global strong solutions of two dimensional navier stokes equations with nonlinear slip boundary conditions abstract the two dimensional time dependent navier stokes equations with nonlinear slip boundary conditions are investigated in this paper since the nonlinear slip boundary conditions of this type include the subdifferential property the weak variational formulation is the variational inequality the existence uniqueness and regularity of global weak solutions are shown using the regularized method moreover the continuous dependence property of the weak solution for given initial data and the behavior of the global weak solution as t are established

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

study of coating flow by the finite element method abstract silliman s analysis of slot coating is extended to accommodate film flows with highly bent menisci as in slide and curtain coating by combining polar and cartesian coordinate parametrizations of meniscus shape the nonlinear algebraic equations from the subdomain galerkin weighted residual method and finite element basis functions are solved by newton s method which is perfected for the free boundary problem convergence behavior is examined slot coating results reveal slow flow zones and show that at high capillary number and low metering rate the downstream meniscus can no longer attach to the slot comparison of coyne and elrod s approximate solutions with the finite element solutions shows that their assumption of a parabolic velocity profile leads to reasonable approximations of meniscus shapes

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

generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities the infinite dimensional version of sard s theorem due to smale 9 is very useful in the study of boundary value problems of the form here a is a cl fredholm map from a banach space e to a banach space f in some applications the existence of a solution is proved only when

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

optimal control of evolutionary quasi newtonian fluids we are interested in the necessary optimality conditions for optimal control problems of systems describing the unsteady flow of incompressible shear thickening fluids to overcome the difficulties related to the differentiability of the control to state mapping consequence of the lack of regularity of the state variable we first establish some intermediate optimality conditions and then pass to the limit

on the global regularity of shear thinning flows in smooth domains abstract in some recent papers we have been pursuing regularity results up to the boundary in w 2 l spaces for the velocity and in w 1 l spaces for the pressure for fluid flows with shear dependent viscosity to fix ideas we assume the classical non slip boundary condition from the mathematical point of view it is appropriate to distinguish between the shear thickening case p 2 and the shear thinning case p 2 and between flat boundaries and smooth arbitrary boundaries the p 2 non flat boundary case is still open the aim of this work is to extend to smooth boundaries the results proved in reference h beirao da veiga on non newtonian p fluids the pseudo plastic case j math anal appl 344 1 2008 175 185 this is done here by appealing to a quite general method introduced in reference h beirao da veiga on the ladyzhenskaya smagorinsky turbulence model of the navier stokes equations in smooth domains the regularity problem j eur math soc in press suitable for considering non flat boundaries

reducing slip boundary value problems from the half to the whole space applications to inviscid limits and to non newtonian fluids abstract the study of a very large class of linear and non linear stationary and evolutive partial differential problems in the half space or similar under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space the approach is particularly suitable for proving new results in strong norms to determine whether this extension is available turns out to be a simple exercise the verification depends on a few general features of the functional space x related to the space variables hence we present an approach as much as possible independent of the particular space x we appeal to a reflection technique hence a crucial assumption is to be in the presence of flat boundaries see below instead of stating general theorems we rather prefer to illustrate how to apply our results by considering a couple of interesting problems as a main example we show that sharp vanishing viscosity limit results that hold for the evolution navier stokes equations in the whole space can be extended to the slip boundary value problem in the half space we also show some applications to non newtonian fluid problems

on the c1 γ ω w2 2 ω regularity for a class of electro rheological fluids abstract we are concerned with a system of nonlinear partial differential equations with p x structure 1 p p x p 0 and no slip boundary conditions we prove the existence and uniqueness of a c 1 w 2 2 solution corresponding to small data without further restrictions on the bounds p p 0 in particular this result is applicable to the steady motion of shear dependent electro rheological fluids

on the regularity of flows with ladyzhenskaya shear dependent viscosity and slip or non slip boundary conditions navier stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by o a ladyzhenskaya and in the case of gradient dependent viscosity by j l lions a particular case is the well known smagorinsky turbulence model this is nowadays a central subject of investigation on the other hand boundary conditions of slip type seems to be more realistic in some situations in particular in numerical applications they are a main research subject the existence of weak solutions u to the above problems with slip or nonslip type boundary conditions is well known in many cases however regularity up to the boundary still presents many open questions in what follows we present some regularity results in the stationary case for weak solutions to this kind of problem see theorem 3 1 and theorem 3 2 the evolution problem is studied in a forthcoming paper 6 a cornerstone in our proof is the classical nirenberg translation method see 38

global regularity properties of steady shear thinning flows abstract in this paper we study the regularity of weak solutions to systems of p stokes type describing the motion of shear thinning fluids in certain steady regimes in particular we address the problem of regularity up to the boundary improving previous results especially in terms of the allowed range for the parameter p

vanishing viscosity limit for the 3d magnetohydrodynamic system with a slip boundary condition abstract this work investigates the solvability regularity and vanishing viscosity limit of the 3d viscous magnetohydrodynamic system in a class of bounded domains with a slip boundary condition

on the number of solutions for the forced pendulum equation soit e e t c 0 t une fonction t periodique avec 0 t e s ds 0 on considere l equation x t x t ksin x t e t c x est t periodique ou k c r et k 0

on non newtonian p fluids the pseudo plastic case abstract in the following we study a class of stationary navier stokes equations with shear dependent viscosity under the non slip dirichlet boundary condition we consider pseudo plastic fluids a fluid is said pseudo plastic or shear thinning if in eq 1 1 below one has p 2 we are interested in global i e up to the boundary regularity results in dimension n 3 for the second order derivatives of the velocity and the first order derivatives of the pressure we consider a cubic domain and impose the non slip boundary condition only on two opposite faces on the other faces we assume periodicity as a device to avoid effective boundary conditions this choice is made so that we work in a bounded domain and simultaneously with a flat boundary

time decay rates of non newtonian flows in rn abstract this paper is concerned with time decay rates of the weak solutions of an incompressible non newtonian fluid motion model in half spaces r n for n 3 with the use of the spectral decomposition of the stokes operator and l p l q estimates it is shown that the weak solutions decay in l 2 norm like t n 2 1 r 1 2 when the initial velocity u 0 l 2 l r for 1 r 2 the higher decay rates t n 2 1 r 1 2 1 2 are obtained if u 0 satisfies the additional moment condition r n x n u 0 x r d x 1 r 2 moreover the error estimates between the non newtonian flow and the navier stokes flow are discussed

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

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

global regularity of a class of p fluid flows in cylinders we consider the motion of an incompressible non newtonian fluid with shear dependent viscosity we extend and improve the results obtained in the recent paper by crispo f crispo shear thinning viscous fluids in cylindrical domains regularity up to the boundary j math fluid mech in press concerning the case of the motion between two coaxial cylinders to the case of a full cylinder actually we prove boundary regularity for solutions to the stationary dirichlet problem with zero boundary data

periodic solutions of the forced pendulum exchange of stability and bifurcations abstract we study the t periodic solutions of the forced pendulum equation u cu a sin u f t where f satisfies f t t 2 f t we prove that this equation always has at least two geometrically distinct t periodic solutions u0 and u1 we then investigate the dependence of the set of t periodic solutions on the forcing strength we prove that under some restriction on the parameters a c the periodic solutions found before can be smoothly parameterized by and that there are some intervals for which u0 u1 are the only t periodic solutions up to geometrical equivalence but there are other intervals in which additional t periodic solutions bifurcate off the branches u0 u1 we characterize the global structure of the bifurcating branches related to this bifurcation phenomenon is the phenomenon of exchange of stability in some intervals u0 is dynamically stable and u1 is unstable while in other intervals the reverse is true a phenomenon which has important consequences for the dynamics of the forced pendulum as we show by both theoretical analysis and numerical simulation

existence of the solution to stationary navier stokes equations with nonlinear slip boundary conditions abstract the stationary navier stokes equations with nonlinear slip boundary conditions are investigated in this paper because the boundary conditions include the subdifferential property on the part boundary the variational formulation of this problem is the variational inequality problem of the second kind with navier stokes operator the main purpose of the paper is to study the existence of the weak solution and the strong solution to this variational inequality problem in terms of the yosida s regularity method

a note on the regularity of flows with shear dependent viscosity abstract we consider a non newtonian fluid governed by stationary incompressible navier stokes equations with shear dependent viscosity using a fixed point argument in an appropriate functional setting we establish the existence of a strong solution for small and suitably regular data uniqueness results are obtained under similar conditions

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

multiple solutions of the periodic boundary value problem for some forced pendulum type equations if f x c 0 and e is a constant t then integrating 0 1 over 0 272 after multiplication of both members of the equation by x shows that the only possible solutions are constant and hence satisfy a sin x c consequently if we exclude the trivial translation by a multiple of 275 0 1 will have no solution if dj a one solution if lp a and two solutions if lc l a in 0th er t erms the intersection of the range of the operator d2 dt2 c d dt f u sin acting on zn periodic functions of class g with the subspace of constant functions in the space c o 2711 of real continuous functions on 0 27r is the closed interval a a whose interior points are images of two distinct solutions and boundary points of one we try to obtain similar results for the case where e is no more constant and

continued fractions and brjuno functions abstract for 0 1 given we consider the modified continued fraction expansion of the real number x defined by x a 0 e 0 x 0 a 0 z and x n 1 1 a n e n x n a n n for n 0 where 1 e n x n e n 1 for n 0 with xn 0 the usual gaussian case is 1 whereas 1 2 is the continued fraction to the nearest integer the brjuno function b x is then defined by b x ln x0 n 1 x0x1 xn 1ln xn these functions were introduced by yoccoz in the 1 and 1 2 cases in his work on the holomorphic conjugacy to a rotation of an analytic map with an indifferent fixed point we will review some properties of these functions namely all these functions are 1 periodic and belong to l ioc p r for 1 p t in this communication we will mainly report on some of the technical tools related to the continued fraction expansions required by the above mentioned results these results deal with the growth of the denominator of the reduced fractions pn qn of the above continued fraction expansion which gives the maximal error rate of approximation the relation between these continued fraction and the usual gaussian case and finally the invariant density generalising the classical result of gauss for the usual case

multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force abstract we prove a multiplicity result for forced oscillations of a spherical pendulum that is a massive point moving on a sphere subject to a periodic action with or without friction allowed to depend on the whole past of the motion the approach is based on topological methods in particular when the unperturbed forcing term is the gravity we obtain two harmonic forced oscillations regardless of the presence of friction and of the form of the perturbing force field

on periodic solutions of forced pendulum like equations the existence and multiplicity of t periodic solutions of the forced pen dulum equation y t q t a sin j t e t 1 have been considered recently by a number of authors and we refer to 2 3 for the history and a survey of this problem the main techniques used up to now have been upper and lower solutions topological degree and variational methods a precise and explicit characterizarion of the set of e e

regularity criterion for generalized newtonian fluids in bounded domains abstract the regularity of a non newtonian flow in a domain with a boundary cannot be treated easily because of the nonlinear viscosity term and the global property of the pressure a distance function was used in a previous study to obtain a strong solution in the present study using the distance function idea we obtain serrin type regularity criteria for the vorticity and the velocity with regard to non newtonian equations with shear dependent viscosity in a bounded domain

a p q coupled system in elliptic nonlinear problems with nonstandard boundary conditions we state an abstract variational formulation to a coupled system constituted by an inequality and an equality motivated by the motion and energy equations and the constitutive laws for the stress tensor and the heat flux respectively when non newtonian fluids are taken care of here the existence of a weak solution is proven via a fixed point argument to multivalued mappings the nonstandard boundary conditions correspond to friction wall laws and energy transfer condition considered on a part of the boundary whereas there exists the presence of the frictional work due to the friction of the fluid motion we conclude by formulating the corresponding stationary heat conducting viscous incompressible flow problem and we establish an existence result

regularity of solutions to a non homogeneous boundary value problem for general stokes systems in r n we give a simple and very complete proof of the existence of a strong h2 solution to the non homogeneous problem 1 1 under the non homogeneous boundary conditions 1 6 here we consider the half space case rn n 3 see theorem 1 2 this regularity result was previously obtained by solonnikov and scadilov in reference 33 for the classical stokes system 0 g x 0 in the 3 d homogenous case a 0 b 0 and a suitable open subset of r3

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions