gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

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

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

a stability property for locally uniformly rotund renorming abstract it is proved that c k e the space of all continuous functions on a hausdorff compact space k taking values in a banach space e admits an equivalent locally uniformly rotund norm if c k and e do so moreover if the equivalent lur norms on c k and e are lower semicontinuous with respect to some weak topologies the lur norm on c k e can be chosen to be lower semicontinuous with respect to an appropriate weak topology as a consequence we prove that if x and y are two hausdorff compacta and c x c y admit equivalent pointwise lower semicontinuous lur norms then so does c x y

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

on the incompressible fluid limit and the vortex motion law of the nonlinear schrodinger equation the nonlinear schrodinger equation nls has been a fundamental model for understanding vortex motion in superfluids the vortex motion law has been formally derived on various physical grounds and has been around for almost half a century we study the nonlinear schrodinger equation in the incompressible fluid limit on a bounded domain with dirichlet or neumann boundary condition the initial condition contains any finite number of degree 1 vortices we prove that the nls linear momentum weakly converges to a solution of the incompressible euler equation away from the vortices if the initial nls energy is almost minimizing we show that the vortex motion obeys the classical kirchhoff law for fluid point vortices similar results hold for the entire plane and periodic cases and a related complex ginzburg landau equation we treat as well the semi classical wkb limit of nls in the presence of vortices in this limit sound waves propagate through steady vortices

natural examples of valdivia compact spaces abstract we collect examples of valdivia compact spaces their continuous images and associated classes of banach spaces which appear naturally in various branches of mathematics we focus on topological constructions generating valdivia compact spaces linearly ordered compact spaces compact groups l 1 spaces banach lattices and noncommutative l 1 spaces

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

near integrable systems a two dimensional circular quantum billiard with unusual boundary conditions introduced by berry and dennis 2008 j phys a math theor 41 135203 is considered in detail it is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with neumann boundary conditions deviations from strong localization are also discussed these results agree well with numerical calculations

single parameter scaling in one dimensional localization revisited the variance of the lyapunov exponent is calculated exactly in the one dimensional anderson model with random site energies distributed according to the cauchy distribution we find a new significant scaling parameter in the system and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization the results obtained are applied to the kronig penney model with the potential in the form of periodically positioned ensuremath delta functions with random strength

lorentz space estimates for the ginzburg landau energy abstract in this paper we prove novel lower bounds for the ginzburg landau energy with or without magnetic field these bounds rely on an improvement of the vortex balls construction estimates by extracting a new positive term in the energy lower bounds this extra term can be conveniently estimated through a lorentz space norm on which it thus provides an upper bound the lorentz space l 2 we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results

semiclassical limit of the gross pitaevskii equation in an exterior domain in this paper we study the semiclassical limit of the gross pitaevskii equation a cubic nonlinear schrodinger equation with the neumann boundary condition in an exterior domain we prove that before the formation of singularities in the limit system the quantum density and the quantum momentum converge to the unique solution of the compressible euler equation with the slip boundary condition as the scaling parameter approaches 0

the correlation coefficient for vortices in the plane abstract for point vortices in the plane we consider the correlation coefficient of ovchinnikov and sigal generalising a result by esposito we show that it vanishes for all vortex equilibria

in out decomposition of boundary integral equations we propose a reformulation of the boundary integral equations for the helmholtz equation in a domain in terms of incoming and outgoing boundary waves we obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling these effects are absent in the well known semiclassical transfer operators in the sense of bogomolny it has long been established that transfer operators are equivalent to the boundary integral approach within semiclassical approximation exact treatments have been restricted to specific boundary conditions such as dirichlet or neumann the approach we propose is independent of the boundary conditions and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary as an application we show how the decomposition may be used to calculate goos haenchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities

complex ginzburg landau equations and dynamics of vortices filaments and codimension 2 submanifolds here we study the asymptotic behavior of solutions to the complex ginzburg landau equations and their associated heat flows in arbitrary dimensions when the ginzburg landau parameter tends to infinity we prove that the energies of solutions in the flow are concentrated at vortices in two dimensions filaments in three dimensions and codimension 2 submanifolds in higher dimensions moreover we show the dynamical laws for the motion of these vortices filaments and codimension 2 submanifolds away from the energy concentration sets we use some measure theoretic arguments to show the strong convergence of solutions in both static and heat flow cases 1998 john wiley sons inc

multi vortex solutions to ginzburg landau equations with external potential we consider the existence of multi vortex solutions to the ginzburg landau equations with external potential on mathbb r 2 these equations model equilibrium states of superconductors and stationary states of the u 1 higgs model of particle physics in the former case the external potential models impurities and defects we show that if the external potential is small enough and the magnetic vortices are widely spaced then one can pin one or an arbitrary number of vortices in the vicinity of a critical point of the potential in addition one can pin an arbitrary number of vortices near infinity if the potential is radially symmetric and of an algebraic order near infinity

a gradient flow approach to an evolution problem arising in superconductivity we study an evolution equation proposed by chapman rubinstein and schatzman as a mean field model for the evolution of the vortex density in a superconductor we treat the case of a bounded domain where vortices can exit or enter the domain we show that the equation can be derived rigorously as the gradient flow of some specific energy for the riemannian structure induced by the wasserstein distance on probability measures this leads us to some existence and uniqueness results and energy dissipation identities we also exhibit some entropies that decrease through the flow and allow us to get regularity results solutions starting in lp p 1 remain in lp 2007 wiley periodicals inc

boundary condition varying circle billiards and gratings the dirichlet singularity waves in a two dimensional domain with robin mixed boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena if the variation includes a d point where the boundary condition is dirichlet vanishing wavefunction a variety of arguments indicate that the system is singular for a circle billiard the boundary condition fails to determine a discrete set of levels so the spectrum is continuous for a diffraction grating defined by periodically varying boundary conditions on the edge of a half plane the phase of a diffracted beam amplitude remains undetermined in both cases the wavefunction on the boundary has a singularity at a d point described by the polylogarithm function

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

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

completeness in the mackey topology by norming subspaces abstract we study the class of banach spaces x such that the locally convex space x x y is complete for every norming and norm closed subspace y x where x y denotes the mackey topology on x associated to the dual pair x y such banach spaces are called fully mackey complete we show that fully mackey completeness is implied by efremov s property e and on the other hand it prevents the existence of subspaces isomorphic to l 1 1 this extends previous results by guirao et al 2017 9 and bonet and cascales 2010 3 further examples of banach spaces which are not fully mackey complete are exhibited like c 0 1 and the long james space j 1 finally by assuming the continuum hypothesis we construct a banach space with w sequential dual unit ball which is not fully mackey complete a key role in our discussion is played by the at least formally smaller class of banach spaces x such that y w has the mazur property for every norming and norm closed subspace y x

distances to spaces of baire one functions given a metric space x and a banach space e we use an index of fragmentability for maps f in e x to estimate the distance of f to the space b 1 x e of baire one functions from x into e when x is polish we use our estimations for these distances to give a quantitative version of the well known rosenthal s result stating that in b 1 x mathbb r the pointwise relatively countably compact sets are pointwise relatively compact we also obtain a quantitative version of a srivatsa s result that states that whenever x is metric any weakly continuous function f in e x belongs to b 1 x e our result here says that for an arbitrary f in e x we have d f b 1 x e leq 2 sup x in b e ast rm osc x circ f where osc x circ f stands for the supremum of the oscillations of x circ f at all points x in x as a consequence of the above we prove that for functions in two variables f x times k to mathbb r x complete metric and k compact there exists a g dense set d subset x such that the oscillation of f at each x k in d times k is bounded by the oscillations of the partial functions f x and f k a representative result in this direction that we prove using games is the following if x is a unfavorable space and k is a compact space then there exists a dense g subset d of x such that for each y k in d times k rm osc f y k le 6 sup x in x rm osc f x 8 sup k in k rm osc f k when the right hand side of the above inequality is zero we are dealing with separately continuous functions f x times k to mathbb r and we obtain as a particular case some well known results obtained by the third named author in the mid 1970s

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

some applications of projective resolutions of identity we show that the unit ball k of the bidual of an asplund space is a corson compact or contains 0 1 and that it has the namioka property on separate to joint continuity the same results are shown for k a valdivia compact a by product is that all dyadic compacts have the namioka property some connections with weakly compactly generated dual spaces and renormings are given

travelling waves for the gross pitaevskii equation ii the purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the gross pitaevskii equation in dimensions two and three our arguments based on minimization under constraints yield a full branch of solutions and extend earlier results see 3 4 8 where only a part of the branch was built in dimension three we also show that there are no travelling wave solutions of small energy

singular continuous spectrum for a class of almost periodic jacobi matrices we consider operators parametrized by a o on 2 with matrix ojy i o i fl o y with an cos 2ttcw 6 if ce is a liouville number and 2 we prove that for a e 0 the operator s spectral measures are all singular continuous we consider the operator on l2 z depending upon three parameters x a 0 1 x a 0 u n u n 1 u n 1 x cos 2iran 6 u n in this note we will sketch the proof of the following result whose detailed proof will appear elsewhere 3 theorem 1 fix a an irrational number obeying

rectifiability of flat chains we prove without using federer s structure theorem that a finite mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0 dimensional slices are rectifiable this implies that every flat chain of finite mass and finite size is rectifiable it also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite mass flat

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

trace formula for dielectric cavities general properties the construction of the trace formula for open dielectric cavities is examined in detail using the krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity the contribution of each periodic orbit is the product of the two factors the first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived the coefficient of the perimeter term differs from the one for closed cavities due to unusual high energy asymptotics of the s matrix for the scattering on the cavity corrections to the leading semi classical formula are briefly discussed obtained formulas agree well with numerical calculations for circular dielectric cavities

intrinsic stickiness and chaos in open integrable billiards tiny border effects rounding border effects at the escape point of open integrable billiards are analyzed via the escape time statistics and emission angles the model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form stickiness chaos and self similar structures for the escape times and emission angles are generated inside backgammon like stripes of initial conditions these stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics tiny rounded borders around 0 1 from the whole billiard size are shown to be sufficient to generate the sticky motion with power law decay esc 1 27 while borders larger than 10 are enough to produce escape times related to the chaotic motion escape exponents in the interval 1 esc 2 are generated due to marginal unstable periodic orbits trapping alternately in time regular and chaotic trajectories

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

robust a priori and a posteriori error analysis for the approximation of allen cahn and ginzburg landau equations past topological changes a priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution for typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis the estimates naturally lead to adaptive mesh refinement and coarsening algorithms numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

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

ginzburg landau vortex dynamics driven by an applied boundary current in this paper we study the time dependent ginzburg landau equations on a smooth bounded domain c 2 subject to an electrical current applied on the boundary the dynamics with an applied current are nondissipative but via the identification of a special structure in an interaction energy we are able to derive a precise upper bound for the energy growth we then turn to the study of the dynamics of the vortices of the solutions in the limit e 0 we first consider the original time scale in which the vortices do not move and the solutions undergo a phase relaxation then we study an accelerated time scale in which the vortices move according to a derived dynamical law in the dynamical law we identify a novel lorentz force term induced by the applied boundary current 2010 wiley periodicals inc

convergence and non convergence of many particle evolutions with multiple signs we address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity we consider two types of particles called positive and negative same sign particles repel each other and opposite sign particles attract each other the interaction potential is the same for all particles up to the sign and has a logarithmic singularity at zero the central example of such systems is that of dislocations in crystals because of the singularity in the interaction potential the discrete evolution leads to blow up in finite time we remedy this situation by regularising the interaction potential at a length scale delta n 0 which converges to zero as the number of particles n tends to infinity we establish two main results the first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled pdes which describe the evolution of their continuum densities in the setting of dislocations these pdes are known as the groma balogh equations in the proof we rely on both the theory of lambda convex gradient flows to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta n regularised version of the groma balogh equations and a priori estimates for the groma balogh equations to pass to the small regularisation limit in a functional setting based on orlicz spaces in order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta n to converge to zero sufficiently slowly the second result is a counterexample demonstrating that if delta n converges to zero sufficiently fast then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the groma balogh equations these results show how the validity of the groma balogh equations as the limit of many particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised

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

semiclassical quantization of billiards with mixed boundary conditions the semiclassical theory for billiards with mixed boundary conditions is developed and explicit expressions for the smooth and the oscillatory parts of the spectral density are derived the parametric dependence of the spectrum on the boundary condition is shown to be a very useful diagnostic tool in the semiclassical analysis of the spectrum of billiards it is also used to check in detail some recently proposed parametric spectral statistics the methods are illustrated in the analysis of the spectrum of the sinai billiard and its parametric dependence on the boundary condition on the dispersing arc

convergence of the parabolic ginzburg landau equation to motion by mean curvature for the complex parabolic ginzburg landau equation we prove that asymptotically vorticity evolves according to motion by mean curvature in brakke s weak formulation the only assumption is a natural energy bound on the initial data in some cases we also prove convergence to enhanced motion in the sense of ilmanen