the weighted hardy s inequality for nonincreasing functions the purpose of this paper is to give an alternative proof of recent results of m arino and b muckenhoupt 1 and e sawyer 8 concerning hardy s inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general lorentz spaces our approach will extend the results of 1 8 to the values of the parameters which are inaccessible by the methods of these papers

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

on embeddings of logarithmic bessel potential spaces we present embedding theorems for certain logarithmic bessel potential spaces modelled upon generalized lorentz zygmund spaces and clarify the role of the logarithmic terms involved in the norms of the space mentioned in particular we get refinements of the sobolev embedding theorems trudinger s limiting embedding as well as embeddings of sobolev spaces into space of holder continuous functions including the result of brezis and wainger

techniques for stress and strain controlled homogenization of inelastic periodic composites abstract a framework is developed within the context of the displacement finite element method for applying periodic boundary conditions to uni period basic cells under the influence of applied macroscopically homogeneous stresses the resulting approach supplements the more common strain controlled method which is also presented and which relies on an additive decomposition of the displacement field into linear imposed and periodic unknown contributions as implemented here the stress controlled method employs a penalty formulation to insure that the displacement solution satisfies the linearperiodic decomposition the performance of both the stress and strain driven methods are demonstrated and assessed on a complete set of homogenization computations for an elasto plastic composite

two scale analysis of heterogeneous elastic plastic materials with asymptotic homogenization and voronoi cell finite element model in this paper a multiple scale finite element model vcfem homo has been developed for elastic plastic analysis of heterogeneous porous and composite materials by combining asymptotic homogenization theory with the voronoi cell finite element model vcfem vcfem for microstructural modeling originates from dirichlet tessellation of representative material elements at sampling points in the structure structural modeling is done by the general purpose finite element code abaqus and interfacing with the microscale vcfem analysis is done through the user subroutine in abaqus for material constitutive relation umat asymptotic homogenization in umat generates macroscopic material parameters for abaqus following the macroscopic analysis a local vcfem analysis is invoked to depict the true evolution of microstructural state variables various numerical examples are executed for validating the effectiveness of vcfem homo and the effect of size shape and distribution of heterogeneities on local and global response is examined

variational reduction for ginzburg landau vortices let be a bounded domain with smooth boundary in r2 we construct non constant solutions to the complex valued ginzburg landau equation e2 u 1 u 2 u 0 in as e 0 both under zero neumann and dirichlet boundary conditions we reduce the problem of finding solutions having isolated zeros vortices with degrees 1 to that of finding critical points of a small c1 perturbation of the associated renormalized energy this reduction yields general existence results for vortex solutions in particular for the neumann problem we find that if is not simply connected then for any k 1 a solution with exactly k vortices of degree one exists

a fast iterative solver for the variable coefficient diffusion equation on a disk we present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk the equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions the resultant matrix is symmetric positive definite so the preconditioned conjugate gradient pcg method can be applied different preconditioners have been tested for comparison in particular a fast direct solver derived from the equation and the semi coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly the present elliptic solver has been applied to study the vortex dynamics of the ginzburg landau equation with a variable diffusion coefficient

local minimizers with vortex pinning to ginzburg landau functional in three dimensions abstract we consider the pinning effect for full ginzburg landau functional the existence of local minimizers with vortices locating in the pinning regions is obtained

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

dynamics of localized structures abstract we describe qualitative behaviour of solutions of the gross pitaevskii equation in 2d in terms of motion of vortices and radiation to this end we introduce the notion of the intervortex energy we develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized stable static solutions it yields description of general time dependent solutions in terms of dynamics of those static solutions glued together

numerical methods for simulating ginzburg landau vortices numerical solution to the ginzburg landau gl equation becomes infeasible as the gl parameter kappa and the number of gl vortices increase to a physically interesting regime it is in this regime that we focus our attention to design a simulated annealing approach for minimizing the associated renormalized energy to study the gl vortex configurations this approach applied to a commonly used model produces vortex configurations which are of interest in recent studies

optimal sobolev imbeddings involving rearrangement invariant quasinorms let m and n be positive integers with n 2 and 1 m n 1 we study rearrangement invariant quasinorms r and d on functions f 0 1 view the mathml source such that to each bounded domain in view the mathml sourcen with lebesgue measure there corresponds c c 0 for which one has the sobolev imbedding inequality r u t c d mu t u cm0 involving the nonincreasing rearrangements of u and a certain mth order gradient of u when m 1 we deal in fact with a closely related imbedding inequality of talenti in which d need not be rearrangement invariant r u t c d d dt x view the mathml sourcen u x u t u x dx u c10 in both cases we are especially interested in when the quasinorms are optimal in the sense that r cannot be replaced by an essentially larger quasinorm and d cannot be replaced by an essentially smaller one our results yield best possible refinements of such limiting sobolev inequalities as those of trudinger strichartz hansson brezis and wainger

vortices in complex scalar fields abstract consider a spatially extended field which evolves in time according to a pde the solutions contain particle like defects whose motions are parts of the full dynamics we inquire into the formulation of an asymptotic particle field defect dynamics which derives from the full pde we carry out this program for two complex scalar field equations in two space dimensions the nonlinear schrodinger equation and the nonlinear heat equation the topological defects are zeros of the complex scalar field with nonzero integer winding numbers called vortices vortices evolving under the nonlinear schrodinger equation behave like point vortices in ideal fluid pairs of vortices evolving under the nonlinear heat equation with like opposite winding numbers undergo a repulsive attractive interaction

numerical study of quantized vortex interaction in complex ginzburg landau equation on bounded domains in this paper we study numerically quantized vortex dynamics and their interaction in the two dimensional complex ginzburg landau equation cgle with a dimensionless parameter e 0 on bounded domains under either dirichlet or homogeneous neumann boundary condition we begin with a review of the reduced dynamical laws rdls for time evolution of quantized vortex centers in cgle and show how to solve these nonlinear ordinary differential equations numerically then we present efficient and accurate numerical methods for solving the cgle on either a rectangular or a disk domain under either dirichlet or homogeneous neumann boundary condition based on these efficient and accurate numerical methods for cgle and the rdls we explore rich and complicated quantized vortex dynamics and interaction of cgle with different e and under different initial physical setups including single vortex vortex pair vortex dipole and vortex lattice compare them with those obtained from the corresponding rdls and identify the cases where the rdls agree qualitatively and or quantitatively as well as fail to agree with those from cgle on vortex interaction finally we also obtain numerically different patterns of the steady states for quantized vortex lattices in the cgle dynamics on bounded domains

on the time splitting spectral method for the complex ginzburg landau equation in the large time and space scale limit we are interested in the numerical approximation of the complex ginzburg landau equation in the large time and space limit there are two interesting regimes in this problem one being the large space time limit and one being the nonlinear schrodinger limit these limits have been studied analytically in for example t colin and a soyeur asymptot anal 13 1996 pp 361 372 f h lin comm pure appl math 51 1998 pp 385 441 f h lin and j x xin comm math phys 200 1999 pp 249 274 we study a time splitting spectral method for this problem in particular we are interested in whether such a scheme is asymptotic preserving ap with respect to these two limits our results show that the scheme is ap for the first limit but not the second one for the large space time limit our numerical experiments show that the scheme can capture the correct physical behavior without resolving the small scale dynamics even for transitional problems where small and large scales coexist

structure and dynamics of defects in 2d complex vector field abstract minimal orbits of the energy functional of a complex 2d vector field may belong to the strata with different topology depending on the sign of nonlinear terms breaking the su 2 symmetry this gives rise to either linearly or circularly polarized states lp and cp which possess different sets of defects the cp vortices have two alternative core structures with either vanishing amplitude or reversed polarization in the inner core in the lp state there are two topological charges associated with the rotation of either argument or director vortices carrying a unit argument or director charge have a punched core with a vanishing amplitude while vortices carrying two half unit charges have a circularly polarized core the punched core structure is unstable when the symmetry breaking is not sufficiently strong in the cp state this leads to transition to the alternative repolarized structure while in the lp state it signals splitting of unit charged vortices into a pair of half unit charged vortices vortices separated by distances large compared to the core size interact in the same way as vortices in a complex scalar field and interactions due to argument and director charges are independent to the leading order

rotating 2n vortex solutions to the gross pitaevskii equation on s2 we establish the existence of rotating solutions to the gross pitaevskii equation iut u 1 2 1 u 2 uposed on s2 that is for u s2 r c these solutions possess vortices that for all time follow the vortex paths of known relative equilibria to the point vortex problem on the two sphere in the asymptotic regime 1 the approach is variational based on minimization of the ginzburg landau energy subject to a momentum constraint we also establish orbital stability within a class of symmetric initial data

vortex dynamics of the nonlinear wave equation abstract is a complex scalar field defined on 2 1 d dimensional minkowski space it satisfies the nonlinear wave equation nlwe 1 0 we study the solutions which contain time like world lines of zeros with winding numbers 1 or 1 we call these particle like defects of the field vortices we formulate an asymptotic particle field vortex dynamics which derives from the nlwe it is analogous to particle field electrodynamics in 2 1 d in particular the winding number plays the role of charge and gradients of arg play the role of the electromagnetic field

dynamics of phase singularities in two dimensional oscillating systems abstract we study the dynamics of vortices that arise in two dimensional nonequilibrium systems above a hopf bifurcation to uniform temporal oscillations evolution equations for the vortex positions and for a global phase field are derived and used to study vortex interactions we suggest that the interactions need not be purely attractive or repulsive bound vortex pairs can exist which either precess around fixed points in the plane or drift in directions perpendicular to the inter vortex axes

mixed vortex antivortex solutions of ginzburg landau equations suppose that v e 0 is a family of minimizers of 1 2 then there is a positive number eo g ft such that for any 0 e0 v has precisely idl isolated zeros inside f all of degree sign d where d deg g 0 is the degree of the map g 0ft n 1 see bbh bcp when ft is star shaped the general case relies on an estimate of stl moreover for any sequence of end0 there is a subsequence of v which converges to a map v x of the form

dynamics of defects in the complex ginzburg landau equation abstract we review the derivation of basic equations for the spiral dynamics proposed by us before these coupled equations involving the trajectories and a global phase are solved here in various situations of physical interest

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

quantization and motion law for ginzburg landau vortices we study the vortex trajectories for the two dimensional complex parabolic ginzburg landau equation without a well preparedness assumption we prove that the trajectory set is rectifiable and satisfies a weak motion law in the case of degree 1 vortices the motion law is satisfied in the classical sense moreover dissipation occurs only at a finite number of times away from these times possible collisions and splittings of vortices are constrained by algebraic equations involving their topological degrees

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

on the dynamical law of the ginzburg landau vortices on the plane we study the ginzburg landau equation on the plane with initial data being the product of n well separated 1 vortices and spatially decaying perturbations if the separation distances are o e 1 e 1 we prove that the n vortices do not move on the time scale o e 2 e e o log 1 e instead they move on the time scale o e 2 log 1 according to the law x j x jw w l6 j logjxl x jj x j j j 2 r 2 the location of the j th vortex the main ingredients of our proof consist of estimating the large space behavior of solutions a monotonicity inequality for the energy density of solutions and energy comparisons combining these we overcome the infinite energy difficulty of the planar vortices to establish the dynamical law c 1999 john wiley sons inc

fully implicit formulation of elastoplastic homogenization problem for two scale analysis abstract in this study using a virtual work equation a micro macro kinematic relation and a linearized constitutive relation a boundary value problem is fully implicitly formulated to determine perturbed displacement increment fields in elastoplastic unit cells for two scale analysis it is shown that this implicit homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increment fields in unit cells and that the boundary value problem formulated provides a computational algorithm which is versatile for initial setting of strain increment fields the computational algorithm developed is then examined by performing a two scale analysis of a holed plate with an elastoplastic micro structure subjected to tensile loading this demonstrates that the convergence in iteratively solving the implicit homogenization problem strongly depends on the initial setting of strain increment fields in unit cells

numerical study of hydrodynamics using the nonlinear schro dinger equation abstract the hydrodynamical behavior of the nonlinear schrodinger equation is investigated by fourier pseudo spectral direct numerical simulations its dispersive and nonlinear acoustics are characterized quantitatively and an equation that describes this regime at leading order is derived a technique that allows the preparaion of periodic initial data containing an arbitrary system of point vortices with minimal acoustic excitations is given the eulerian dynamics of a jet made of an array of counter rotating vortices is obtained sinuous and varicose instabilities are shown to take place finally the numerical methods best suited to study vortex sound interactions are discussed

optimal embeddings and compact embeddings of bessel potential type spaces first we establish necessary and sufficient conditions for embeddings of bessel potential spaces h sigma x mathbb r n with order of smoothness less than one modelled upon rearrangement invariant banach function spaces x mathbb r n into generalized holder spaces to this end we derive a sharp estimate of modulus of smoothness of the convolution of a function f in x mathbb r n with the bessel potential kernel g 0 1 such an estimate states that if g sigma belongs to the associate space of x then omega f g sigma t precsim int limits 0 t n s frac sigma n 1 f s ds quad rm for all quad t in 0 1 quad rm and every quad f in x mathbb r n second we characterize compact subsets of generalized holder spaces and then we derive necessary and sufficient conditions for compact embeddings of bessel potential spaces h sigma x mathbb r n into generalized holder spaces we apply our results to the case when x mathbb r n is the lorentz karamata space l p q b mathbb r n in particular we are able to characterize optimal embeddings of bessel potential spaces h sigma l p q b mathbb r n into generalized holder spaces and also compact embeddings of spaces in question applications cover both superlimiting and limiting cases

multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method abstract this paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with voronoi cell vcfem for microstructural modeling the voronoi cell finite element model originates from dirichlet tessellation of a representative material element or a base cell in the microstructure homogenized material coefficients for a global displacement finite element model are generated by vcfem analysis using periodic boundary conditions on the base cell following the macroscopic analysis the local vcfem analysis is implemented to depict the true evolution of microstructural stresses and strains various numerical examples are executed for validating the effectiveness of vcfem macro micro modeling of elastic materials the effect of size shape orientation and distribution of heterogeneities on the local and global response are examined

an u ale formulation of 3 d unsteady viscoelastic flow the arbitrary lagrange euler ale formulation is applied to the analysis of unsteady viscoelastic flow of an upper convected maxwell ucm medium it is shown how the requirement of incremental objectivity affects the stress updating scheme the current implementation of the ale formulation uses the time discontinuous galerkin least squares td gls method to handle the advective parts of the equations further the ale formulation is compared with an updated ale formulation that is shown to give comparable results but is much more cost effective the technique presented is particularly effective for free surface flows three applications are given

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

influence of lateral boundary conditions on a phase defect description of traveling wave convection patterns abstract traveling wave convection patterns in binary fluid mixtures of ethanol and water have been studied experimentally in containers with lateral boundaries of different shapes including circular cells of two different sizes rectangular and stadium shaped cells the nature of the patterns depends qualitatively upon the shape of the lateral boundaries this paper analyzes these patterns in terms of topological phase defects in the complex order parameter providing a reduced description of the system the experimentally measured phase fields and hence the patterns can be reconstructed from the location and charge of the topological defects for good agreement with experiment the structure of individual defects is found to depend on both nearest neighbor interactions of like sign defects and the global structure of the patterns reflected by the net charge of the defects further implications of these results for understanding traveling wave patterns and dynamics are discussed

traveling wave solutions of the schrodinger map equation we first construct traveling wave solutions for the schrodinger map in r 2 m t m m m 3 e 3 in 2 of the form m x 1 x 2 t where m has exactly two vortices at approximately 1 2 0 r 2 of degree 1 we use a perturbative approach that gives a complete characterization of the asymptotic behavior of the solutions with a few modifications a similar construction yields traveling wave solutions of schrodinger map equations in higher dimensions 2010 wiley periodicals inc

dynamics of multiple degree ginzburg landau vortices for the two dimensional complex parabolic ginzburg landau equation we prove that asymptotically vortices evolve according to a simple ordinary differential equation which is a gradient flow of the kirchhoff onsager functional this convergence holds except for a finite number of times corresponding to vortex collisions and splittings which we describe carefully the only assumption is a natural energy bound on the initial data

refined jacobian estimates and gross pitaevsky vortex dynamics we study the dynamics of vortices in solutions of the gross pitaevsky equation i u t delta u 1 over varepsilon 2 u left 1 u 2 right in a bounded simply connected domain omega subset mathbb r 2 with natural boundary conditions on previous rigorous results have shown that for sequences of solutions u varepsilon with suitable well prepared initial data one can determine limiting vortex trajectories and moreover that these trajectories satisfy the classical ode for point vortices in an ideal incompressible fluid we prove that the same motion law holds for a small but fixed varepsilon and we give estimates of the rate of convergence and the time interval for which the result remains valid the refined jacobian estimates mentioned in the title relate the jacobian j u of an arbitrary function u in h 1 omega mathbb c to its ginzburg landau energy in the analysis of the gross pitaevsky equation they allow us to use the jacobian to locate vortices with great precision and they also provide a sort of dynamic stability of the set of multi vortex configurations

prediction of the mechanical behavior of nonlinear heterogeneous systems by multi level finite element modeling an accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented this method is based on the classical homogenization theory assuming local spatial periodicity of the microstructure consequently the microstructure is identified by a representative volume element rve with conformity of opposite boundaries at any stage of the deformation process the local macroscopic stress is obtained by applying the local macroscopic deformation represented by the deformation tensor on a unique rve through imposing appropriate boundary conditions and averaging the resulting rve stress field if the assumption of local periodicity of the morphology is valid this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation the homogenization method was implemented in a multi level finite element program with meshes on macroscopic level mesh of entire structure and microscopic level meshes of rves the performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet