on resolving the multiplicity of arbitrary tensor products of the u n groups representations of u n are realised as right translations on holomorphic hilbert bargmann spaces of n n complex variables r fold tensor product spaces of irreducible representations of u n are shown to be isomorphic to subspaces of the holomorphic hilbert spaces maps are exhibited which carry an irreducible representation of u n into these subspaces the algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity several examples from u 3 are explicitly worked out

local existence of the strong solutions for the full navier stokes poisson equations abstract we study the full navier stokes poisson equations for viscous polytropic fluids we prove the existence and uniqueness of local strong solutions for all initial values satisfying some compatibility conditions it is important that we allow the initial density to be vacuum

very weak generalized and strong solutions to the stokes system in the half space abstract in this paper we study the stokes system in the half space r n with n 2 we give existence and uniqueness results in weighted sobolev spaces after the central case of the generalized solutions we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument

a spectral method for the stokes problem in three dimensional unbounded domain we present a method for solving the stokes problem in unbounded domains it relies on the coupling of the transparent boundary operator and a spectral method in spherical coordinates it is done explicitly by the use of vector valued spherical harmonics a uniform inf sup condition is proved which provides an optimal error estimate

very weak solutions of the stationary navier stokes equations for an incompressible fluid past obstacles abstract we consider the stationary motion of an incompressible navier stokes fluid past obstacles in r 3 subject to the given boundary velocity v b external force f div f and nonzero constant vector k e 1 at infinity our main result is the existence of at least one very weak solution v in k e 1 l 3 for arbitrary large f l 3 2 l 12 7 provided that the flux of v b k e 1 on the boundary of each body is sufficiently small with respect to the viscosity the uniqueness of very weak solutions is proved by assuming that f and v b k e 1 are suitably small moreover we establish weak and strong regularity results for very weak solutions in particular our existence and regularity results enable us to prove the existence of a weak solution v satisfying v l 3 2

on leray s problem of steady navier stokes flow past a body in the plane 2 flow i n t h e u n b o u n d e d d o m a i n 8 0 2 1 properties of the vorticity and its level sets 81 2 2 boundedness of the velocity convergence of 7 and additional properties of the vorticity 89 2 3 convergence and decay rates in sectors 98 2 4 convergence of the speed 103 2 5 some monotonicity results and decay rates for the vorticity 105

unsteady interaction of a viscous fluid with an elastic shell modeled by full von karman equations abstract we study well posedness and asymptotic dynamics of a coupled system consisting of linearized 3d navier stokes equations in a bounded domain and a classical nonlinear full von karman shallow shell equation that accounts for both transversal and lateral displacements on a flexible part of the boundary we also take into account rotational inertia of filaments of the shell our main result shows that the problem generates a semiflow in an appropriate phase space the regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in the lateral displacements and its absence our second result states the existence of a compact global attractor for this semiflow in the case of presence of rotational damping in the transversal component and a particular structure of external forces

poincare birkhoff witt theorems and generalized casimir invariants for some infinite dimensional lie groups ii analogues of the poincare birkhoff witt theorem for several pairs of lie groups acting on complex manifolds are studied these results lead to algebraic analogues of a theorem of i segal on the two sided regular representation of a unimodular locally compact group for dual representations of some infinite dimensional lie groups on generalized bargmann segal fock spaces generalized casimir invariants for these dual representations are also developed

semi group theory for the stokes operator with navier type boundary conditions on l p spaces in this article we consider the stokes problem with navier type boundary conditions on a domain omega not necessarily simply connected since under these conditions the stokes problem has a non trivial kernel we also study the solutions lying in the orthogonal of that kernel we prove the analyticity of several semigroups generated by the stokes operator considered in different functional spaces we obtain strong weak and very weak solutions for the time dependent stokes problem with the navier type boundary condition under different hypotheses on the initial data u0 and external force f then we study the fractional and pure imaginary powers of several operators related with our stokes operators using the fractional powers we prove maximal regularity results for the homogeneous stokes problem on the other hand using the boundedness of the pure imaginary powers we deduce maximal l p l q regularity for the inhomogeneous stokes problem

a variational approach in weighted sobolev spaces to the operator x1 in exterior domains of ir3 consider the dirichlet problem v u k 1 u f withv k 0 in 3 or in an exterior domain of 3 where the skew symmetric differential operator 1 x1 is a singular perturbation of the laplacian because of the inhomogeneity of the fundamental solution we study existence uniqueness and regularity in sobolev spaces with anisotropic weights in these spaces the operator 1 yields an additional positive definite term giving better results than in sobolev spaces with radial weights the elliptic equation v u k 1 u f can be taken as a model problem for the oseen equations a linearized form of the navier stokes equations

vibrations of rectangular plates reinforced by any number of beams of arbitrary lengths and placement angles this paper presents an analytical method for the vibration analysis of plates reinforced by any number of beams of arbitrary lengths and placement angles both the plate and stiffening beams are generally modeled as three dimensional 3 d structures having six displacement components at a point and the coupling at an interface is generically described by a set of distributed elastic springs each of the displacement functions is here invariably expressed as a modified fourier series which consists of a standard fourier cosine series plus several supplementary series functions used to ensure and improve uniform convergence of the series representation unlike most existing techniques the current method offers a unified solution to the vibration problems for a wide spectrum of stiffened plates regardless of their boundary conditions coupling conditions and reinforcement configurations several numerical examples are presented to validate the methodology and demonstrate the effect on modal parameters for a stiffened plate with various boundary conditions coupling conditions and reinforcement configurations

regularity results for elliptic equations in lipschitz domains abstract we develop a simple variational argument based on the usual nirenberg difference quotient technique to deal with the regularity of the solutions of dirichlet and neumann problems for some linear and quasilinear elliptic equation in lipschitz domains we obtain optimal regularity results in the natural family of sobolev spaces associated with the variational structure of the equations in the linear case we obtain in a completely different way some of the results of d jerison and c e kenig about the laplace equation

very weak solutions and large uniqueness classes of stationary navier stokes equations in bounded domains of r2 extending the notion of very weak solutions developed recently in the three dimensional case to bounded domains r2 we obtain a new class of unique solutions u in lq q 2 to the stationary navier stokes system u u u p f divu k u g with data f k g of low regularity as a main consequence we obtain a new uniqueness class also for classical weak or strong solutions indeed such a solution is unique if its lq norm is sufficiently small or the data satisfy the uniqueness condition of a very weak solution

lp theory for stokes and navier stokes equations with navier boundary condition abstract this work was intended as an attempt at studying stationary stokes and navier stokes problem with navier boundary conditions 1 3 we wish to investigate some results of existence uniqueness and regularity of solutions in hilbert case and in l p theory

a class of solutions to stationary stokes and navier stokes equations with boundary data in w 1 q q we develop a theory for a general class of very weak solutions to stationary stokes and navier stokes equations in a bounded domain with boundary of class c2 1 corresponding to boundary data in the distribution space w 1 q q 1 q these solutions exist and are unique for small data in the nonlinear case in their class of existence and satisfy a correponding estimate in terms of the data moreover they become regular if the data are regular to our knowledge the only existence result for solutions attaining such boundary data is due to giga 16 proposition 2 2 for the stokes case however the methods and the approach used in the present paper are different than giga s and cover more general issues including the nonlinear navier stokes equations and the precise way in which the boundary data are attained by the solutions we also introduce in the last section a further generalization of the solution class

l q estimates for the stationary oseen equations on exterior domains we study the dirichlet problem for the stationary oseen equations on exterior smooth domains in rn n 2 our main results are the existence and uniqueness of weak and very weak solutions of the oseen equations satisfying appropriate lq estimates the uniqueness of very weak solutions is shown by utilizing recent results for very weak solutions of the stokes equations on bounded domains then for n 1 n 1 q the existence and lq estimates of very weak solutions in lq n is deduced by a duality argument from our existence result and d2 r estimates for strong solutions these results combined with some regularity results for very weak solutions enable us to prove the existence uniqueness and d1 r estimates of weak solutions in d1 r n l n 1 r n 1 r n where n 1 n r n 1

invariant theory of the block diagonal subgroups of gl n c and generalized casimir operators abstract let c be the field of complex numbers and let gl n c denote the general linear group of order n over c let n be the sum of m positive integers p 1 p m and consider the block diagonal subgroup gl p 1 c x x gl p m c the adjoint representation of the lie group gl n c on its lie algebra gives rise to the coadjoint representation of gl n c on the symmetric algebra of all polynomial functions on gl n c the polynomials that are fixed by the restriction of the coadjoint representation to the block diagonal subgroup form a subalgebra called the algebra of invariants a finite set of generators of this algebra is explicitly determined and the connection with the generalized casimir invariant differential operators is established

solving two dimensional second order elliptic equations in exterior domains using the inverted finite elements method in this paper inverted finite element method is used for solving two dimensional second order elliptic equations with a dirichlet boundary condition in an exterior domain after laying down the method and after giving an estimate of the error we detail how its implementation can be accomplished numerical results show the high efficiency and the accuracy of the method especially for equations with infinitely varying coefficients

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

pointwise decay of stationary rotational viscous incompressible flows with nonzero velocity at infinity we consider a stationary viscous incompressible flow around a translating and rotating body optimal rates of decay are derived for the velocity and its gradient on the basis of a representation formula involving a fundamental solution constructed by r b guenther and e a thomann the fundamental solution of the linearized navier stokes equations for spinning bodies in three spatial dimensions time dependent case j math fluid mech 8 2006 77 98 for a linearized system

initial and boundary value problem of the unsteady navier stokes system in the half space with holder continuous boundary data abstract we study the initial boundary value problem of the navier stokes system in the half space we prove the unique solvability of the weak solution for some short time interval 0 t with the velocity in c 2 r n 0 t 0 1 when the given initial data in c r n and the given boundary data in c 2 r n 1 0 t satisfy the compatibility conditions our result generalizes the result in 29 considering nonhomogeneous dirichlet boundary data

a well posed problem for the exterior stokes equations in two and three dimensions this paper treats the stokes problem in exterior lipschitz continuous domains of 2 and 3 using the weighted sobolev spaces of hanouzet in 3 and giroire in 2 we establish the inf sup condition between the velocity and pressure spaces this fundamental result shows that the variational stokes problem is well posed in those spaces in the last paragraph we obtain additional regularity of the solution when the data are smoother

the existence and uniqueness of very weak solutions of the stationary boussinesq system abstract we consider the dirichlet problem for the stationary boussinesq system in a bounded smooth domain in r n where n 2 or 3 we first prove the existence of very weak solutions for arbitrarily large boundary data in w 1 q q w 1 r r provided that 3 q r if n 3 and 2 q r if n 2 the uniqueness of very weak solutions is also shown for sufficiently small data

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

instability of an inverse problem for the stationary navier stokes equations this paper provides a theoretical study of the detection of an object immersed in a fluid when the fluid motion is governed by the stationary navier stokes equations with nonhomogeneous dirichlet boundary conditions to solve this inverse problem we make a boundary measurement on a part of the exterior boundary first we present an identifiability result we then use a shape optimization method in order to identify the obstacle we minimize a nonlinear least squares thus we prove the existence of the first order shape derivative of the state characterize it and deduce the gradient of the least squares functional finally we study the stability of this setting doing a shape sensitivity analysis of order two hence we prove the existence of the second order shape derivatives and we give the expression of the shape hessian at possible solutions of the original inverse problem then the compactness of the riesz operator corresponding to this shape hessian is shown and the ill posedness of the ident

the navier stokes equations in the weak l n space with time dependent external force abstract we consider the navier stokes equations with time dependent external force either in the whole time or in positive time with initial data with domain either the whole space the half space or an exterior domain of dimension n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n space bounded for all time in particular this result gives sufficient conditions for the unique existence and the stability of small time periodic solutions or almost periodic solutions this result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak l n space with time independent external force

an energy residual method for detection of the causes of vibration hypersensitivity this work deals with hypersensitive vibration behavior of plates the aim of the proposed method is to detect structural zones inducing such behavior it is based on a residual calculation which takes into account structural uncertainties and has a low numerical cost since it requires only the resolution of the problem for the nominal structure then this solution is used to calculate energy residuals on different parts of perturbed structures in order to detect which zones will produce an hypersensitive behavior the basis of the method is first developed on a simple problem a rod and then applied to a typical hypersensitive structure a plates network finally one can show that the proposed tool is able to detect which zones of the plates network are responsible for hypersensitive behavior

weighted hilbert spaces for the stationary exterior stokes problem with navier slip boundary conditions abstract in this paper we analyze the three dimensional exterior stokes problem with the navier slip boundary conditions describing the flow of a viscous and incompressible fluid past an obstacle where it is assumed that the fluid may slip at the boundary because the flow domain is unbounded we set the problem in weighted spaces in order to control the behavior at infinity of the solutions this functional framework also allows to prescribe various behaviors at infinity of the solutions growth or decay existence and uniqueness of solutions are shown in a hilbert setting which gives the tools for a possible numerical analysis of the problem weighted korn s inequalities are the key point in order to study the variational problem

the stationary oseen equations in an exterior domain an approach in weighted sobolev spaces abstract in this work we study the linearized navier stokes equations in an exterior domain of r 3 at the steady state that is the oseen equations we are interested in the existence and the uniqueness of weak strong and very weak solutions in l p theory which makes our work more difficult our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space r 3 and the properties in bounded domains our approach rests on the use of weighted sobolev spaces

casimir operators of semidirect sum lie algebras a new method for constructing the casimir operators of a class of non semisimple lie algebras is presented this class includes the semidirect sums of the unitary orthogonal or symplectic algebras with a heisenberg weyl algebra w n the construction procedure is illustrated with the simple example of the wu n n u n algebras its power is then demonstrated on the more difficult case of the wsp 2n r w n sp 2n r algebras detailed results are given for the eigenvalues of the wsp 6 r casimir operators relevant to a microscopic theory of nuclear collective motions

existence and regularity of very weak solutions of the stationary navier stokes equations we consider the stationary navier stokes equations in a bounded domain in rn with smooth connected boundary where n 2 3 or 4 in case that n 3 or 4 existence of very weak solutions in ln is proved for the data belonging to some sobolev spaces of negative order moreover we obtain complete lq regularity results on very weak solutions in ln if n 2 then similar results are also proved for very weak solutions in l q 0 omega with any q0 2 we impose neither smallness conditions on the external force nor boundary data for our existence and regularity results

stationary stokes oseen and navier stokes equations with singular data the concept of very weak solution introduced by giga math z 178 287 329 1981 for the stokes equations has hardly been studied in recent years for either the navier stokes equations or the navier stokes type equations we treat the stationary stokes oseen and navier stokes systems in the case of a bounded open set connected of class mathcal c 1 1 of mathbb r 3 taking up once again the duality method introduced by lions and magenes problemes aus limites non homogenes et applications vols 1 2 dunod paris 1968 and giga math z 178 287 329 1981 for open sets of class mathcal c infty see also chapter 4 of necas les methodes directes en theorie des equations elliptiques french masson et cie ed paris academia editeurs prague 1967 which considers the hilbertian case p 2 for general elliptic operators we give a simpler proof of the existence of a very weak solution for stationary oseen and navier stokes equations when data are not regular enough based on density arguments and a functional framework adequate for defining more rigourously the traces of non regular vector fields in the stationary navier stokes case the results will be valid for external forces not necessarily small which lets us extend the uniqueness class of solutions for these equations considering more regular data regularity results in fractional sobolev spaces will also be discussed for the three systems all these results can be extended to other dimensions

vibration sensitive behaviour of a connecting angle case of coupled beams and plates in this paper semi infinite one and two dimensional 1 d and 2 d coupled structures are considered in order to find conditions under which coupling effects can induce hypersensitive vibrating behaviour the approach is based on classical wave decomposition and the first point is to show that high sensitivity can exist in such simple systems then for the case of beams it is shown that two critical coupling angles can be defined and that their values depend only on wave number ratio a similar study is then performed on semi infinite coupled plates and the existence of a critical coupling angle is shown its value can be determined using three structural parameters the results are finally compared to finite coupled structures

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

casimir operators of semidirect products of semisimple with heisenberg groups casimir operators for semidirect products of some semisimple groups with heisenberg groups are computed the analysis is carried out using dual representations on fock space wherein the action of the semidirect products are related to their dual groups namely certain unitary orthogonal and symplectic groups the compact symplectic group chain is also investigated by passing to the complexification groups between the symplectic groups are constructed which are of the form of semidirect products of symplectic groups with heisenberg groups