the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

averaged lagrangians and the mean effects of fluctuations in ideal fluid dynamics we begin by placing the generalized lagrangian mean glm equations for a compressible adiabatic fluid into the euler poincare ep variational framework of fluid dynamics for an averaged lagrangian we then state the ep averag ing result that glm equations arise from glm hamilton s principles in the ep framework next we derive a new set of approximate small amplitude glm equations gm equations at second order in the fluctuating displacement of a lagrangian trajectory from its mean position these equations express the linear and nonlinear back reaction effects on the eulerian mean fluid quantities by the fluctuating displacements of the lagrangian trajectories in terms of their eulerian second moments the derivation of the gm equations uses the linearized relations between eulerian and lagrangian fluctuations in the tradition of lagrangian stability analysis for fluids the gm derivation also uses the method of averaged lagrangians in the tradition of wave mean flow interaction wmfi the gm ep motion equations for compressible and incompressible ideal fluids are compared with the euler alpha turbulence closure equations an alpha model is a glm or gm fluid theory with a taylor hypothesis closure thc such closures are based on the linearized fluctuation relations that determine the dynamics of the lagrangian statistical quantities in the euler alpha closure equations we use the ep averaging result to bridge between the glm equations and the euler alpha closure equations hence combining the small amplitude approximation with thc yields in new gm turbulence closure equations for compressible fluids in the ep variational framework 2002 elsevier science b v all rights reserved

the camassa holm equation as a geodesic flow on the diffeomorphism group misiolek j geom phys 24 203 208 1998 has shown that the camassa holm equation is a geodesic flow on the bott virasoro group in this paper it is shown that the camassa holm equation for the case 0 is the geodesic spray of the weak riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the h1 inner product over the entire group this paper uses the right trivialization technique to rigorously verify that the euler poincare theory for lie groups can be applied to diffeomorphism groups the observation made in this paper has led to physically meaningful generalizations of the ch equation to higher dimensional manifolds

necessary conditions for existence of non degenerate hamiltonian structures the necessary criteria are pointed out for the exisence of hamiltonian and bi hamiltonian non degenerate structures for a nonlinear system of partial differential equations of first order the results are formulated in terms of the new invariants of the intrinsic geometry introduced in this paper connected with the nijenhuis and haantjes tensors of a 1 1 tensor field

constructing an atlas for the diffeomorphism group of a compact manifold with boundary with application to the analysis of image registrations this paper considers the problem of defining a parameterization chart on the group of diffeomorphisms with compact support motivated primarily by a problem in image registration where diffeomorphic warps are used to align images constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population we construct a chart for particular choices of boundary conditions on the space on which the diffeomorphism acts and for a particular class of metrics on the diffeomorphism group which define a class of diffeomorphic interpolating splines the geodesic equation is computed for this class of metrics and we show how it can be solved in the spline representation furthermore we demonstrate that the spline representation generates submanifolds of the diffeomorphism group and we study this mapping explicit computational examples are included showing how this chart can be constructed in practice and that the use of the geodesic distance allows better classification of variation than those obtained using just a euclidean metric on the space of warps

objective fluxes in a multi scale continuum description of sparse medium dynamics abstract we discuss reasons justifying a multi scale continuum description of sparse media which do not admit a choice of a representative volume element remaining permanent in time with constant mass we suggest the choice of objective derivatives for time varying terms in the balance equations pertinent to the scheme that we analyze to avoid some problems connected with s o 3 based changes in observers which emerge within the setting of the standard theory of gases when we start from it to get continuum models the scheme discussed here can be reduced to versions of the averaged regularizations of the navier stokes equations in our approach we have primarily in mind the continuum description of bodies like macro molecular fluids or granular ones even if what we propose can be significant in a broader setting

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

mel nikov analysis of numerically induced chaos in the nonlinear schro dinger equation abstract certain hamiltonian discretizations of the periodic focusing nonlinear schrodinger equation nls have been shown to be responsible for the generation of numerical instabilities and chaos in this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the nls using heuristic mel nikov methods the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system the structure of the persistent homoclinic orbits that are predicted by the mel nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos the mel nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments in a neighborhood of the persistent homoclinic orbits the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study

navier stokes equation and diffusions on the group of homeomorphisms of the torus a stochastic variational principle for the two dimensional navier stokes equation is established the velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold navier stokes equation is reinterpreted as a perturbed equation of geodesics for the l 2 norm the method described here should hold as well in higher dimensions

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

diagonalizing matrices over c x a complete characterization of those compact hausdorff spaces is given such that for every n each normal element in the algebra c x mn of continuous functions from x to mn can be continuously diagonalized the conditions are that x be a sub stonean space with dim x 2 and carries no nontrivial g bundles over any closed subset for g a symmetric group or the circle group in particular diagonalization is assured on every totally disconnected sub stonean space but also on connected spaces of the form y y where y is a simply connected noncompact graph

stability criteria for the 2d α euler equations abstract we derive analogues of the classical rayleigh fjortoft and arnold stability and instability theorems in the context of the 2d euler equations

monopoles and baker functions the work in this paper pertains to the solutions of nahm s equations which arise in the atiyah drinfield hitchin manin nahm construction of solutions to the bogomol nyi equations for static monopoles this paper provides an explicit construction of the solution of nahm s equations which satisfy regularity and reality conditions the lax form of nahm s equations is reduced to a standard eigenvalue problem by a special gauge transformation these equations may then be solved by the method of baker krichever this leads to a compact representation of the solutions of nahm s equations the regularity condition is shown to be related to the monodromy of the gauge reduced linear operator hitchin showed that the solutions of nahm s equations can be characterized by an algebraic curve and some data on that curve here this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential donaldson coordinatized the moduli space ofk monopoles by a class of rational maps from the riemann sphere to itself the data of a baker function is equivalent to this map this method gives an apriori construction of the known two monopole solutions we also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge these solutions correspond to reducible curves with elliptic components and the associated donaldson rational function has a simple partial fraction expansion

on unique ergodicity in nonlinear stochastic partial differential equations we illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes examples exhibiting parabolic and hyperbolic structure are studied in detail in the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the krylov bogolyubov procedure and compactness fails

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

the supersymmetric camassa holm equation and geodesic flow on the superconformal group we study a family of fermionic extensions of the camassa holm equation within this family we identify three interesting classes a equations which are inherently hamiltonian describing geodesic flow with respect to an h1 metric on the group of superconformal transformations in two dimensions b equations which are hamiltonian with respect to a different hamiltonian structure and c supersymmetric equations classes a and b have no intersection but the intersection of classes a and c gives a system with interesting integrability properties we demonstrate the painleve property for some simple but nontrivial reductions of this system and also discuss peakon type solutions

simultaneous space time adaptive wavelet solution of nonlinear parabolic differential equations dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time these methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time the same time step is used for all spatial locations and all scales this approach clearly does not fully exploit space time intermittency we propose an adaptive wavelet collocation method for solving highly intermittent problems e g turbulence on a simultaneous space time computational domain which naturally adapts both the space and time resolution to match the solution besides generating a near optimal grid for the full space time solution this approach also allows the global time integration error to be controlled the efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent 1d t dimensional and 2d t dimensional test problems in particular we found that the space time method uses roughly 18 times fewer space time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method while achieving similar global accuracy

blow up phenomenon for a periodic rod equation in this paper firstly we find the optimal constant for a kind of sobolev inequality on the unit circle via fourier series then we apply the optimal constant on a nonlinear rod equation to give sufficient conditions on the initial datum which guarantee finite time singularity formation for the corresponding solution

the two dimensional periodic b equation on the diffeomorphism group of the torus in this paper the two dimensional periodic b equation is discussed under geometric aspects i e as a geodesic flow on the diffeomorphism group of the torus in the framework of arnold s 1966 ann inst fourier grenoble 16 319 famous approach we achieve some well posedness results for the b equation and we perform explicit curvature computations for the 2d camassa holm equation which is obtained for b 2 finally we explain the special role of the choice b 2 by giving a rigorous proof that b 2 is the only case in which the associated geodesic flow is weakly riemannian

new perspectives in turbulence scaling laws asymptotics and intermittency intermittency a basic property of fully developed turbulent flow decreases with growing viscosity therefore classical relationships obtained in the limit of vanishing viscosity must be corrected when the reynolds number is finite but large these corrections are the main subject of the present paper they lead to a new scaling law for wall bounded turbulence which is of key importance in engineering and to a reinterpretation of the kolmogorov obukhov scaling for the local structure of turbulence which has been of paramount interest in both theory and applications the background of these results is reviewed in similarity methods in the statistical theory of vortex motion and in intermediate asymptotics and relevant experimental data are summarized

a modified wavelet approximation for multi resolution awcm in simulating nonlinear vibration of mdof systems abstract wavelet methods have been presented recently to solve the initial value problems of time dependent odes before applying the current wavelet methods a nonlinear vibration equation usually needs to be transformed to state equations which may raise the storage and computation cost in this paper a modified form of wavelet approximation for the vibration solutions of linear nonlinear mdof systems is presented based on which a multi resolution awcm adaptive wavelet collocation method is established for mdof systems the modified wavelet approximation s wavelet coefficients explicitly include the boundary first order time derivatives of the solutions besides values of solutions on inner time locations of the time domain on which the solutions are defined this can avoid to transform the dynamical equations of a linear nonlinear vibrational mdof system to state equations therefore by using the modified wavelet approximation the computation and storage cost required for wavelet collocation method may be reduced to at most 1 2 of that needed by those based on state equations and the computation efficiency may be at least twice faster than wavelet collocation methods based on state equations because the number of solution components that are needed to be computed is only 1 2 of the corresponding state equations on the other hand the implementation of initial conditions is straightforward by using the modified wavelet approximation proposed in this paper the effectiveness of the modified wavelet approximation is tested by applying it to a linear and a nonlinear 10 dof vibrational system

global well posedness for the averaged euler equations in two dimensions abstract we prove global well posedness of the two dimensional averaged euler or euler equations for initial potential vorticity of class l2 this model generalizes the one dimensional fokas fuchssteiner camassa holm equation which describes the propagation of unidirectional waves on the surface of shallow water as such it can be realized as a geodesic equation for the h1 metric on the lie algebra of vector fields moreover in two dimensions the model obeys an advection equation for the so called potential vorticity in close analogy to the vorticity form of the euler equations we construct solutions to the weak form of the potential vorticity equation by taking the inviscid limit of solutions to a system regularized by artificial viscosity since the streamfunction vorticity relation is of order four we can show uniqueness even for potential vorticities in l2

exponential mixing for stochastic model of two dimensional second grade fluids abstract in this paper we establish the exponential mixing property of stochastic model for the incompressible second grade fluids the general criterion established by odasso 2008 plays an important role

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

padded continued fraction absorbing boundary conditions for dispersive waves continued fraction absorbing boundary conditions cfabcs are new arbitrarily high order local absorbing boundary conditions that are highly effective in simulating wave propagation in unbounded domains the current versions of cfabcs are developed for the non dispersive acoustic wave equation with convex polygonal computational domains in this paper the cfabcs are modified through augmentation of special padding elements and are effective for absorbing evanescent waves occurring in dispersive wave problems while retaining their absorption properties for propagating waves the padded cfabcs for dispersive wave equations result in fourth order evolution equations which are solved using an efficient combination of crank nicholson method newmark time stepping scheme and operator splitting ideas the effectiveness of the padded cfabcs and their implementation is illustrated through numerical examples with varying levels of dispersion

the generation of modulated wavetrains in the solution of the korteweg de vries equation on demontre une relation entre la methode des ondes modulees de whitham et la methode de lax et levermore contenant la limite de petite dispersion

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

low regularity solutions of the periodic fornberg whitham equation this paper studies low regularity solutions of the periodic fornberg whitham equation with initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic fornberg whitham equation

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

two phase nonlinear waves in the long josephson junction abstract the electrodynamic properties of the long josephson junction for the case of two phase wave train are investigated by means of asymptotic solutions of the perturbed sine gordon equation in the form of the whitham equations describing the modulation of its period matrix within the power balance approximation the corresponding current voltage characteristic is calculated in terms of riemann functions and analyzed numerically for the zero magnetic field the resonance peaks of the characteristic are shown to correspond to the reduction condition for the period matrix

dynamic analysis of a poroelastic layered half space using continued fraction absorbing boundary conditions abstract a methodology for dynamic analysis of a poroelastic layered half space is proposed a poroelastic layered half space is discretized with regular thin layers to a certain depth below the layers continued fraction absorbing boundary conditions cfabcs which are accurate and effective in modeling wave propagation in various unbounded domains are applied in order to represent the infinite extent of the half space with the representation a spectral decomposition which is an effective approach in the solution of wave propagation problems is utilized green functions of a poroelastic layered half space are obtained accurately with the proposed numerical model the green functions can be applied to various dynamic problems in the half space including foundation dynamics and soil structure interaction problems

stochastic euler poincare reduction we prove a euler poincare reduction theorem for stochastic processes taking values on a lie group which is a generalization of the reduction argument for the deterministic case j e marsden and t s ratiu introduction to mechanics and symmetry a basic exposition of classical mechanical systems texts in applied mathematics springer 2003 we also show examples of its application to so 3 and to the group of diffeomorphisms which includes the navier stokes equation on a bounded domain and the camassa holm equation

analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid we establish the existence of three new subgroups of the group of volumepreserving diffeomorphisms of a compact n dimensional n 2 riemannian manifold which are associated with the dirichlet neumann and mixed type boundary conditions that arise in second order elliptic pdes we prove that when endowed with the hs hilbert class topologies for s n 2 1 these subgroups are c differential manifolds we consider these new diffeomorphism groups with an h1 equivalent right invariant metric and prove the existence of unique smooth geodesics t of this metric as well as existence and uniqueness of the jacobi equations associated to this metric geodesics on these subgroups are in fact the flows of a timedependent velocity vector field u t x so that t t u t t with 0 x x and remarkably the vector field u t x solves the so called lagrangian averaged euler lae equations on m these equations and their viscous counterparts the lagrangian averaged navier stokes lans equations model the motion of a fluid at scales larger than an a priori fixed parameter 0 while averaging or filtering out the small scale motion and this is achieved without the use of artificial viscosity we prove that for divergence free initial data satisfying u 0 on m the lae equations are well posed globally when n 2 we also find the boundary conditions that make the lans equations well posed globally when n 3 and prove that solutions of the lans equations converge when n 2 3 for almost all t in some fixed time interval 0 t in hs s n 2 1 3 to solutions of the lae equations thus confirming the scaling arguments of barenblatt chorin received april 26 2000 1991 mathematics subject classification 35q35 35q53 58b20 58d05

the existence of solution for viscous camassa holm equations on bounded domain in five dimensions abstract the existence of global weak solution and local existence of strong solution for five dimensional viscous camassa holm equations on bounded domain are proved in this note the global existence of strong solution is also proved when small initial data is given

a finite element alternative to infinite elements abstract in this paper a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior and not in the far field by developing a finite element mesh that approximates the stiffness of an unbounded domain directly without approximating the far field displacement profile first the current formulation provides a superior alternative to infinite elements ies that have long been used to incorporate unbounded domains into the finite element method fem in contrast to most ies the present formulation a requires no new shape functions or special integration rules b is proved to be both accurate and efficient and c is versatile enough to handle a large variety of domains including those with anisotropic stratified media and convex polygonal corners in addition to this the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved this error expression can be used to calculate the accuracy of a given mesh a priori this in turn allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem we formulate such an optimization problem solve it and use the results to develop a practical mesh generation methodology this methodology does not require any additional computation on the part of the user and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method

well posedness and blow up solution for the stochastic dullin gottwald holm equation in this paper to describe the uncertainty from external environment we propose a stochastic dullin gottwald holm sdgh equation by adding an additive noise to the well known dgh equation using regularization the local existence and uniqueness of the solution of sdgh are proved by means of a priori estimates we find that the local solution may blow up in a finite time if the initial value satisfies certain conditions in this paper to describe the uncertainty from external environment we propose a stochastic dullin gottwald holm sdgh equation by adding an additive noise to the well known dgh equation using regularization the local existence and uniqueness of the solution of sdgh are proved by means of a priori estimates we find that the local solution may blow up in a finite time if the initial value satisfies certain conditions

fredholm properties of riemannian exponential maps on diffeomorphism groups we prove that exponential maps of right invariant sobolev h r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear fredholm maps of index zero as long as r is sufficiently large this generalizes the result of ebin et al geom funct anal 16 2006 for the l 2 metric on the group of volume preserving diffeomorphisms important in hydrodynamics in particular our results apply to many other equations of interest in mathematical physics we also prove an infinite dimensional morse index theorem settling a question raised by arnold and khesin topological methods in hydrodynamics springer new york 1998 on stable perturbations of flows in hydrodynamics finally we include some applications to the global geometry of diffeomorphism groups

geodesic warps by conformal mappings in recent years there has been considerable interest in methods for diffeomorphic warping of images with applications in e g medical imaging and evolutionary biology the original work generally cited is that of the evolutionary biologist d arcy wentworth thompson who demonstrated warps to deform images of one species into another however unlike the deformations in modern methods which are drawn from the full set of diffeomorphisms he deliberately chose lower dimensional sets of transformations such as planar conformal mappings in this paper we study warps composed of such conformal mappings the approach is to equip the infinite dimensional manifold of conformal embeddings with a riemannian metric and then use the corresponding geodesic equation in order to obtain diffeomorphic warps after deriving the geodesic equation a numerical discretisation method is developed several examples of geodesic warps are then given we also show that the equation admits totally geodesic solutions corresponding to scaling and translation but not to affine transformations

uniform time of existence for the alpha euler equations we consider the euler equations on a bounded three dimensional domain with frictionless navier boundary conditions our main result is the existence of a strong solution on a positive time interval uniform in for sufficiently small combined with the convergence result in 6 this implies convergence of solutions of the euler equations to solutions of the incompressible euler equations when 0 in addition we obtain a new result on local existence of strong solutions for the incompressible euler equations on bounded three dimensional domains the proofs are based on new a priori estimates in conormal spaces

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

stability of peakons and linear dispersion limit for the periodic dullin gottwald holm equation the dullin gottwald holm equation can be described as the unidirectional propagation of surface waves in a shallow water regime in this paper we study the orbital stability problem of the peaked solitons peakons for the periodic dullin gottwald holm equation on the line by constructing a function only depending on three important conservative laws we prove that the shapes of peakons are stable under small perturbations we also demonstrate that the solutions of the cauchy problem for this equation converge to those of the corresponding periodic camassa holm equation as the linear dispersive parameter converges to zero

boundary layer for the navier stokes alpha model of fluid turbulence we study boundary layer turbulence using the navier stokes alpha model obtaining an extension of the prandtl equations for the averaged flow in a turbulent boundary layer in the case of a zero pressure gradient flow along a flat plate we derive a nonlinear fifth order ordinary differential equation which is an extension of the blasius equation we study it analytically and prove the existence of a two parameter family of solutions satisfying physical boundary conditions matching these parameters with the skin friction coefficient and the reynolds number based on momentum thickness we get an agreement of the solutions with experimental data in the laminar and transitional boundary layers as well as in the turbulent boundary layer for moderately large reynolds numbers

the lagrangian averaged navier stokes equation with rough data in sobolev spaces abstract the lagrangian averaged navier stokes equation is a recently derived approximation to the navier stokes equation in this article we prove the existence of short time solutions to the incompressible isotropic lagrangian averaged navier stokes equation with low regularity initial data in sobolev spaces w s p r n for 1 p for l 2 based sobolev spaces we obtain global existence results more specifically we achieve local existence with initial data in the sobolev space h n 2 p p r n for initial data in h 3 4 2 r 3 we obtain global existence improving on previous global existence results which required data in h 3 2 r 3

from simplicial chern simons theory to the shadow invariant ii this is the second of a series of papers in which we introduce and study a rigorous simplicial realization of the non abelian chern simons path integral for manifolds m of the form m s1 and arbitrary simply connected compact structure groups g more precisely we introduce for general links l in m a rigorous simplicial version wlorig l of the corresponding wilson loop observable wlo l in the so called torus gauge by blau and thompson nucl phys b 408 2 345 390 1993 for a simple class of links l we then evaluate wlorig l explicitly in a non perturbative way finding agreement with turaev s shadow invariant l

consistent transmitting boundary with continued fraction absorbing boundary conditions for analysis of soil structure interaction in a layered half space abstract a continued fraction absorbing boundary condition cfabc is combined with a consistent transmitting boundary and applied to an analysis of soil structure interaction in a layered half space in order to be a perfect absorber for in plane waves with an arbitrary horizontal wavenumber in a layered half space it is proposed that the cfabc be used in the form of a 2 layer a method for determination of parameters of the 2 layers was proposed for evanescent waves as well as propagating waves combined with consistent transmitting boundaries the 2 layers of cfabc are applied to problems of dynamics of rigid strip and circular foundations on a layered half space the results demonstrate that the proposed 2 layers of the cfabc can represent a layered half space accurately in the entire frequency range and absorb both evanescent and propagating p and s waves effectively

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

on diagonalization in map m g motivated by some questions in the path integral approach to topological gauge theories we are led to address the following question given a smooth map from a manifoldm to a compact groupg is it possible to smoothly diagonalize it i e conjugate it into a map to a maximal torust ofg

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

variational principles for lagrangian averaged fluid dynamics the lagrangian average la of the ideal fluid equations preserves their fundamental transport structure this transport structure is responsible for the kelvin circulation theorem of the la flow and hence for its potential vorticity convection and helicity conservation we show that lagrangian averaging also preserves the euler poincar e variational framework that implies the exact ideal fluid equations in the eulerian representation this is expressed in the lagrangian averaged euler poincar e laep theorem proved here we illustrate the laep theorem by applying it to incompressible ideal fluids to derive the lagrangian averaged euler equations and thereby recover the generalized lagrangian mean motion equation finally we discuss recent progress in applications of these equations as the basis for new la closure models of fluid turbulence

on geometric phases for soliton equations this paper develops a new complex hamiltonian structure forn soliton solutions for a class of integrable equations such as the nonlinear schrodinger sine gordon and korteweg de vries hierarchies of equations that yields amongst other things geometric phases in the sense of hannay and berry for example one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons the main new tools are complex angle representations that linearize the corresponding hamiltonian flows on associated noncompact jacobi varieties this new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi periodic solutions a method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters in particular the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated riemann surface in this setting soliton geometric asymptotics are constructed for studying geometric phases in the quantum case the general approach is worked out in detail for the three specific hierarchies of equations mentioned some links with functions the braid group and geometric quantization are pointed out as well

hamiltonian structure for the modulation equations of a sine gordon wavetrain on etudie l equation sinus gordon e 2 t 2 x 2 u sin u 0 ou e est un petit parametre on construit une solution asymptotique sous la forme d un train d onde n phase variant lentement

low regularity solutions of the periodic general degasperis procesi equation abstract this paper studies low regularity solutions of the periodic general degasperis procesi equation with an initial value the existence and the uniqueness of solutions are proved the results are illustrated by considering the periodic peakons of the periodic general degasperis procesi equation

on the weak limit of rapidly oscillating waves etant donnee une onde non lineaire oscillant rapidement u x t on caracterise et on deduit des equations d evolution pour la limite faible de u

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

structure preserving model reduction for mechanical systems this paper focuses on methods of constructing of reduced order models of mechanical systems which preserve the lagrangian structure of the original system these methods may be used in combination with standard spatial decomposition methods such as the karhunen loeve expansion balancing and wavelet decompositions the model reduction procedure is implemented for three dimensional finite element models of elasticity and we show that using the standard newmark implicit integrator significant savings are obtained in the computational costs of simulation in particular simulation of the reduced model scales linearly in the number of degrees of freedom and parallelizes well

blow up of solutions to the dgh equation in this paper firstly we find best constants for two convolution problems on the unit circle via a variational method then we apply the best constants on a nonlinear integrable shallow water equation the dgh equation to give sufficient conditions on the initial data which guarantee finite time singularity formation for the corresponding solutions finally we discuss the blow up phenomena for the nonperiodic case

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

the stationary navier stokes system in nonsmooth manifolds the poisson problem in lipschitz and c 1 domains we consider the linearized version of the stationary navier stokes equations on a subdomain of a smooth compact riemannian manifold m the emphasis is on regularity the boundary of is assumed to be only c1 and even lipschitz and the data are selected from appropriate sobolev besov scales our approach relies on the method of boundary integral equations suitably adapted to the variable coefficient setting we are considering here applications to the stationary nonlinear navier stokes equations in this context are also discussed

boundary effects on exact solutions of the lagrangian averaged navier stokes α equations in order to clarify the behavior of solutions of the lagrangian averaged navier stokes lans equations in the presence of solid walls we identify a variety of exact solutions of the full equations and their boundary layer approximations the solutions demonstrate that boundary conditions suggested for the lans equations in the literature 1 for a bounded domain do not apply in a semi infinite domain the convergence to solutions of the navier stokes equations as 0 is elucidated for infinite energy solutions in a semi infinite domain and non uniqueness of these solutions is discussed we also study the boundary layer approximation of lans equations denoted the prandtl equations and report solutions for turbulent jets and wakes our version of the prandtl equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of cheskidov 2

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

local and global well posedness of the fractional order epdiff equation on r d abstract of concern is the study of fractional order sobolev type metrics on the group of h diffeomorphism of r d and on its sobolev completions d q r d it is shown that the h s sobolev metric induces a strong and smooth riemannian metric on the banach manifolds d s r d for s 1 d 2 as a consequence a global well posedness result of the corresponding geodesic equations both on the banach manifold d s r d and on the smooth regular frechet lie group of all h diffeomorphisms is obtained in addition a local existence result for the geodesic equation for metrics of order 1 2 s 1 d 2 is derived

global regularity for a birkhoff rott α approximation of the dynamics of vortex sheets of the 2d euler equations abstract we present an regularization of the birkhoff rott equation induced by the two dimensional euler equations for the vortex sheet dynamics we show that an initially smooth self avoiding vortex sheet remains smooth for all times under the regularized dynamics provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure

modulations of perturbed kdv wavetrains the modulations of n phase korteweg devries kdv wavetrains in the presence of external perturbations is investigated an invariant representation of these modulation equations in terms of differentials on a riemann surface is derived from averaged perturbed conservation laws in particular the explicit dependence of the representation on the external perturbation is obtained this invariant representation is used to place the equation in a riemann diagonal form whose dependence on the external perturbation is explicitly computed

twisting with a flip the art of pestunization we construct mathcal n 2 supersymmetric yang mills theory on 4d manifolds with a killing vector field with isolated fixed points it turns out that for every fixed point one can allocate either instanton or anti instanton contributions to the partition function and that this is compatible with supersymmetry the equivariant donaldson witten theory is a special case of our construction we present a unified treatment of pestun s calculation on s 4 and equivariant donaldson witten theory by generalizing the notion of self duality on manifolds with a vector field we conjecture the full partition function for a mathcal n 2 theory on any 4d manifold with a killing vector using this new notion of self duality to localize a supersymmetric theory is what we call pestunization

topological phenomena in the real periodic sine gordon theory the set of real finite gap sine gordon solutions corresponding to a fixed spectral curve consists of several connected components a simple explicit description of these components obtained by the authors recently is used to study the consequences of this property in particular this description allows to calculate the topological charge of solutions the averaging of the x derivative of the potential and to show that the averaging of other standard conservation laws is the same for all components

geometry and curvature of diffeomorphism groups withh1metric and mean hydrodynamics holm marsden and ratiu adv in math 137 1998 1 81 derived a new model for the mean motion of an ideal fluid in euclidean space given by the equationv t u t v t 2 u t t u t grad p t where div u 0 andv 1 2 u in this model the momentumvis transported by the velocityu with the effect that nonlinear interaction between modes corresponding to length scales smaller than is negligible we generalize this equation to the setting of ann dimensional compact riemannian manifold the resulting equation is the euler poincare equation associated with the geodesic flow of theh1right invariant metric on ds the group of volume preserving hilbert diffeomorphisms of classhs we prove that the geodesic spray is continuously differentiable fromtds m intottds m so that a standard picard iteration argument proves existence and uniqueness on a finite time interval our goal in this paper is to establish the foundations for lagrangian stability analysis following arnold ann inst grenoble16 1966 319 361 to do so we use submanifold geometry and prove that the weak curvature tensor of the right invarianth1metric on ds is a bounded trilinear map in thehstopology from which it follows that solutions to jacobi s equation exist using such solutions we are able to study the infinitesimal stability behavior of geodesics

well posedness of the epdiff equation with a pseudo differential inertia operator abstract in this article we study the class of right invariant fractional order sobolev type metrics on groups of diffeomorphisms of a compact manifold m our main result concerns well posedness properties for the corresponding euler arnold equations also called the epdiff equations which are of importance in mathematical physics and in the field of shape analysis and template registration depending on the order of the metric we will prove both local and global well posedness results for these equations as a result of our analysis we will also obtain new commutator estimates for elliptic pseudo differential operators

integrable vs nonintegrable geodesic soliton behavior abstract we study confined solutions of certain evolutionary partial differential equations pde in 1 1 space time the pde we study are lie poisson hamiltonian systems for quadratic hamiltonians defined on the dual of the lie algebra of vector fields on the real line these systems are also euler poincare equations for geodesic motion on the diffeomorphism group in the sense of the arnold program for ideal fluids but where the kinetic energy metric is different from the l2 norm of the velocity these pde possess a finite dimensional invariant manifold of particle like measure valued solutions we call pulsons we solve the particle dynamics of the two pulson interaction analytically as a canonical hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum the result of this two pulson interaction for rear end collisions is elastic scattering with a phase shift as occurs with solitons the results for head on antisymmetric collisions of pulsons tend to be singularity formation numerical simulations of these pde show that their evolution by geodesic dynamics for confined or compact initial conditions in various nonintegrable cases possesses the same type of multi soliton behavior elastic collisions asymptotic sorting by pulse height as the corresponding integrable cases do we conjecture this behavior occurs because the integrable two pulson interactions dominate the dynamics on the invariant pulson manifold and this dynamics dominates the pde initial value problem for most choices of confined pulses and initial conditions of finite extent

the geometry of real sine gordon wavetrains the characterization ofreal n phase quasiperiodic solutions of the sine gordon equation has been an open problem in this paper we achieve this result employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature a significant by product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds and an understanding of how the real isospectral manifolds are embedded by placing the problem in this general context these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions

a rate of convergence for the lansα regularization of navier stokes equations abstract a rate of convergence of the solutions of the lans equations with periodic boundary to the solutions of the navier stokes equations as 0 is obtained in a mixed l 1 l 2 time space norm for small initial data in besov type function spaces in which global existence and uniqueness of solutions can also be established

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

a random walk interpretation of the incompressible navier stokes equations we introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid and we consider the force per unit volume exerted by such a molecule on the rest of the fluid averaging this force over an ensemble of such molecules and taking an appropriate limit we obtain the navier stokes equations the origin of time irreversibility in these equations is discussed in terms of the model

lagrangian averaging with geodesic mean this paper revisits the derivation of the lagrangian averaged euler lae or euler equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume preserving diffeomorphism group under the additional assumption that first order fluctuations are statistically isotropic and transported by the mean flow as a vector field averaging of the kinetic energy lagrangian of an ideal fluid yields the lae lagrangian the derivation presented here assumes a euclidean spatial domain without boundaries

convergence of the 2d euler α to euler equations in the dirichlet case indifference to boundary layers abstract in this article we consider the euler system as a regularization of the incompressible euler equations in a smooth two dimensional bounded domain for the limiting euler system we consider the usual non penetration boundary condition while for the euler regularization we use velocity vanishing at the boundary we also assume that the initial velocities for the euler system approximate in a suitable sense as the regularization parameter 0 the initial velocity for the limiting euler system for small values of this situation leads to a boundary layer which is the main concern of this work our main result is that under appropriate regularity assumptions and despite the presence of this boundary layer the solutions of the euler system converge as 0 to the corresponding solution of the euler equations in l 2 in space uniformly in time we also present an example involving parallel flows in order to illustrate the indifference to the boundary layer of the 0 limit which underlies our work