduce that critical points of the willmore energy or the frame energy for tori are smooth analytic surfaces away possibly from isolated branched points under the condition that either the genus is at most 2 or if the sub manifold does not intersect the subspace of hyper elliptic points using a compactness result from 19 we can conclude that each closed sub manifold of the teichmuller space including points under the previous assumptions possess a possibly branched smooth willmore minimizer satisfying the conformal constrained willmore equation

the large genus limit of the infimum of the willmore energy the willmore energy of a surface immersed into bbb r n is the integral of its squared mean curvature it is known that the infimum beta n p of the willmore energy among all closed oriented surfaces of genus p in bbb r n is attained by a smooth embedded surface and that 4 pi beta n p 8 pi for p geq 1 we show that beta n p converges to 8 pi as p goes to infinity

computational parametric willmore flow with spontaneous curvature and area difference elasticity effects a new stable continuous in time semidiscrete parametric finite element method for willmore flow is introduced the approach allows for spontaneous curvature and area difference elasticity effects which are important for many applications in particular in the context of membranes the method extends ideas from dziuk and the present authors to obtain an approximation that allows for a tangential redistribution of mesh points which typically leads to better mesh properties moreover we consider volume and surface area preserving variants of these schemes and in particular we obtain stable approximations of helfrich flow we also discuss fully discrete variants and present several numerical computations

higher order asymptotic attraction of pullback attractors for a reaction diffusion equation in non cylindrical domains abstract in this paper we consider the higher order asymptotic attraction of pullback attractors for a reaction diffusion equation with domains expanding in time firstly to make the test functions meaningful a maximum principle is proved for the corresponding variational solution then we establish some higher order integrability results about the difference of variational solutions and finally we prove the main result that the known l 2 l 2 pullback d attractor can attract indeed the d class in l q norm for any q 2

distribution of magnetization in the finite ising chain an isomorphism is shown to hold between the persistent random walk and the ising chain as well as between the biased persistent random walk and the closed ising chain in the presence of an external magnetic field the analytical expressions for the distributions of spontaneous and induced magnetization for the ising chain are derived using this isomorphism we show that this derivation is significantly more laborious using the partition function

some results about the existence of critical points for the willmore functional using a perturbative approach it is shown existence and multiplicity of critical points for the willmore functional in ambient manifold mathbb r 3 g epsilon where g epsilon is a metric close and asymptotic to the euclidean one with the same technique it is proved a non existence result in general riemannian manifold m g of dimension three

l2 lp random attractors for stochastic reaction diffusion equation on unbounded domains abstract in this paper the existence of l 2 l p random attractor is established for a stochastic reaction diffusion equation on the whole space r n this random attractor is a compact and invariant tempered set which attracts every tempered random subset of l 2 in the topology of l p the nonlinearity f is supposed to satisfy some growth of arbitrary order p 1 where p 2 the l 2 l p asymptotic compactness of the random dynamical system is proved by an asymptotic a priori estimate of the unbounded part of solutions

concentration and energy fluctuations in a critical polymer mixture a semi grand canonical monte carlo algorithm is employed in conjunction with the bond fluctuation model to investigate the critical properties of an asymmetric binary ab polymer mixture by applying the equal peak weight criterion to the concentration distribution the coexistence curve separating the a rich and b rich phases is identified as a function of temperature and chemical potential to locate the critical point of the model the cumulant intersection method is used the accuracy of this approach for determining the critical parameters of fluids is assessed attention is then focused on the joint distribution function of the critical concentration and energy which is analysed using a mixed field finite size scaling theory that takes due account of the lack of symmetry between the coexisting phases the essential ising character of the binary polymer critical point is confirmed by mapping the critical scaling operator distributions onto independently known forms appropriate to the 3d ising universality class in the process estimates are obtained for the field mixing parameters of the model which are compared both with those yielded by a previous method and with the predictions of a mean field calculation

pullback attractors for a non autonomous homogeneous two phase flow model abstract this article studies the pullback asymptotic behavior of solutions for a non autonomous homogeneous two phase flow model in a two dimensional domain we prove the existence of pullback attractors a v in v the velocity has the h 1 regularity and a y in y the velocity has the l 2 regularity then we verify the regularity of the pullback attractors by proving that a v a y which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data the method used in this article is similar to the one used in zhao and zhou 2007 42 in the case of the non autonomous incompressible non newtonian fluid in a two dimensional domain let us mention that the nonlinearity involved in the model considered in this article is stronger than the one in the two dimensional non newtonian flow studied in zhao and zhou 2007 42

on the pullback attractor for the non autonomous sir equations with diffusion abstract in this paper we study the dynamics of a non autonomous sir model with diffusion we first prove that the l 2 l 2 pullback attractor attracts any bounded subset of l 2 in the topology of l 2 for any 0 then we show that the l 2 l 2 pullback attractor is indeed a l 2 l 2 pullback attractor

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

a numerical method for simulation dynamics of incompressible lipid membranes in viscous fluid a numerical scheme is presented for modeling dynamics of incompressible elastic lipid membranes imbedded into viscous fluid a new elliptic equation for tension in the membrane implying its local incompressibility is derived the membranes dynamics is approximated in a semi implicit way the lattice boltzmann method is used to approximate the fluid flow forces acting on the fluid from the lipid membrane are implemented using the immersed boundary method by peskin the method is illustrated by examples of axisymmetric membranes with deformations and flows typical for experiments with lipid vesicles and nanotubes where deformations and flows can be modulated by external forces applied to the membrane