viscosity versus vorticity stretching global well posedness for a family of navier stokes alpha like models abstract we study global well posedness and regularity of solutions for a family of incompressible three dimensional navier stokes alpha like models that employ fractional laplacian operators this family of equations depends on two parameters 1 and 2 which affect the strength of non linearity vorticity stretching and the degree of viscous smoothing varying 1 and 2 interpolates between the incompressible navier stokes equations and the incompressible lagrangian averaged navier stokes model our main result which contains previously established results of j l lions and others provides a relationship between 1 and 2 that is sufficient to guarantee global existence uniqueness and regularity of solutions

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

time harmonic bem for 2 d piezoelectricity applied to eigenvalue problems abstract we will derive the fundamental generalized displacement solution using the radon transform and present the direct formulation of the time harmonic boundary element method bem for the two dimensional general piezoelectric solids the fundamental solution consists of the static singular and the dynamics regular parts the former evaluated analytically is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle the static bem is a component of the time harmonic bem which is formulated following the physical interpretation of somigliana s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the stroh lekhnitskii sl formalism the time harmonic bem is obtained by adding the boundary integrals for the dynamic regular part which from the original double integral representation over the boundary element and the unit circle are reduced to simple line integrals along the unit circle the bem will be applied to the determination of the eigen frequencies of piezoelectric resonators the eigenvalue problem deals with full non symmetric complex valued matrices whose components depend non linearly on the frequency a comparative study will be made of non linear eigenvalue solvers qz algorithm and the implicitly restarted arnoldi method iram the fem results whose accuracy is well established serve as the basis of the comparison it is found that the iram is faster and has more control over the solution procedure than the qz algorithm the use of the time harmonic fundamental solution provides a clean boundary only formulation of the bem and when applied to the eigenvalue problems with iram provides eigen frequencies accurate enough to be used for industrial applications it supersedes the dual reciprocity bem and challenges to replace the fem designed for the eigenvalue problems for piezoelectricity

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

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

hypersingular bem for dynamic fracture in 2 d piezoelectric solids fracture behavior of piezoelectric solids under time harmonic loading is numerically analyzed in this paper a 2 d boundary element method bem based on both displacement and traction boundary integral equations is presented the time harmonic green s functions for the infinite plane are split into singular plus regular terms the singular ones coinciding with the static green s functions in this manner the singular and hypersingular integrals arising in the formulation may be treated by the same simple regularization procedure proposed by the authors for static piezoelectricity quarter point elements are used for the direct evaluation of stress and electric displacement intensity factors from nodal values several numerical examples for the scattering of waves by different crack configurations including branched and curved cracks and crack interaction problems are given to demonstrate the performance of the proposed method 2006 elsevier b v all rights reserved

on the quadratic type of some simple self dual modules over fields of characteristic two let g be a finite group and let k be an algebraically closed field of characteristic 2 let v be a non trivial simple self dual kg module we say that v is self dual if it is isomorphic to its dual v it is a theorem of w x fong 4 lemma 1 that in this case there is a non degenerate g invariant alternating bilinear form f say defined on v v we say that v is a kg module of quadratic type if f is the polarization of a non degenerate w x g invariant quadratic form defined on v in a previous paper 6 the present authors described some methods to decide if such a module v is of w x quadratic type one of the main results of 6 is the following suppose that g is a group with a split b n pair of odd characteristic p let v be a simple self dual kg module that is not of quadratic type and let w be the brauer character of v then there exists a complex irreducible character of g that occurs as a constituent of the induced character 1 and contains b w as a modular constituent with non zero multiplicity this result suggests that we should investigate the decomposition modulo 2 of the irreducible characters in 1 when g is a group of lie type of odd characteristic and b see which real valued irreducible brauer characters occur as constituents the decomposition modulo 2 of these characters is not known in complete

dynamic green s functions in anisotropic piezoelectric thermoelastic and poroelastic solids a procedure is described to generate fundamental solutions or green s functions for time harmonic point forces and sources the linearity of the field equations permits the green s function to be represented as an integral over the surface of a unit sphere where the integrand is the solution of a one dimensional impulse response problem the method is demonstrated for the theories of piezoelectricity thermoelasticity and poroelasticity time domain analogues are discussed and compared with known expressions for anisotropic elasticity

on the existence and uniqueness of solutions to stochastic three dimensional lagrangian averaged navier stokes equations we prove the existence and uniqueness of solutions for a stochastic version of the threedimensional lagrangian averaged navier stokes equation in a bounded domain to this end we previously prove some existence and uniqueness results for an abstract stochastic equation and justify that our model falls within this framework

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

3d bem for the general piezoelectric solids in this paper we present a new scheme of three dimensional boundary element method for the general anisotropic piezoelectric solids we use the radon transform representation of the three dimensional fundamental solutions of piezoelectricity and integrate them analytically over the triangular boundary element with the linear interpolation this reduces the computation for the system matrices g and h from the standard singular surface integrations to the simple regular line integrations and enables a drastic reduction of the computation time the integrand of the line integral consists of the product of a function dependent and another function independent on the location vectors representing the source and observation points the latter function depends only on the material and element properties and thus calculated only once for each element and saved for a repeated use in the calculation of g and h matrices and in the post processing exploitation of this favorable structure results in the further reduction of the computation time for very large systems the implementation of the proposed method with numerical examples will be presented

3d bem for general anisotropic elasticity abstract a new method for computing the system matrices g and h of the three dimensional boundary element method is presented for the elastostatic problems of general anisotropy green s functions expressed by the line integrals over a semi circle have a simple structure that allows analytical integration over the boundary elements the computation for the system matrices g and h is reduced to numerical integration of regular line integrals over the unit semi circle the proposed method has been implemented and tested with numerical examples

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

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

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

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

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

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

2 d time harmonic bem for solids of general anisotropy with application to eigenvalue problems abstract we present the direct formulation of the two dimensional boundary element method bem for time harmonic dynamic problems in solids of general anisotropy we split the fundamental solution obtained by radon transform into static singular and dynamics regular parts we evaluate the boundary integrals for the static singular part analytically and those for the dynamic regular part numerically over the unit circle we apply the developed bem to eigenvalue analysis we determine eigenvalues of full non symmetric complex valued matrices depending non linearly on the frequency by first reducing them to the generalized linear eigenvalue problem and then applying the qz algorithm we test the performance of the qz algorithm thoroughly in comparison with the fem solution the proposed bem is not only a strong candidate to replace the fem for industrial eigenvalue problems but it is also applicable to a wider class of two dimensional time harmonic problems

efficient form of the lans α turbulence model in a primitive equation ocean model the lagrangian averaged navier stokes alpha email protected model is a turbulence parameterization that has been shown to capture some of the most important features of high resolution ocean modeling at lower resolution the email protected model improves turbulence statistics with an additional nonlinear term and a smoothed advecting velocity in this work we investigate two smoothing techniques helmholtz inversions and filters and their effect on the email protected model s results and efficiency we show that convolution filters are an effective smoothing method that filters and helmholtz inversions produce similar trends statistics like higher resolution email protected simulations as the smoothing parameter is increased and that the filter is computationally more efficient filters must be constructed such that a pressure velocity numerical instability is not excited we show analytically that certain ranges of filter weights are unstable and confirm this with numerical experiments our stability criterion also guarantees that the kinetic energy is well defined and that the filtered velocity is smoother that the original velocity simulations of email protected using the largest filter width nine in the pop primitive equation ocean model resemble doubled resolution simulations of standard pop in statistics such as kinetic energy eddy kinetic energy and potential temperature fields the computational cost of adding email protected with this filter is only 27 as compared to a factor of 8 10 for a doubling of resolution

time harmonic green s functions for anisotropic magnetoelectroelasticity abstract two dimensional 2 d and three dimensional 3 d time harmonic green s functions for linear magnetoelectroelastic solids are derived in this paper by means of radon transform displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time harmonic point force point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2 d case and surface integrals over a unit sphere in 3 d case this dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms the singular terms coincide with the green s functions for the static problem and may be further reduced to closed form expressions the proposed green s functions can be used in the corresponding boundary element method bem formulation

an efficient meshless technique for the solution of transversely isotropic two dimensional piezoelectricity in this paper a new meshless technique is presented for the solution of two dimensional piezoelectricity problem the technique derivation is based on the solution of the corresponding analog equation by transforming the original set of differential equations into three poisson equations with unknown right hand side terms boundary discretisation of the resulting integral equations is eliminated by the use of the method of fundamental solutions the right hand side terms are represented in the solution as particular solutions expressed in terms of radial basis functions the problem solution is then rewritten in its new form which involves complementary solution and particular solution the governing partial differential operator for piezoelectricity is applied on the obtained solution form and forced to be satisfied at a set of domain points whereas the prescribed boundary conditions are satisfied at another set of boundary points the proposed technique is implemented into computer code where several numerical examples with different boundary conditions are tested the results demonstrated excellent agreement with those obtained from analytical and fem solutions

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

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 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

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

2 d transient dynamic analysis of cracked piezoelectric solids by a time domain bem transient dynamic crack analysis of two dimensional 2 d homogeneous and linear piezoelectric solids is presented in this paper for this purpose a time domain boundary element method bem is developed the method uses a combination of the strongly singular displacement boundary integral equations bies and the hypersingular traction bies on the external boundary of the cracked solid strongly singular displacement bies are used while on the crack faces hypersingular traction bies are applied a convolution quadrature formula is applied for the temporal discretization while a collocation method is adopted for the spatial discretization the accuracy of the present time domain bem is verified by using numerical results obtained by the finite element method fem numerical examples are presented and discussed to show the effects of the mechanical and the electrical impact loading on the dynamic intensity factors

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

discriminants of polynomials and of quadratic forms abstract in this paper we describe the quadratic forms over any field k which admit a similarity with a given separable characteristic polynomial f x as the transfer of some binary quadratic form associated to the polynomial f x using this description we show that the discriminant of the quadratic form is the discriminant of this polynomial

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

dynamic crack analysis in piezoelectric solids with non linear electrical and mechanical boundary conditions by a time domain bem abstract this paper presents advanced transient dynamic crack analysis in two dimensional 2d homogeneous and linear piezoelectric solids using non linear mechanical and electrical crack face boundary conditions stationary cracks in infinite and finite piezoelectric solids subjected to impact loadings are considered for this purpose a time domain boundary element method tdbem is developed a galerkin method is implemented for the spatial discretization while a collocation method is applied for the temporal discretization an explicit time stepping scheme is obtained to compute the unknown boundary data including the generalized crack opening displacements cods numerically an iterative solution algorithm is developed to consider the non linear semi permeable electrical crack face boundary conditions furthermore an additional iteration scheme for crack face contact analysis is implemented at time steps when a physically meaningless crack face intersection occurs several numerical examples are presented and discussed to show the effects of the electrical crack face boundary conditions on the dynamic intensity factors