asymptotic behavior for the navier stokes equations with nonzero external forces abstract we estimate the asymptotic behavior for the stokes solutions with external forces first we found that if there are external forces then the energy decays slowly even if the forces decay quickly then we also obtain the asymptotic behavior in the temporal spatial direction for weak solutions of the navier stokes equations we also provide a simple example of external forces which shows that the stokes solution does not decay quickly

on pole assignability over polynomial rings by an explicit counterexample it is shown that the polynomial ring in more than one variables over an arbitrary algebraically closed field is not pole assignable that is one can find a pair of matrices over the ring which is completely reachable but not pole assignable

simulation of stopped diffusions in this work we study standard euler updates for simulating stopped diffusions as an immediate application we discuss the computation of first exit times of diffusions from a domain we focus on one dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem in particular we give a fully implementable algorithm to compute the first exit time from an interval numerically the brownian motion case is treated in detail and extensions to general diffusions are given special emphasis is given to numerical experiments for every ansatz we include numerical experiments confirming the conjectured accuracy of our methods our algorithm is of order one in a weak sense comparisons with other algorithms are shown results that are superior to those obtained with other methods are presented when approximating a first hitting time distribution the results obtained with our algorithm are much better than those achieved with other methods

decay rate for the incompressible flows in half spaces we show that the time decay rate of l 2 norm of weak solution for the stokes equations and for the navier stokes equations on the half spaces are t frac n 2 frac 1 r frac 1 2 frac 1 2 if the initial data u 0 in l 2 cap l r and int mathbb r n y n u 0 y r dy infty for 1 r 2 we also show that the decay rate is determined by the linear part of the weak solution we use the heat kernel and ukai s solution formula for the stokes equations it has been known up to now that the decay rate on the half space was t frac n 2 frac 1 r frac 1 2 which was obtained by borchers and miyakawa 1 and ukai 9

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

stabilization of polynomially parametrized families of linear systems the single input case abstract given a continuous time family of finite dimensional single input linear systems parametrized polynomially such that each of the systems in the family is controllable there exists a polynomially parametrized control law making each of the systems in the family stable

lower bound of l2 decay of the navier stokes flow in the half space r n and its asymptotic behavior in the frequency space abstract we consider the asymptotic behavior of weak solutions of the navier stokes equations in the half space r n we obtain the lower bound of the energy decay of the navier stokes flow by means of the profile of the initial data indeed we construct a class of the initial data which causes the slow decay of the navier stokes flow with an explicit rate furthermore we investigate the asymptotic behavior of concentration in the frequency space

algebraic l 2 decay for weak solutions of a viscous boussinesq system in exterior domains abstract the algebraic decay rates for the total kinetic energy of weak solutions of the n dimensional viscous boussinesq system in exterior domains are established by means of the spectral decomposition method of fractional powers of the laplacian and stokes operators the results reveal that l 2 decay of the velocity field with zero divergence becomes slower than that of the temperature

notes on the space time decay rate of the stokes flows in the half space in this paper a stokes equations in the half space r n n 2 has been considered we derive a rapid decay rate of the stokes flow in space and time when the initial data decreases fast enough and satisfies some additional condition initial data decreasing too slowly to be x h l1 r n are also considered

decay results of solutions to the incompressible navier stokes flows in a half space abstract in a half space we consider the asymptotic behavior of the strong solution for the non stationary navier stokes equations in particular the decay rates of the second order derivatives of the navier stokes flows in l r r n n 2 with 1 r are derived by using l q l r estimates and a clever analysis on the fractional powers of the stokes operator in addition we prove that the strong solution and its first and second derivatives decay in time more rapidly than observed in general if the initial datum lies in a suitable weighted space

asymptotic behavior for the navier stokes equations in 2d exterior domains we show that the lp spatial temporal decay rates of solutions of incompressible flow in an 2d exterior domain when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in he xin c he z xin weighted estimates for nonstationary navier stokes equations in exterior domain methods appl anal 7 3 2000 443 458 and our previous results h o bae b j jin asymptotic behavior of stokes solutions in 2d exterior domains j math fluid mech in press h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains submitted for publication for the spatial decay rate estimate we first extend temporal decay rate result of the navier stokes solutions for general lp space when the initial velocity is in l 2 lr 1 r q 1 r q

the initial boundary value problem for thin plate flow of incompressible non newtonian viscous fluids abstract in this paper the particular model that describes the thin plate flow of incompressible non newtonian viscous fluids is investigated this model which arises from polymer processing concerns the initial boundary problem of nonstationary thin plate flow of non newtonian viscous incompressible fluids by using monotone operator theory and schauder s fixed point theorem an existence and uniqueness theorem for a generalized solution is obtained and the regularity for this solution is tested

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

an exit probability approach to solving high dimensional dirichlet problems we present an approach to solving high dimensional dirichlet problems in bounded domains which is based on a variant of the feynman kac formula that connects solutions to elliptic partial differential equations and functional integration we integrate the resulting system of stochastic differential equations numerically with the euler scheme to correct for possible intermediate excursions of the simulated paths and to find good approximations for first exit times we extend to higher dimensions a strategy which we introduced in f m buchmann j comput phys 202 2005 pp 446 462 for stopping problems in one dimension in addition for dirichlet problems good approximations of the first exit points are needed to evaluate the boundary condition to this end we sample a random point on the tangent hyperplane of the boundary of the domain once a first exit is estimated to detect excursions and adequately compute first exit times we apply the same local half space approximation we therefore assume that the domain is smooth enough such that these approximations are possible numerical experiments in dimensions up to 128 show that resulting approximations are of very high quality in particular we observe that first order convergence behavior of the euler scheme can be maintained

some results on pole placement and reachability it is proved that a 2 dimensional system f g over a projective free ring is pole assignable if and only if im g contains a unimodular further it is shown that im g contains a unimodular iff f 1 im g contains a unimodular some observations on reachability are also mentioned in the last section

weighted estimates for the incompressible fluid in exterior domains for a strong solution u x t of the navier stokes equations in exterior domain in rn where n 2 3 we study the time decay of x u t lp for n when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 3 2007 547 567 h o bae b j jin asymptotic behavior for the navier stokes solutions in 2d exterior domains j funct anal 240 2006 508 529 and we will extend bae and jin s results by modifying their methods

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

an improved l2 decay for weak solutions of a boussinesq system in exterior domains abstract the algebraic decay for the total kinetic energy of weak solutions of the viscous boussinesq system in an exterior domain is given by means of the spectral analysis method of the stokes operator which gives a positive answer to the l 2 decay question in 11

asymptotic behavior for the stokes flow and navier stokes equations in half spaces abstract using the solution formula in ukai 1987 27 for the stokes equations we find asymptotic profiles of solutions in l 1 r n n 2 for the stokes flow and non stationary navier stokes equations since the projection operator p l 1 r n l 1 r n is unbounded we use a decomposition for p u u to overcome the difficulty and prove that the decay rate for the first derivatives of the strong solution u of the navier stokes system in l 1 r n is controlled by t 1 2 1 t n 2 2 for any t 0

decay properties of solutions to the non stationary magneto hydrodynamic equations in half spaces abstract l 1 decay properties of the strong solution including the first and second order spacial derivatives to the non stationary magneto hydrodynamic mhd equations are established in a half space and the weighted cases are also considered additionally

spatial stability of 3d exterior stationary navier stokes flows abstract in this paper we study the stability of stationary solutions w for the navier stokes flows in an exterior domain with zero velocity at infinity with suitable assumptions of w by the works of chen 1993 kozono ogawa 1994 and borchers miyakawa 1995 if u 0 w l r l 3 then one can obtain u t w p o t 3 2 1 r 1 p for 1 r p u t w p o t 3 2 1 r 1 p 1 2 for 1 r p 3 where u x t is a solution of the navier stokes equations with the initial condition u 0 in this paper we will prove that for any 0 3 if x u 0 w belongs to l r then one has x u t w l p o t 3 2 1 r 1 p 2 for p 3 r 3 r

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

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

domain decomposition solution of elliptic boundary value problems via monte carlo and quasi monte carlo methods domain decomposition of two dimensional domains on which boundary value elliptic problems are formulated is accomplished by probabilistic monte carlo as well as by quasi monte carlo methods generating only a few interfacial values and interpolating on them continuous approximations for the trace of solution are thus obtained to be used as boundary data for the subproblems the numerical treatment can then proceed by standard deterministic algorithms separately in each of the so obtained subdomains monte carlo and quasi monte carlo simulations may naturally exploit multiprocessor architectures leading to parallel computing as well as the ensuing domain decomposition does the advantages such as scalability obtained by increasing the number of processors are shown both theoretically and experimentally in a number of test examples and the possibility of using clusters of computers grid computing is emphasized

domain decomposition solution of nonlinear two dimensional parabolic problems by random trees a domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension based on the probabilistic representation of solutions as an average of suitable multiplicative functionals such a direct probabilistic representation requires generating a number of random trees whose role is that of the realizations of stochastic processes used in the linear problems first only few values of the sought solution inside the space time domain are computed by a monte carlo method on the trees an interpolation is then carried out in order to approximate interfacial values of the solution inside the domain thus a fully decoupled set of sub problems is obtained the algorithm is suited to massively parallel implementation enjoying arbitrary scalability and fault tolerance properties pruning the trees is shown to increase appreciably the efficiency of the algorithm numerical examples conducted in 2d including some for the kpp equation are given

decay results of higher order norms for the navier stokes flows in 3d exterior domains using the nonstationary stokes solution formula and the regularity theory of the steady stokes equations we establish the decay results including the weighted cases of higher order spatial derivatives for the navier stokes flows in 3d exterior domains which is a long standing problem the higher order norm weighted decay results in this article are inspired by the work in bae and jin proc r soc edinburgh sect a 135 461 477 2005 schonbek and wiegner proc r soc edinburgh sect a 126 677 685 1996 respectively which are considered in the whole space

temporal decays in l 1 and l for the stokes flow we estimate the time decay rates in l1 in the hardy space and in l of solutions for the stokes equations on the half spaces for the estimates in the hardy space we adopt the ideas in math z 231 1999 383 temporal and spatial decays for the stokes flows submitted for publication and also use the heat kernel and the solution formula for the stokes equations

new monte carlo schemes for simulating diffusions in discontinuous media we introduce new monte carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity these schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains this is achieved using either stochastic process techniques or an approach based on finite differences numerical tests on elliptic parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

pointwise decay estimate of navier stokes flows in the half space with slowly decreasing initial value abstract in this paper we study the spatial and temporal decay estimate of the navier stokes flow in the half space corresponding to auniformly but slowly decreasing initial velocity we show the local in time solvability of the navier stokes equations with u x t c 0 1 x t and u x t c 0 t 1 2 1 x t when 1 x t e t a h l r n 0 and t 1 2 1 x t e t a h l r n 0 0 n asymptotically it holds that u x t e t a h x o t x 0 n as x the solution exists globally in time for 1 when the data is small enough and it holds that u x t e t a h x o x t 1 n as x t

multidimensional exponential timestepping with boundary test exponential timestepping algorithms are efficient for exit time problems because a boundary test can be performed at the end of each timestep giving high order convergence in numerical evaluation of mean exit times successive time increments are independent random variables with an exponential distribution we show how to perform exact timestepping for brownian motion in more than one dimension and consider hitting times of curved surfaces taking as examples the exit times from a circle from a ball from an ellipse and from the region bounded by the two lines of a hyperbola we report the results of numerical experiments performed with the algorithms developed in this work

exponential timestepping with boundary test for stochastic differential equations we present new numerical methods for scalar stochastic differential equations successive time increments are independent random variables with an exponential distribution we perform numerical experiments using a double well potential exponential timestepping algorithms are efficient for escape time problems because a simple boundary test can be performed at the end of each step

on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain the exterior nonstationary problem is studied for the 3d navier stokes equations the l1 summability is proved for smooth solutions which correspond to initial data satisfying certain symmetry and moment conditions the result is then applied to show that such solutions decay in time more rapidly than observed in general furthermore an asymptotic expansion is deduced and a lower bound estimate is given for the rates of decay in time

decay rate of solutions of navier stokes equations in 3 dimensional half space abstract in this article we derive the l 2 spatial decay rate of a weak solution of the incompressible flow in the half space when the half space or some other fluid region with boundary is concerned pressure estimate is main obstacle since we do not have enough information of the pressure on the boundary in this paper we give an idea which does not require any pressure information on the boundary and we hope that our idea could be applied to other problems such as exterior domain problem

feedback control of nonlinear systems by extended linearization for single input multiple output nonlinear systems we consider a design method based on the family of linearizations of the system parameterized by constant operating points nonlinear state feedback control laws and observer state feedback control laws are designed such that the eigenvalues of the family of linearized closed loop systems are placed at specified values that are locally invariant with respect to the closed loop operating point the method is illustrated by application to the problem of automatically balancing an inverted pendulum

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

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

simulating diffusion processes in discontinuous media in this article we propose new monte carlo techniques for moving a diffusive particle in a discontinuous media in this framework we characterize the stochastic process that governs the positions of the particle the key tool is the reduction of the process to a skew brownian motion sbm in a zone where the coefficients are locally constant on each side of the discontinuity the new position of the particle after a constant time step is sampled from the exact distribution of the sbm process at the considered time to do so we propose two different but equivalent algorithms a two steps simulation with a stop at the discontinuity and a one step direct simulation of the sbm dynamic some benchmark tests illustrate their effectiveness

upper and lower bounds of temporal and spatial decays for the navier stokes equations we obtain the lower bounds of the temporal spatial decays for weak solutions of the navier stokes equations c0 1 t 5 2i4 1 2 i 2u t l2 c1 1 t 5 2i4 for i 1 2 the upper bound parts are estimated in several papers for example bae and jin submitted and he and xin proc roy soc edinburg section a 131 3 2001 597 by the interpolation arguments we also have c0 1 t 54 2 1 x 2 2u t l2 c1 1 t 54 2 for 0 2

parametric stabilization is easy a polynomially parametrized family of continuous time controllable linear systems is always stabilizabte by polynomially parametrized feedback

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

long time behavior for navier stokes flows in a two dimensional exterior domain abstract there are several long standing problems on the incompressible navier stokes flows in 2d exterior domains which claim how to characterize l 1 summability of the 2d n s flows whether the total net force exerted on the boundary is finite and how to establish decay results of higher order spatial derivatives including the weighted cases in order to solve these questions we firstly find some types of new technical inequalities which are used to overcome the difficulties caused by the domain boundary using l q l r properties for nonstationary stokes flows together with elliptic estimates for the steady stokes system we can avoid the strong singularity and answer these mentioned problems completely it should be pointed out that main results in this article are motivated by the works in 5 37 respectively

exact asymptotics for the probability of exit from a domain and applications to simulation we study the asymptotics of the exit probability p x s e t where is the exit time from an open set and p x s e is the law of a diffusion process with a small parameter e multiplying the diffusion coefficient we consider the case of the brownian bridge in many dimensions this choice being motivated by applications to numerical simulation the method uses recent results reducing the problem to the solution of a system of linear first order pde s

decay rates for the incompressible navier stokes flows in 3d exterior domains abstract the exterior nonstationary problem is studied for the 3d navier stokes equations for which the associated total net force to the boundary may not vanish the time decay properties of the strong solution including the first and second derivatives are shown in l q and weighted spaces in particular the relation of weighted l 1 summability for smooth solutions is discussed in details between the time decay and the total net force exerted by the fluid to the body the conclusions in this article improve and extend results in h bae b jin asymptotic behavior for the navier stokes equations in 2d exterior domains j funct anal 240 2006 508 529 h bae b jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 2007 547 567 and in c he t miyakawa on l 1 summability and asymptotic profiles for smooth solutions to navier stokes equations in a 3d exterior domain math z 245 2003 387 417 c he t miyakawa on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain j differential equations 246 2009 2355 2386 respectively

space time decay of navier stokes flows invariant under rotations we show that the solutions to the non stationary navier stokes equations in open image in new window d 2 3 which are left invariant under the action of discrete subgroups of the orthogonal group o d decay much faster as open image in new window than in generic case and we compute for each subgroup the precise decay rates in space time of the velocity field

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

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

large time behavior for the nonstationary navier stokes flows in the half space abstract asymptotic behavior of higher order spatial derivatives in l r is established for incompressible navier stokes flows in the half space which is a long time unsolved problem the main tools employed for solving this problem are the nonstationary stokes solution formula and the a priori estimates of the steady stokes system in the half space another main result is devoted to studying time l 1 decay and a partial answer to this open problem is given by means of a crucial and new estimate for the stokes flow two theorem 1 1 theorem 1 2 in this article can be regarded as a great improvement and powerful extension of the work in 6 38

lower bounds of rates of decay for solutions to the navier stokes equations where a p 3 2 p 1 and the constant c depends on the l2 and lnorms of the initial data u0 this paper deals with the more subtle problem of deriving lower bounds on the energy decay rates we show that for a certain class of initial data the solutions u x t to the 2d and 3d navier stokes equations admit an algebraic lower bound on the energy decay specifically there are two cases to consider in the first case the average of the initial data f u0 dx is nonzero this case was treated in the earlier paper 5 where it was established that

weighted decay properties for the incompressible stokes flow and navier stokes equations in a half space abstract using the stokes solution formula and l q l r estimates of the stokes operator semigroup we establish the weighted decay properties for the stokes flow and navier stokes equations including their spatial derivatives in half spaces in addition the unboundedness of the projection operator p l r n l r n is overcome by employing a decomposition for the nonlinear term and l asymptotic behavior for the second derivatives of navier stokes flows in half spaces is given

asymptotic behavior of solutions to the nonstationary magneto hydrodynamic equations abstract asymptotic behavior of the higher order spacial derivatives of the strong solution to the incompressible non stationary magneto hydrodynamic mhd equations is given in a half space which is a long time difficult question the main tools employed in this article are the non stationary stokes solution formula and some a priori estimates of the steady stokes system in the half space

sequential control variates for functionals of markov processes using a sequential control variates algorithm we compute monte carlo approximations of solutions of linear partial differential equations connected to linear markov processes by the feynman kac formula it includes diffusion processes with or without absorbing reflecting boundary and jump processes we prove that the bias and the variance decrease geometrically with the number of steps of our algorithm numerical examples show the efficiency of the method on elliptic and parabolic problems

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0