on the convergence of the two dimensional second grade fluid model to the navier stokes equation we consider the equations governing the motion of incompressible second grade fluids in a bounded two dimensional domain with navier slip boundary conditions we first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus in the l h1 and the l2 h2 time space norms next we study their asymptotic behavior when tends to zero prove that they converge to regular solutions of the navier stokes equations and give the rate of convergence in terms of

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

bounding the equivariant betti numbers of symmetric semi algebraic sets abstract let r be a real closed field the problem of obtaining tight bounds on the betti numbers of semi algebraic subsets of r k in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to ole nik and petrovski thom and milnor these bounds are all exponential in the number of variables k motivated by several applications in real algebraic geometry as well as in theoretical computer science where such bounds have found applications we consider in this paper the problem of bounding the equivariant betti numbers of symmetric algebraic and semi algebraic subsets of r k we obtain several asymptotically tight upper bounds in particular we prove that if s r k is a semi algebraic subset defined by a finite set of s symmetric polynomials of degree at most d then the sum of the s k equivariant betti numbers of s with coefficients in q is bounded by s k d o d unlike the classical bounds on the ordinary betti numbers of real algebraic varieties and semi algebraic sets the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant as an application we improve the best known bound on the ordinary betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to gabrielov vorobjov and zell

from second grade fluids to the navier stokes equations abstract we consider the limit 0 for a second grade fluid on a bounded domain with dirichlet boundary conditions we show convergence towards a solution of the navier stokes equations under two different types of hypothesis on the initial velocity u 0 if the product u 0 l 2 u 0 h 1 is sufficiently small we prove global in time convergence if there is no smallness assumption we obtain local in time convergence up to the time c u 0 h 1 4

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

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

existence of weak solutions for steady viscous fluids with general slip boundary conditions abstract we study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper applying monotone operator theory we first establish the existence result of weak solutions for an approximation problem then using the compactness methods and the point wise convergence property of velocity gradients we get the desired results

boundary layers associated with incompressible navier stokes equations the noncharacteristic boundary case abstract the goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic i e there is injection and or suction everywhere at the boundary following earlier work on the boundary layer for linearized navier stokes equations in the case where the boundaries are characteristic no slip at the boundary and non permeable we consider here the case where the boundary is permeable and thus noncharacteristic the form of the boundary layer and convergence results are derived in two cases linearized equation and full nonlinear equations we prove that there exists a boundary layer at the outlet downwind of the form e uz e where u is the speed of injection suction at the boundary z is the distance to the outlet of the channel and e is the kinematic viscosity we improve an earlier result of s n alekseenko 1994 siberian math j 35 no 2 209 230 where the convergence in l 2 of the solutions of the navier stokes equations to that of the euler equations at vanishing viscosity was established in the two dimensional case we are able to derive the physically relevant uniform in space l norm estimates of the boundary layer the uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic sobolev imbedding to the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full nonlinear navier stokes equations for incompressible fluids

on the roughness induced effective boundary conditions for an incompressible viscous flow abstract we consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities it is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel after studying the corresponding boundary layers we obtain rigorously the navier friction condition it is valid when the size and amplitude of the imperfections tend to zero furthermore the coefficient in the law is determined through an auxiliary boundary layer type problem and the tangential drag force and the effective mass flow are determined up to order o e3 2 the value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of o e

a note on the blowup criterion of the lagrangian averaged euler equations abstract in this note we consider the existence of the 3d lagrangian averaged euler equations we obtain local existence in w s p and a sufficient condition for the global existence

some remarks on a certain class of axisymmetric fluids of differential type we prove global existence and uniqueness for axisymmetric solutions without swirl for the three dimensional second grade fluid and the euler equations the domain considered is either a bounded domain invariant with respect to rotations about some axis or the full space r 3 for a certain class of stationary solutions of the euler equations lyapunov stability is proved

average of the navier s law on the rapidly oscillating boundary we study the flow of newtonian fluid in a domain with periodically wrinkled boundary with slip navier s boundary condition the goal of this paper is to replace a microscopic boundary condition posed on the rough boundary by some macroscopic boundary condition posed on the middle surface of the oscillating boundary depending on the shape of wrinkles and the friction coefficient we get four different effective models

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

on the positivity of symmetric polynomial functions part i general results abstract we prove that a real symmetric polynomial inequality of degree d 2 holds on r n if and only if it holds for elements with at most d 2 distinct non zero components which may have multiplicities we establish this result by solving a cauchy problem for ordinary differential equations involving the symmetric power sums this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function in the final section extensions of our results to the whole space r n are outlined the main results are theorems 5 1 and 5 2 with corollaries 2 1 and 5 2 and the corresponding results for r n from the last subsection part ii will contain a discussion on the ordered vector space h d n in general as well as on the particular cases of degrees d 4 and d 5 finite test sets for positivity in the homogeneous case and other sufficient criteria

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

heat transfer in unsteady axisymmetric second grade fluid this work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet the axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet mathematical analysis has been carried out for two heating processes namely i with prescribed surface temperature pst case and ii prescribed surface heat flux phf case the modelled non linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable the resulting non linear partial differential equations are solved analytically using homotopy analysis method ham the series solutions are developed and the convergence is properly discussed the series solutions and graphs of velocity and temperature are constructed particular attention is given to the variations of emerging parameters such as second grade parameter prandtl and eckert numbers

slip with friction and penetration with resistance boundary conditions for the navier stokes equations numerical tests and aspects of the implementation we consider slip with friction and penetration with resistance boundary conditions in the steady state navier stokes equations this paper describes some aspects of the implementation of these boundary conditions for finite element discretizations numerical tests on two and three dimensional channel flows across a step using the slip with friction boundary condition study the influence of the friction parameter on the position of the reattachment point and the reattachment line of the recirculating vortex respectively

weak convergence for weighted empirical processes of dependent sequences in this paper we establish weak convergence theorems for weighted empirical processes of strong mixing p mixing and associated sequences we apply these results to obtain weak convergence of integral functionals of empirical processes and of mean residual life processes in reliability theory to carry out the proofs we develop two rosenthal type inequalities for strong mixing and associated sequences

on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 abstract we prove that homogeneous symmetric polynomial inequalities of degree p 4 5 in n positive 1 variables can be algorithmically tested on a finite set depending on the given inequality theorem 13 the test set can be obtained by solving a finite number of equations of degree not exceeding p 2 section 3 discusses the structure of the ordered vector spaces h p n and h p n in section 4 positivity criteria for degrees 4 and 5 are stated and proved the main results are theorems 10 14 part iii of this work will be concerned with the construction of extremal homogeneous symmetric polynomials best inequalities of degree 4 in n positive variables

cramer type moderate deviations for self normalized ψ mixing sequences abstract let i i 1 be a sequence of mixing random variables let m n 0 1 k n 2 m and y j i 1 m m j 1 i 1 j k set s k o j 1 k y j and s o k i 1 k y j 2 we prove a cramer type moderate deviation expansion for p s k o s o k x as n our result is similar to the recent work of chen et al 2016 4 where the authors established cramer type moderate deviation expansions for mixing sequences comparing to the result of chen et al our results hold for mixing coefficients with polynomial decaying rate and wider ranges of validity

well posedness of stochastic second grade fluids abstract the theory of turbulent newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer in this study we consider stochastic second grade fluids filling a two dimensional bounded domain with the navier slip boundary condition with friction we prove the well posedness of this problem and establish a stability result our stochastic model involves a multiplicative white noise and a convective term with third order derivatives which significantly complicate the analysis

zero viscosity limit of the linearized navier stokes equations for a compressible viscous fluid in the half plane the zero viscosity limit for an initial boundary value problem of the linearized navier stokes equations of a compressible viscous fluid in the half plane is studied by means of the asymptotic analysis with multiple scales we first construct an approximate solution of the linearized problem of the navier stokes equations as the combination of inner and boundary expansions next by carefully using the technique on energy methods we show the pointwise estimates of the error term of the approximate solution which readily yield the uniform stability result for the linearized navier stokes solution in the zero viscosity limit 1999 john wiley sons inc

effective boundary conditions for laminar flows over periodic rough boundaries effective boundary conditions or wall laws are proposed for a laminar flow over a rough wall with periodic roughness elements these effective conditions are posed on a regularized boundary which allows the details of the wall to be avoided and dramatically reduces the computational cost the effective conditions stem from an asymptotic expansion of the solution which is presented here both first and second order conditions are discussed and tested numerically on bidimensional cases

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

on moderate deviations for martingales let x n x t n f t n 0 t 1 be square integrable martingales with the quadratic characteristics x n n 1 2 we prove that the large deviations relation p x 1 n r 1 r 1 holds true for r growing to infinity with some rate depending on l 2 n z 0 t 1 x t n 2 28 and n 2 n e x n 1 1 1 where 0 and l 2 n 0 n 2 n 0 as n the exact bound for the remainder is also obtained

on the positivity of symmetric polynomial functions part iii extremal polynomials of degree 4 abstract in this paper which is a continuation of v timofte on the positivity of symmetric polynomial functions part i general results j math anal appl 284 2003 174 190 and v timofte on the positivity of symmetric polynomial functions part ii lattice general results and positivity criteria for degrees 4 and 5 j math anal appl in press we study properties of extremal polynomials of degree 4 and we give the construction of some of them the main results are theorems 9 13 15 16 and 18

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

a coherent analysis of stokes flows under boundary conditions of friction type this paper is concerned with the stationary and nonstationary flow of viscous incompressible fluid under boundary conditions of friction type which are certain nonlinear boundary conditions similar to the so called signorini boundary condition in elasticity we assume that the flow is governed by the linear stokes equation while the boundary condition is nonlinear from the methodological viewpoint the analysis is carried out in a coherent way starting from study of the related boundary value problems for the stationary flow by means of the theory of variational inequalities and getting to wellposedness of the initial boundary value problems for the nonstationary flow by means of the nonlinear semigroup theory from the viewpoint of applications we mention original motivations and include some new generalizations like the cases of anisotropic friction and inhomogeneous boundary value

incompressible euler as a limit of complex fluid models with navier boundary conditions in this article we study the limit 0 of solutions of the euler equations and the limit 0 of solutions of the second grade fluid equations in a bounded domain both in two and in three space dimensions we prove that solutions of the complex fluid models converge to solutions of the incompressible euler equations in a bounded domain with navier boundary conditions under the hypothesis that there exists a uniform time of existence for the approximations independent of and this additional hypothesis is not necessary in 2d where global existence is known and for axisymmetric flows without swirl for which we prove global existence our conclusion is strong convergence in l2 to a solution of the incompressible euler equations assuming smooth initial data

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

finite element solution for flow of a third grade fluid past a horizontal porous plate with partial slip the steady flow of a third grade fluid past a horizontal porous plate with partial slip is investigated the arising non linear problem is solved numerically using a finite element method the results of no slip and slip conditions are presented graphically and the effects of fluid parameters suction velocity and slip parameter have been discussed comparison of the present analysis is made with the existing results

on the global existence of solutions to the prandtl s system abstract in this paper we establish a global existence of weak solutions to the two dimensional prandtl s system for unsteady boundary layers in the class considered by oleinik j appl math mech 30 1966 951 provided that the pressure is favourable this generalizes the local well posedness results due to oleinik 1966 mathematical models in boundary layer theory chapman hall london 1999 for the proof we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations

a general class of exponential inequalities for martingales and ratios in this paper we introduce a technique for obtaining exponential inequalities with particular emphasis placed on results involving ratios our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance or its quadratic variation for continuous martingales we provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales the spirit of this application is that when going from results for sums of independent random variables to martingales one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance the decoupling inequalities used to attain our goal are of independent interest they include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets finally we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares

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

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

self normalized cramer type moderate deviations under dependence we establish a cramer type moderate deviation result for self normalized sums of weakly dependent random variables where the moment requirement is much weaker than the non self normalized counterpart the range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes we consider three types of self normalization the equal block scheme the big block small block scheme and the interlacing scheme simulation study shows that the latter can have a better finite sample performance our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra high dimensional time series mean vectors

real even symmetric ternary forms let sen m denote the set of all real symmetric forms of degree m 2d let psen m and sen m denote the cones of positive semidefinite psd and sum of squares sos elements of sen m respectively for m 2 or 4 these cones coincide for m 6 they do not and were analyzed in even symmetric sextics by m d choi t y lam and b reznick 1987 math z 195 pp 559 580 we present an easily checked necessary and sufficient condition for an even symmetric n ary octic to be in psen 8 and for an even symmetric ternary decic to be in pse3 10 we also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10 we proceed to discuss the extremal elements of the cones pse3 8 and pse3 10 this leads to the question how many of these extremal forms have sos representations we prove that pse3 8 se3 8 and demonstrate that pse3 10 se3 10 is nonempty providing many new examples of psd forms which are not sos we give a graphic representation of ternary forms which also indicates whether or not an element of se3 8 or se3 10 is psd we interpret elements of pse3 m as inequalities in particular we give all symmetric polynomial inequalities of degree 5 satisfied by the sides of a triangle

existence and uniqueness of the flow of second grade fluids with slip boundary conditions this paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply connected domain the existence of a unique classical solution global in time is proved under suitable regularity and growth restrictions on the initial data the slip law and the body and surface forces the method is based on a fixed point formulation of the problem

the effect of the slip condition on flows of an oldroyd 6 constant fluid the steady flows of a non newtonian fluid are considered when the slippage between the plate and the fluid is valid the constitutive equations of the fluid are modeled by those for an oldroyd 6 constant fluid they give rise to non linear boundary value problems the analytical solutions are obtained using powerful easy to use analytic technique for non linear problems the homotopy analysis method it is shown that solutions exist for all values of non newtonian parameters the solutions valid for no slip condition for all values of non newtonian parameters can be derived as the special cases of the present analysis finally graphs are plotted and critical assessment is made for the cases of slip and no slip conditions

upwind discretization of a time dependent two dimensional grade two fluid model in this paper we propose a finite element scheme for solving numerically the equations of a transient two dimensional grade two non newtonian rivlin ericksen fluid model this system of equations is considered an appropriate model for the motion of a water solution of polymers by introducing a new variable denoted z we split the problem into a coupled one with a transport equation as one of our aims is to derive unconditional a priori estimates from the discrete analogue of the transport equation we stabilize our scheme by adding a consistent stabilizing term we use the p 2 p 1 taylor hood finite element scheme for the velocity v and the pressure p and the discontinuous p 1 finite element for an auxiliary variable z the error is of the order of h 3 2 k considering that the discretization of the transport equation loses inevitably a factor h 1 2