information entropy of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy e n of the classical orthogonal polynomials e email protected p 2 n x logp 2 n x w x dx here an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of jacobi matrix associated with the orthogonal polynomials p n x and the weighted moments of p 2 n x m n email protected x kp 2 n x w x dx k 1 2

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

standard and entropic uncertainty relations of the finite well this paper deals with the standard and entropic uncertainty relations for the finite potential well with different length and potential depth we calculate the position and momentum dispersions and entropies that are dependent on the quantum states of the finite well and determine the corresponding standard and entropic uncertainty relations then we calculate the entropy of occurrence for the mass point inside and outside the well we discuss the results and determine a class of probability distributions with a parameter for which the standard deviation becomes infinite independent of the value of this parameter

spectra of weighted lb algebras of entire functions on banach spaces abstract given a decreasing family of continuous weights on a banach space x we consider the weighted inductive limits of spaces of entire functions v h x and v h 0 x motivated by recent research by d carando and p sevilla peris on weighted frechet algebras of entire functions on banach spaces we determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial schauder decompositions we study hormander algebras of entire functions defined on a banach space and we give a description of them in terms of sequence spaces we also focus on algebra homomorphisms between these spaces and obtain a banach stone type theorem for a particular decreasing family of weights finally we study the spectra of these weighted algebras endowing them with an analytic structure

power series for solutions of the 3 mathcal d navier stokes system on r 3 in this paper we study the fourier transform of the 3 mathcal d navier stokes system without external forcing on the whole space r 3 the properties of solutions depend very much on the space in which the system is considered in this paper we deal with the space phi alpha alpha of functions v k frac c k k alpha where alpha 2 epsilon epsilon 0 and c k is bounded sup k in r 3 smallsetminus 0 c k infty we construct the power series which converges for small t and gives solutions of the system for bounded intervals of time these solutions can be estimated at infinity in k space by exp rm const sqrt t k

the method of space time conservation element and solution element a new approach for solving the navier stokes and euler equations a new numerical framework for solving conservation laws is being developed this new framework differs substantially in both concept and methodology from the well established methods i e finite difference finite volume finite element and spectral methods it is conceptually simple and designed to overcome several key limitations of the above traditional methods a two level scheme for solving the convection diffusion equation u t a u x 2u x2 0 0 is constructed and used to illuminate major differences between the present method and those mentioned above this explicit scheme referred to as the a scheme has two independent marching variables unj and ux nj which are the numerical analogues of u and u x at j n respectively the a scheme has the unusual property that its stability is limited only by the cfl condition i e it is independent of also it can be shown that the amplification factors of the a scheme are identical to those of the leapfrog scheme if o and to those of the dufort frankel scheme if a o these coincidences are unexpected because the a scheme and the above classical schemes are derived from completely different perspectives and the a scheme does not reduce to the above classical schemes in the limiting cases the a scheme is extended to solve the 1d time dependent navier stokes equations of a perfect gas stability of this explicit solver also is limited only by the cfl condition in spite of the fact that it does not use i any techniques related to the high resolution upwind methods and ii any ad hoc parameter the current navier stokes solver is capable of generating highly accurate shock tube solutions particularly for high reynolds number flows shock discontinuities can be resolved within one mesh interval the inviscid 0 a scheme is reversible in time it also is neutrally stable i e free from numerical dissipation such a scheme generally cannot be extended to solve the euler equations thus the inviscid version is modified stability of this modified scheme referred to as the a scheme is limited by the cfl condition and 0 1 where is a special parameter that controls numerical dissipation moreover if 0 the amplification factors of the a scheme are identical to those of the leapfrog scheme which has no numerical dissipation on the other hand if 1 the two amplification factors of the a scheme become the same function of the courant number and the phase angle unexpectedly this function also is the amplification factor of the highly diffusive lax scheme note that because the lax scheme is very diffusive and it uses a mesh that is staggered in time a two level scheme using such a mesh is often associated with a highly diffusive scheme the a scheme which also uses a mesh staggering in time demonstrates that it can also be a scheme with no numerical dissipation the euler extension of the a scheme has stability conditions similar to those of the a epsiv scheme itself it has the unusual property that numerical dissipation at all mesh points can be controlled by a set of local parameters moreover it is capable of generating accurate shock tube solutions with the cfl number ranging from close to 1 to 0 022

on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

on a particular analytical solution to the 3d navier stokes equations and its peculiarity for high reynolds numbers a special class of axially symmetric nonstationary flows of incompressible viscous fluids is examined for it the 3d navier stokes equations are reduced to a nonlinear partial differential equation of the third order and a linear partial differential equation of the second order these equations are studied and their particular analytical solutions are found the obtained particular solution to the navier stokes equations could be used to describe some types of turbulent flows of viscous fluids in the case of high reynolds numbers

ultradistributions of roumieu type and projective descriptions we show that the topology of the weighted inductive limit of frechet spaces of entire functions which is obtained as the fourier laplace transform of the space of ultradistributions with compact support of roumieu type cannot be described by means of canonical weighted sup seminorms this is a natural weighted lf space of entire functions for which the problem of projective description of bierstedt meise and summers has a negative answer however the projective hull of the lf space coincides algebraically with the inductive limit the behaviour of the corresponding inductive limits of spaces of continuous functions is completely opposite the topology can be described by weighted sup seminorms but the lf space is a proper subspace of its projective hull

local regularity criteria of the 3d navier stokes and related equations abstract in this paper we are concerned with the local regularity of suitable weak solutions to the 3d navier stokes equations and its generalized case with fractional power of the laplacian 3 4 1 in the first part we derive some regularity criteria in terms of the deformation tensor d u for the classical navier stokes equations in the second part for the fractional case we obtain some regularity conditions for suitable weak solutions including the velocity u the gradient of the velocity u the rotation tensor u and the deformation tensor d u this generalizes the results obtained by gustafson et al 2007

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

wavelets and orthogonal polynomials based on harmonic oscillator eigenstates we present a new method for the construction of wavelets the main tool is the harmonic oscillator eigenstates with the help of their properties we are able to construct wavelets which have explicit expressions and have exponential decay in both time domain and frequency domain moreover we get a new family of polynomials which are orthogonal in a weighted squarely integrable function space

interpolation of entire functions and projective descriptions abstract special entire functions of completely regular growth with additional properties are utilized to interpolate entire functions with certain bounds and to give an example of a weighted inductive limit of banach spaces of entire functions such that its topology cannot be described by the canonical weighted sup seminorms associated with the weights of the steps this provides a new more natural counterexample to a problem of bierstedt meise and summers

on weighted inductive limits of spaces of continuous functions the general problem of a projective description of weighted inductive limits of spaces of continuous functions was raised by bierstedt eise and summers in 3 similar questions were studied in 4 in the setting of sequence spaces on a general index set i the study was recently continued by bierstedt and meise in 5 the aim of the present article is to treat some questions proposed to the author by k d bierstedt on this subject our notation for locally convex spaces 1 c s is standard and we refer the reader to 13 and 14 if e is a 1 c s cs e denotes the family of all the continuous seminorms on e following floret 10 a 1 c s is said to satisfy the countable neighbourhood property c n p if given p ces e there are a 0 and p cs e such that p x 0 such that u 0 there is v v such that avi 0 i e if given any x x there is v v with v x 0 then v will be refered to as a nachbin family on x in what follows if e is a 1 c s c x e indicates the collection of all continuous e valued functions on x and cc x e the subspace of c x e of all the functions with compact support corresponding to a nachbin family v and to a 1 c s e we associate the following nachbin spaces

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

existence and regularity of a class of weak solutions to the navier stokes equations abstract we construct a class of weak solutions to the navier stokes equations which have second order spatial derivatives and one order time derivatives of p power summability for 1 p 5 4 meanwhile we show that u l s 0 t w 2 r with 1 s 3 2 r 2 for 1 r 5 4 r can be relaxed not to exceed 3 2 if we consider only in the interior of in the end we extend the classical regularity theorem our results show that u is a regular solution if u l s 0 t l r with 1 s 3 2 r 1 for satisfying 1 3 with 1 s 1 r 5 6 for arbitrary domain in r 3 and 1 s 2 for r n with n 3 this result was previously obtained byh beirao da veiga chinese ann math ser b 16 1995 407 412

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

nonsymmetric bifurcations of solutions of the 2d navier stokes system abstract we consider the 2d navier stokes system written for the stream function with periodic boundary conditions and construct a set of initial data such that initial critical points bifurcate from 1 to 2 and then to 3 critical points in finite time the bifurcation takes place in a small neighborhood of the origin our construction does not require any symmetry assumptions or the existence of special fixed points for another set of initial data we show that 3 critical points merge into 1 critical point in finite time we also construct a set of initial data so that bifurcation can be generated by the navier stokes flow and do not require the existence of an initial critical point

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

projective limits of weighted lb spaces of holomorphic functions los limites proyectivos de limites inductivos de espacios de banach tambien llamados espacios plb surgen de forma natural en el analisis matematico en esta tesis estudiamos espacios plb cuyos bloques de construccion son espacios de banach de funciones holomorfas definidas por normas supremo ponderadas el estudio de estos espacios extiende la investigacion de agethen bierstedt bonet quienes han considerado recientemente espacios plb ponderados de funciones continuas desde otra perspectiva extiende la investigacion de limites inductivos ponderados de espacios de banach de funciones holomorfas los cuales han sido analizados intensamente por varios autores los ultimos anos nuestro proposito es estudiar las propiedades localmente convexas de los espacios descritos arriba en particular investigamos cuando son ultrabornologicos o tonelados ademas investigamos bajo que circunstancias se pueden intercambiar el limite proyectivo y el inductivo y por lo tanto el espacio plb coincide con el limite inductivo de espacios de frechet definidos por la misma sucesion espacios de este ultimo tipo has sido investigados por bierstedt bonet probamos condiciones necesarias para las propiedades de los espacios antes mencionadas bajo hipotesis muy poco restrictivas en cuanto a condiciones suficientes usamos metodos homologicos cuya exploracion fue iniciada por palamodov al final de los sesenta y continuada por vogt wengenroth y otros a lo largo de los ultimos 40 anos presentamos tambien un criterio para decidir si los espacios son tonelados adaptado a estas situaciones no obstante parece ser inevitable descomponer funciones holomorfas para probar cualquier resultado relativo a a las condiciones suficientes por lo tanto introducimos varios contextos en los cuales lo ultimo es posible dentro de estos contextos conseguimos la descomposicion de diferentes formas es decir por descomposicion de polinomios en el disco y en el espacio un metodo conectado con la teoria de proyecciones de bergman dos tipos de representaciones del espacio de sucesiones y el metodo de hormander bajo algunas hipotesis adicionales satisfechas como mostramos por muchos ejemplos damos en casi todos los contextos mencionados anteriormente unas caracterizaciones completas de cuando el espacio es ultrabornologico cuando es tonelado y cuando los limites inductivo y proyectivo son intercambiables para finalizar nuestra investigacion de espacios plb ponderados presentamos dos resultados que muestran que espacios de este tipo se pueden escribir en algunos casos como el producto tensorial de un espacio de frechet y un espacio df el segundo resultado acerca de representaciones de productos tensoriales muestra que algunos espacios de ultradistribuciones introducidos recientemente por schmets y valdivia resultan ser espacios plb ponderados de funciones holomorfas

remarks on the regularity criterion to the navier stokes equations via the gradient of one velocity component abstract in this paper we consider the classical three dimensional 3d incompressible navier stokes equations by making use of a new anisotropic sobolev inequality we improve a regularity criterion for the weak solution to the 3d navier stokes equations involving the gradient of one velocity component this result improves the previous work of zhou and pokorn 2010

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

logarithmic potential of hermite polynomials and information entropies of the harmonic oscillator eigenstates the problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential vn t hn x 2 ln x t e x2 dx at the zeros of the hermite polynomial hn x here a closed analytical expression for vn t is obtained which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of hn x is known an inequality for the values of vn t at the zeros of hn x is conjectured which leads to a new nonvariational upper bound for the entropies finally the exact formula for vn t is written in an alternative way which allows the entropies to be expressed in terms of the even order spectral moments of the hermite polynomials the asymptotic n 1 limit of this alternative expression for the entropies is discussed and the conjectured upper bound for the entropies is proved to be asymptotically valid

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

topological properties of convolutor spaces via the short time fourier transform we discuss the structural and topological properties of a general class of weighted l 1 convolutor spaces our theory simultaneously applies to weighted d l1 spaces as well as to convolutor spaces of the gelfand shilov spaces k m p in particular we characterize the sequences of weight functions m p p is an element of n for which the space of convolutors of k m p is ultrabornological thereby generalizing grothendieck s classical result for the space o c of rapidly decreasing distributions our methods lead to the first direct proof of the completeness of the space o c of very slowly increasing smooth functions

note on loss of regularity for solutions of the 3 d incompressible euler and related equations one of the central problems in the mathematical theory of turbulence is that of breakdown of smooth indefinitely differentiable solutions to the equations of motion in 1934 j leray advanced the idea that turbulence may be related to the spontaneous appearance of singularities in solutions of the 3 d incompressible navier stokes equations the problem is still open we show in this report that breakdown of smooth solutions to the 3 d incompressible slightly viscous i e corresponding to high reynolds numbers or highly turbulent navier stokes equations cannot occur without breakdown in the corresponding solution of the incompressible euler ideal fluid equation we prove then that solutions of distorted euler equations which are equations closely related to the euler equations for short term intervals do breakdown

hypercyclic operators on non normable frechet spaces abstract every infinite dimensional separable non normable frechet space admits a continuous hypercyclic operator a large class of separable countable inductive limits of banach spaces with the same property is given but an example of a separable complete inductive limit of banach spaces which admits no hypercyclic operator is provided it is also proved that no compact operator on a locally convex space is hypercyclic

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

bifurcation of critical points for solutions of the 2d euler and 2d quasi geostrophic equations we consider the 2d euler and 2d quasi geostrophic equations with periodic boundary conditions for both systems we will use the stream function formulation and study the bifurcation problem for the critical points of the stream function in a small neighborhood of the origin we construct a set of initial data such that initial critical points of the stream function bifurcate from 1 to 2 and then to 3 critical points in finite time for the quasi geostrophic equation the whole bifurcation process takes place strictly within the lifespan of the constructed local solution

regularity and singularity in solutions of the three dimensional navier stokes equations higher moments of the vorticity field m t in the form of l 2 m norms graphic 1 are used to explore the regularity problem for solutions of the three dimensional incompressible navier stokes equations on the domain graphic 2 it is found that the set of quantities formula 3 provide a natural scaling in the problem resulting in a bounded set of time averages d m t on a finite interval of time 0 t the behaviour of d m 1 d m is studied on what are called good and bad intervals of 0 t which are interspersed with junction points neutral i for large but finite values of m with large initial data m 0 o gr 4 it is found that there is an upper bound formula 4 which is punctured by infinitesimal gaps or windows in the vertical walls between the good bad intervals through which solutions may escape while this result is consistent with that of leray leray 1934 acta math 63 193 248 doi 10 1007 bf02547354 5 and scheffer scheffer 1976 pacific j math 66 535 552 this estimate for m corresponds to a length scale well below the validity of the navier stokes equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed graphic 1 gif 4 embed graphic 2 gif 5 http dx doi org doi 10 1007 bf02547354

nachbin spaces as cq algebras abstract we introduce a new class of commutative c q algebras whose topologies generalize the uniform and the strict topology we consider multipliers of these modules and describe their submodule structure a key role is played by nachbin spaces

regularity criteria for suitable weak solutions of the navier stokes equations near the boundary abstract we present some new regularity criteria for suitable weak solutions of the navier stokes equations near the boundary in dimension three we prove that suitable weak solutions are holder continuous up to the boundary provided that the scaled mixed norm l x t p q with 3 p 2 q 2 2 q p q 3 2 is small near the boundary our methods yield new results in the interior case as well partial regularity of weak solutions is also analyzed under some additional integral conditions

the algebraic equalities and their topological consequences in weighted spaces abstract we study algebraic equalities and their topological consequences in weighted banach frechet or lb spaces of holomorphic like functions on a locally compact and compact hausdorff space x our main results are the following 1 the algebraic equality v a x v 0 a x for lb spaces with o and o growth conditions given by a weight sequence v v n n always implies that these spaces are dfs the converse statement is valid under the additional condition cd which is a weakened version of the typical biduality condition for the steps a v n x and a v n 0 x generating v a x and v 0 a x respectively 2 under the same condition cd the algebraic equality a v x a v 0 x between the projective hulls of v a x and v 0 a x is equivalent to a v x semi montel thus we completely remove or significantly weaken some stringent conditions used before in many papers studying the similar problems see e g bierstedt and bonet 2006 5 and references therein

the convenient setting for non quasianalytic denjoy carleman differentiable mappings abstract for quasianalytic denjoy carleman differentiable function classes c q where the weight sequence q q k is log convex stable under derivations of moderate growth and also an l intersection see 1 6 we prove the following the category of c q mappings is cartesian closed in the sense that c q e c q f g c q e f g for convenient vector spaces applications to manifolds of mappings are given the group of c q diffeomorphisms is a regular c q lie group but not better

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

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

on the weak strong uniqueness of koch tataruʼs solution for the navier stokes equations abstract we prove the weak strong uniqueness between koch tataru s solution and leray s weak solution for three dimensional incompressible navier stokes equations

on the l2 decay of weak solutions for the 3d asymmetric fluids equations abstract we study the long time behavior of weak solutions for the asymmetric fluids equations in the whole space r 3 we prove that u w t l 2 r 3 2 c t 1 3 2 for all t 0 through fourier splitting moreover we use a direct method to show how to take advantage of the particular form of the equation for the angular velocity and obtain the so far unknown faster decay rate w t l 2 r 3 2 c t 1 5 2 for all t 0

regularity criterion of weak solutions to the 3d navier stokes equations via two velocity components abstract this paper is concerned with the regularity criterion of leray hopf weak solutions to the 3d navier stokes equations with respect to serrin type condition on two velocity filed components it is shown that the weak solution u u 1 u 2 u 3 is regular on 0 t if there exist two solution components for example u 2 and u 3 satisfying the condition u 2 u 3 l p 1 0 t l q 1 r 3 for 2 p 1 3 q 1 2 3 2 q 1

weighted lb spaces of holomorphic functions vh g v0h g and completeness of v0h g abstract in this article we show that algebraic equalities between weighted inductive limits of spaces of holomorphic functions and or between their projective hulls sometimes have strong consequences for the locally convex properties of the spaces involved in these equalities for these results we impose the typical conditions which imply biduality between the spaces with the o and o growth conditions and use interpolating sequences for the step spaces h v n g n n in section 1 we show that v h g v 0 h g holds algebraically if and only if this space is dfs and that h v g h v 0 g is true if and only if the space is semi montel in section 2 we provide a new characterization of the semi montel property of h v g which is much simpler than the one given before in k d bierstedt j bonet j taskinen associated weights and spaces of holomorphic functions studia math 127 1998 137 168 section 3 proves that the completeness of v 0 h g sometimes implies that the inductive limit ind n h v n 0 g must be boundedly retractive

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

long time behavior for the nonstationary navier stokes flows in l1 r n the large time decay for the navier stokes flows in l1 r n is a long standing unsolved question the main difficulties are that usual lq lr estimates for the stokes flow fail in this case and the projection operator p l1 r n l 1 r n becomes unbounded using the stokes solution formula we find a crucial and new estimate for the stokes flow in l1 r n which plays a fundamental role in studying the time l1 behavior for the navier stokes equations in addition we decompose the operator p into two parts and reduce its unboundedness to establish an l1 estimate for an elliptic problem with neumann boundary condition which is overcome by using the weighted estimates of the gaussian kernel s convolution the main results in this article are motivated by bae s works 2006 2008 1 2 and brandolese s work 2001 11

uncertainty relationships in many body systems rigorous relationships among radial and logarithmic expectation values of an n body system are obtained by means of information theoretic methods especially interesting are the uncertainty expressions in terms of the product r n pb p which generalize the well known relation r2 p2 2 9n2 4 additionally an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown for illustration the aocmcy of the inequalities found are numerically analysed in a hameefock framework for atomic systems finally it is pointed out how these inequalities are improved in atoms by raking into account other physical quantities or by considering some semiempirical results on atomic information entropies the one particle density p r is a fundamental quantity in the study of the physical and chemical properties of many fermion systems parr and yang 1989 since hohenberg and kohn 1964 formulated their theorem about the existence of a universal functional e pl for the energy of the many fermion system much effort has been paid in the study of the function p

existence and decay of solutions to the dissipative quasi geostrophic equation with delays abstract we consider the dissipative quasi geostrophic equation with forcing and advective terms containing delays we prove the existence of weak solutions and algebraic decay of the energy provided the initial data 0 is in l p l 2 1 p 2 and the forcing term satisfies some mild conditions on its long time behavior

toeplitz operators on the space of analytic functions with logarithmic growth abstract continuous and compact toeplitz operators for positive symbols are characterized on the space h v of analytic functions with logarithmic growth on the open unit disc of the complex plane the characterizations are in terms of the behaviour of the berezin transform of the symbol the space h v was introduced and studied by taskinen the bergman projection is continuous on this space in a natural way which permits to define toeplitz operators sufficient conditions for general symbols are also presented

on the spaces ocm of john horvath abstract in the table on p 442 of john horvath s textbook topological vector spaces and distributions the questions for an explicit description of the topology of o c m by seminorms for the completeness and for the metrizability of the space o c m remained open whereas for m the completeness was shown by a grothendieck in his thesis we resolve these questions in proposition 1 proposition 2 proposition 3 in proposition 4 proposition 5 proposition 6 proposition 7 we investigate multiplication and convolution in the spaces s m o c m s m o c m analogously to the treatment in the spaces e m d m d m e m in j horvath s book ch 4 6 multiplication 9 convolution 10 regularization in section 6 the multiplication and convolution of holomorphic functions with values in s m o c m s m o c m are investigated analogously to j horvath s treatment of holomorphic functions with values in the spaces e m d m d m e m

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

unbounded normal derivative for the stokes system near boundary we study local boundary regularity for the stokes system we show that unlike in the interior case non local effects can lead to a violation of local regularity in the spatial variables near the boundary