a numerical continuation method based on pade approximants a continuation algorithm is presented with a new predictor which is based on a rational representation of the solution path this algorithm belongs to the class of asymptotic numerical methods that connect perturbation techniques with a discretization principle without the use of a correction process several examples from shell buckling and from contact mechanics are analyzed to assess the efficiency and the reliability of the method

analyticity and decay estimates of the navier stokes equations in critical besov spaces in this paper we establish analyticity of the navier stokes equations with small data in critical besov spaces dot b frac 3 p 1 p q the main method is gevrey estimates the choice of which is motivated by the work of foias and temam contemp math 208 151 180 1997 we show that mild solutions are gevrey regular that is the energy bound e sqrt t lambda v t e p infty holds in e p tilde l infty 0 t dot b frac 3 p 1 p q cap tilde l 1 0 t dot b frac 3 p 1 p q globally in time for p we extend these results for the intricate limiting case p in a suitably designed e space as a consequence of analyticity we obtain decay estimates of weak solutions in besov spaces finally we provide a regularity criterion in besov spaces

characterization of initial data in the homogeneous besov space for solutions in the serrin class of the navier stokes equations abstract consider the cauchy problem of the navier stokes equations in r n with initial data a in the homogeneous besov space b p q 1 n p r n for n p and 1 q we show that the stokes flow e t a can be controlled in l q 0 b r 1 0 r n for 2 n r 1 with p r where l q denotes the lorentz space as an application the global existence theorem of mild solutions for the small initial data is established in the above class which is slightly stronger than serrin s conversely if the global solution belongs to the usual serrin class l q 0 l r r n for r and as above with 1 q then the initial data a necessarily belongs to b r q 1 n r r n moreover we prove that such solutions are analytic in the space variables our method for the proof of analyticity is based on a priori estimates of higher derivatives of solutions in l p r n with holder continuity in time

stationary navier stokes equations with critically singular external forces existence and stability results abstract we show the unique existence of solutions to stationary navier stokes equations with small singular external forces belonging to a critical space to the best of our knowledge this is the largest critical space that is available up until now for this kind of existence this result can be viewed as the stationary counterpart of the existence result obtained by h koch and d tataru for the free non stationary navier stokes equations with initial data in bmo 1 the stability of the stationary solutions in such spaces is also obtained by a series of sharp estimates for resolvents of a singularly perturbed operator and the corresponding semigroup

ill posedness of the stationary navier stokes equations in besov spaces abstract the solutions of the stationary navier stokes equations in r n for n 3 in the scaling invariant besov spaces are investigated it is proved that a sequence of bounded smooth external forces whose b 1 3 norms converges to zero can produce a sequence of bounded smooth solutions whose b 1 norms never converges to zero such norm inflation phenomena are shown by constructing the sequence of external forces as similar to those of initial data proposed by bourgain pavlovic in the non stationary problem

global well posedness for navier stokes equations in critical fourier herz spaces abstract we prove the global well posedness for the 3d navier stokes equations in critical fourier herz spaces by making use of the fourier localization method and the littlewood paley theory the advantage of working in fourier herz spaces lies in that they are more adapted than classical besov spaces for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero our result is an improvement of a recent theorem by lei and lin 2011 10

on time analyticity of the navier stokes equations in a rotating frame with spatially almost periodic data abstract we consider the navier stokes equations with the coriolis force when initial data may not decrease at spatial infinity so that almost periodic data is allowed we prove that the local in time solution is analytic in time when initial data are in f m 0 the fourier preimage of the space of all finite radon measures with no point mass at the origin when the initial data are almost periodic this implies that the complex amplitude is analytic in time in particular a new mode cannot be created at any positive time

an efficient mixed variational reduced order model formulation for nonlinear analyses of elastic shells summary the koiter newton method had recently demonstrated a superior performance for non linear analyses of structures compared to traditional path following strategies the method follows a predictor corrector scheme to trace the entire equilibrium path during a predictor step a reduced order model is constructed based on koiter s asymptotic post buckling theory which is followed by a newton iteration in the corrector phase to regain the equilibrium of forces in this manuscript we introduce a robust mixed solid shell formulation to further enhance the efficiency of stability analyses in various aspects we show that a hellinger reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy we demonstrate that extremely large step sizes with a reasonable out of balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path more importantly the numerical effort of the corrector phase involving a newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation we study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems this article is protected by copyright all rights reserved

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

global well posedness for keller segel system in besov type spaces abstract the cauchy problems for keller segel system are studied using homogeneous besov spaces with the homogeneous besov spaces b p 2 n p r n which is the scaling critical case for keller segel system global solutions for small initial data are obtained in the space in addition ill posedness for keller segel system is also studied

well posedness of the viscous boussinesq system in besov spaces of negative regular index s 1 this paper is concerned with well posedness of the natural convection in a viscous incompressible fluid we first prove that the n dimensional boussinesq system is well posed for initial data u0 0 either belonging to b 1 1 b 1 1 bp r 1 or to b 1 1 bp 1 with 1 r n2 0 and then we prove that this system is well posed for initial data belonging to b 1 1 b 1 1 bn2 1 1 bn2 1 1 well posedness bp rs lt bp rs

uniform global well posedness of the navier stokes coriolis system in a new critical space we prove global well posedness for the navier stokes coriolis system nsc in a critical space whose definition is based on fourier transform namely the fourier besov morrey space mathcal fn 1 mu infty mu 1 with 0 mu 3 the smallness condition on the initial data is uniform with respect to the angular velocity omega our result provides a new class for the uniform global solvability of nsc and covers some previous ones for mu 0 nsc is ill posedness in mathcal fn 1 mu infty mu 1 which shows the optimality of the results with respect to the space parameter mu 0 the lack of hausdorff young inequality in morrey spaces suggests that there are no inclusions between mathcal fn 1 mu infty mu 1 and the largest previously known existence classes of kozono yamazaki besov morrey space and koch tataru textit bmo 1 for navier stokes equations 3dns so taking in particular omega 0 we obtain a critical initial data class that seems to be new for global existence of solutions of 3dns

regularity criteria for 3d navier stokes equations in terms of finite frequency parts of velocity in b 1 abstract in this paper we prove that a leray hopf weak solution u to 3d navier stokes equations is regular if l 0 t b 1 norm of a suitable low frequency part of u is bounded by a scaling invariant constant depending on the kinematic viscosity and initial value u 0 moreover we prove that a leray hopf weak solution is regular if its medium frequency part with fourier modes between k 2 and k for a sufficiently high wavenumber k has small b 1 norm our results imply that energy concentration at sufficiently high wavenumber bands bringing about singularity of the incompressible navier stokes flow can be prevented by an energy threshold at a lower wavenumber band

ill posedness of the 3d navier stokes equations in a generalized besov space near bmo 1 abstract the ill posedness of the 3d navier stokes equations in a generalized besov space which is smaller than b q 1 q 2 is considered in 2008 bourgain pavlovic proved that the 3d navier stokes equation is ill posed in b 1 by showing norm inflation phenomena of the solution for some initial data on the other hand in 2008 germain proved that the flow map is not c 2 in the space b q 1 for q 2 however he did not treat ill posed problem in such spaces thus our result is an extension of these previous results

global well posedness and ill posedness for the navier stokes equations with the coriolis force in function spaces of besov type abstract we consider the initial value problems for the navier stokes equations in the rotational framework we introduce function spaces b p q s r 3 of besov type and prove the global in time existence and the uniqueness of the mild solution for small initial data in our space b 1 2 1 r 3 near bmo 1 r 3 furthermore we also discuss the ill posedness for the navier stokes equations with the coriolis force which implies the optimality of our function space b 1 2 1 r 3 for the global well posedness

dynamic buckling of cylindrical shells subject to an axial impact in a symplectic system this paper discusses the dynamic pre buckling of finite cylindrical shells in the propagation and reflection of axial stress waves by introducing the hamiltonian system into dynamic buckling of structures the problem can be described mathematically in a symplectic space the solutions of hamiltonian dual equations shown in canonical variables are obtained the problem is reduced to the determination of eigenvalues and eigensolutions with the former indicating critical buckling loads and the latter buckling modes numerical example presented shows phenomena of axisymmetric and non axisymmetric dynamic buckling subject to impacts of axial load

boundedness almost periodicity and stability of certain navier stokes flows in unbounded domains abstract we investigate the existence uniqueness and stability of bounded and almost periodic mild solutions to several navier stokes flow problems in our strategy we propose a general framework for studying the semi linear evolution equations with certain smoothing properties of the linear part and with the local lipschitz continuity of the nonlinear operator our method is based on interpolation functors combined with differential inequalities our abstract results are applied to navier stokes oseen equations describing flows of incompressible viscous fluid passing a translating and rotating obstacle and to navier stokes equations on aperture domains and or in besov spaces

the second iterate for the navier stokes equation we consider the iterative resolution scheme for the navier stokes equation and focus on the second iterate more precisely on the map from the initial data to the second iterate at a given time t we investigate boundedness properties of this bilinear operator this new approach yields very interesting results a new perspective on koch tataru solutions a first step towards weak strong uniqueness for koch tataru solutions and finally an instability result in b q 1 for q 2

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

separating invariants for 2 2 matrices abstract a minimal separating set is found for the algebra of matrix invariants of several 2 2 matrices over an infinite field of arbitrary characteristic

computing finite rotations of shells by an asymptotic numerical method abstract we present an asymptotic numerical algorithm for the computation of elastic shells with large rotations the theoretical formulation involves a three field hu washizu functional which allows us to put the problem into a quadratic framework the spatial discretization is based on geometrically exact element recently presented by buchter et al buchter et al three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept int j numer methods engrg 37 1994 2551 1568 several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method as compared to more classical iterative algorithms

strong solutions of the navier stokes equations based on the maximal lorentz regularity theorem in besov spaces abstract we show existence and uniqueness theorem of local strong solutions to the navier stokes equations with arbitrary initial data and external forces in the homogeneous besov space with both negative and positive differential orders which is an invariant space under the change of scaling if the initial data and external forces are small then the local solutions can be extended globally in time our solutions also belong to the serrin class in the usual lebesgue space the method is based on the maximal lorentz regularity theorem of the stokes equations in the homogeneous besov spaces as an application we may handle such singular data as the dirac measure and the single layer potential supported on the sphere

ill posedness of the navier stokes equations in a critical space in 3d abstract we prove that the cauchy problem for the three dimensional navier stokes equations is ill posed in b 1 in the sense that a norm inflation happens in finite time more precisely we show that initial data in the schwartz class s that are arbitrarily small in b 1 can produce solutions arbitrarily large in b 1 after an arbitrarily short time such a result implies that the solution map itself is discontinuous in b 1 at the origin

blowup for the nonlinear heat equation with small initial data in scale invariant besov norms abstract we consider the cauchy problem of the nonlinear heat equation u t u u b u 0 x u 0 with b 2 and b n we prove that initial data u 0 s r n the schwartz class arbitrarily small in the scale invariant besov norm b n b 1 b 2 q 2 b r n can produce solutions that blow up in finite time the case b 3 answers a question raised by yves meyer our result also proves that the smallness assumption put in an earlier work by c miao b yuan and b zhang for the global in time solvability is essentially optimal

high order predictor corrector algorithms new predictor corrector algorithms are presented for the computation of solution paths of non linear partial differential equations the predictors and the correctors are based on perturbation techniques and pade approximants this extends the asymptotic numerical method anm which is an efficient high order continuation technique without corrector the efficiency and the reliability of the new technique are assessed by several examples within thin shell theory and navier stokes equations many variants have been tested to establish an optimal algorithm

existence and uniqueness of solutions for navier stokes equations with hyper dissipation in a large space abstract we study cauchy problem of the 3d navier stokes equations with hyper dissipation by using the fourier localization technique we prove that the system has a unique global solution for large initial data in a critical fourier herz space more importantly the energy of this solution is infinite

koiter analysis of folded structures using a corotational approach the paper deals with geometrically nonlinear finite element analysis of folded plate and shell structures a koiter asymptotic approach is proposed based on the reuse of a linear element in the nonlinear context through a corotational formulation the corotational approach represents a simple and effective way to satisfy the basic requirement of koiter analysis i e full objectivity in the finite element modeling in fact starting simply from a suitable linear finite element and implementing the corotational algebra proposed in garcea et al 2009 zagari 2009 lead to objective explicit expressions for the first four variations of the strain energy which are needed by asymptotic analysis the shell element used here is the flat shell quadrangular element with 4 nodes and 6 dofs per node proposed in madeo et al 2012 and called miss 4 a mixed element based on the reissner mindlin plate theory with an allman like quadratic interpolation for displacements and an equilibrated isostatic interpolation for the stress resultants the element is free from locking and spurious zero energy modes so it appears a suitable candidate for nonlinear corotational analysis the results of the numerical validation show the effectiveness and accuracy of the proposed approach and its excellent overall robustness for both mono and multi modal buckling problems also in the presence of strong nonlinear pre critical behavior

the uniqueness of nondecaying solutions for the navier stokes equations the uniqueness of solutions of the navier stokes equations in the whole space is established when the velocity field is bounded and the pressure field is a bmo valued locally integrable in time function for bounded initial data here the velocity field may not decay at space infinity although there are a few results concerning uniqueness without the decay assumption our result is new and applicable for solutions constructed by solving the integral equations

global well posedness for the generalized magneto hydrodynamic equations in the critical fourier herz spaces abstract this paper concerns the cauchy problem of the n dimensional generalized incompressible magneto hydrodynamic gmhd equations with 1 2 1 by using the fourier localization argument and the littlewood paley theory we get the global well posedness of the gmhd equations with small initial data u 0 b 0 belongs to the critical fourier herz spaces b q 2 1 with q 1 2 in addition for 2 q ill posedness for the case 1 in b q 1 is also established

computing invariants of algebraic groups in arbitrary characteristic let g be an affine algebraic group acting on an affine variety x we present an algorithm for computing generators of the invariant ring k x g in the case where g is reductive furthermore we address the case where g is connected and unipotent so the invariant ring need not be finitely generated for this case we develop an algorithm which computes k x g in terms of a so called colon operation from this generators of k x g can be obtained in finite time if it is finitely generated under the additional hypothesis that k x is factorial we present an algorithm that finds a quasi affine variety whose coordinate ring is k x g along the way we develop some techniques for dealing with nonfinitely generated algebras in particular we introduce the finite generation ideal

navier stokes equations with external forces in lorentz spaces and its application to the self similar solutions abstract we show existence theorem of global mild solutions with small initial data and external forces in lorentz spaces with scaling invariant norms if the initial data have more regularity in another scaling invariant class then our mild solution is actually the strong solution the result on local existence of solutions for large data is also discussed our method is based on the maximal regularity theorem on the stokes equations in lorentz spaces then we apply our theorem to prove existence of self similar solutions provided both initial data and external forces are homogeneous functions since we construct the global solution by means of the implicit function theorem as a byproduct its stability with respect to the given data is necessarily obtained

weak solutions for the navier stokes equations with b 1 ln bxr 1 r 21 r l2 initial data one of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained in the book recent developments in the navier stokes problems lemarie rieusset proved that the navier stokes equations have global weak solutions for initial data in the space bxr 1 r 21 r rn l2 rn 0 r 1 where xr is the space of functions whose pointwise products with hr functions belong to l2 xr denotes the closure of c0 rn in xr and bxr 1 r 21 r rn is the besov space over xr in this paper we partially extend this result of lemarie rieusset to the larger initial value space b 1 ln rn bxr 1 r 21 r rn l2 rn 0 r 1 where b 1 ln rn is a logarithmically modified version of the usual besov space b 1 rn

separating invariants and finite reflection groups abstract a separating algebra is roughly speaking a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants in this paper we introduce a geometric notion of separating algebra this allows us to prove that only groups generated by reflections may have polynomial separating algebras and only groups generated by bireflections may have complete intersection separating algebras

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

asymptotic numerical method for buckling analysis of shell structures with large rotations in this paper the buckling and post buckling of elastic structures taking into account large rotations are investigated using an asymptotic numerical method the critical points are detected by two different ways first by a bifurcation indicator second by analysing the poles of a pade approximant the first step of the post buckling branch is computed starting from the bifurcation point and using an extended system the remaining bifurcating branch is followed by the same algorithm as for the fundamental path several examples are tested to show the effectiveness of the proposed method

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

constructing modular separating invariants abstract we consider a finite dimensional modular representation v of a cyclic group of prime order p we show that two points in v that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation simultaneously we describe an explicit construction for a separating set consisting of polynomials with these properties

temporal decay for the generalized navier stokes equations abstract in this paper we study the asymptotic behavior of solutions to the generalized incompressible navier stokes equations t u u u u 0 div u 0 we show that some weighted negative besov norms of solutions are preserved along time evolution and we obtain the optimal time decay rates of the higher order spatial derivatives of solutions both in the subcritical case 1 2 1 and the critical case 1 2 by using the fourier splitting approach and the interpolation techniques

discontinuity of weak solutions to the 3d nse and mhd equations in critical and supercritical spaces abstract we demonstrate that the three dimensional incompressible magneto hydrodynamics mhd system is ill posed due to the discontinuity of weak solutions in a wide range of spaces specifically we construct initial data which has finite energy and is small in certain spaces such that any leray hopf type of weak solution to the mhd system starting from this initial data is discontinuous at time t 0 in such spaces an analogous result is also obtained for the navier stokes equation which extends the previous result of ill posedness in b 1 by cheskidov and shvydkoy to spaces that are not necessarily critical the region of the spaces where the norm inflation occurs almost touches l 2

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects