an infinite dimensional purification principle without saturation abstract we establish a purification process for the min expected utilities coming from economy theory involved in the phenomenon of aversion to the ambiguity in game and decision theories we use instead of the saturation property a characterization of lyapunov infinite dimensional vector measures established in g knowles lyapunov vector measures siam j control 13 1975 294 303

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

nonatomic games on loeb spaces in the setting of noncooperative game theory strategic negligibility of individual agents or diffuseness of information has been modeled as a nonatomic measure space typically the unit interval endowed with lebesgue measure however recent work has shown that with uncountable action sets for example the unit interval there do not exist pure strategy nash equilibria in such nonatomic games in this brief announcement we show that there is a perfectly satisfactory existence theory for nonatomic games provided this nonatomicity is formulated on the basis of a particular class of measure spaces hyperfinite loeb spaces we also emphasize other desirable properties of games on hyperfinite loeb spaces and present a synthetic treatment embracing both large games as well as those with incomplete information

navier stokes flow in the weighted hardy space with applications to time decay problem abstract the asymptotic expansions of the navier stokes flow in r n and the rates of decay are studied with aid of weighted hardy spaces fujigaki and miyakawa 12 miyakawa 28 proved the nth order asymptotic expansion of the navier stokes flow if initial data decays like 1 x n 1 and if nth moment of initial data is finite in the present paper it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem the second author 39 established the weighted estimates of the strong solutions in the weighted hardy spaces with small initial data which belongs to l n and a weighed hardy space firstly the refinement of the previous work 39 is achieved with alternative proof then the existence time of the solution in the weighted hardy spaces is characterized without any hardy norm as a result in two dimensional case the smallness condition on initial data is completely removed as an application the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by brandolese 3

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

multilevel algorithms considered as iterative methods on semidefinite systems for the representation of piecewise d linear functions instead of the usual finite element basis a generating system is introduced that contains the nodal basis functions of the finest level and of all coarser levels of discretization this approach enables the author to work directly with multilevel decompositions of a function for a partial differential equation the galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the one dimensional 1d case only about twice in the two dimensional 2d case about 4 3 times and in the three dimensional 3d case about 8 7 times as many unknowns as the usual system furthermore the semidefinite system possesses not just one but many solutions however the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem the author shows that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite sy

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

approximate purification of mixed strategies under relatively weak conditions on the information structure of a game it is shown that any mixed strategy of a player can be replaced by a certain pure strategy without affecting any player s expected payoff appreciably no matter what strategies the other players use under even weaker conditions the existence of a nash equilibrium in mixed strategies is shown to guarantee the existence of a pure strategy combination which is an approximate equilibrium

algebraic multigrid for higher order finite elements two related approaches for solving linear systems that arise from a higher order finite element discretization of elliptic partial differential equations are described the first approach explores direct application of an algebraic based multigrid method amg to iteratively solve the linear systems that result from higher order discretizations while the choice of basis used on the discretization has a significant impact on the performance of the solver results indicate that amg is capable of solving operators from both poisson s equation and a first order system least squares fosls formulation of stoke s equation in a scalable manner nearly independent of basis order p for 3

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

components reduction regularity results for the navier stokes equations in general dimensions abstract we consider the cauchy problem of the navier stokes equations in arbitrary dimensions and establish several new components reduction regularity criteria

weighted estimate of the asymptotic profiles of the navier stokes flow in rn abstract we study the asymptotic profiles of the navier stokes flow and the weighted estimate of the spatial and temporal decay rate of the error term this paper extends the result of y fujigaki and t miyakawa in y fujigaki t miyakawa asymptotic profiles of nonstationary incompressible navier stokes flows in the whole space siam j math anal 33 3 2001 523 544 where they derived the asymptotic profiles of the navier stokes flow and estimated the temporal decay rate of the error term

multilevel first order system least squares for nonlinear elliptic partial differential equations a fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods the equations are converted to a first order system that is then linearized via newton s method first order system least squares fosls is used to formulate and discretize the newton step and the resulting matrix equation is solved using algebraic multigrid amg the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation a general theory is developed that confirms the usual full multigrid efficiency accuracy comparable to the finest level discretization is achieved at a cost proportional to the number of finest level degrees of freedom in a companion paper the theory is applied to elliptic grid generation egg and supported by numerical results

mild solutions to the time fractional navier stokes equations in r n abstract this paper addresses the existence and uniqueness of mild solutions to the navier stokes equations with time fractional differential operator of order 0 1 several interesting properties about the solution are also highlighted like regularity and decay rate in lebesgue spaces which will depend on the fractional exponent moreover it is shown that the l p exponent range which the solution belongs to is different from the range for the solution of the classical problem with 1

some remarks on the navier stokes equations in r3 we study various existence and uniqueness results for solutions of the navier stokes equations in connection with function spaces related to real harmonic analysis

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

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

on the navier stokes equations in the half space with initial and boundary rough data in morrey spaces abstract we consider the initial boundary value problem for the navier stokes equations in the half space with data in morrey spaces existence of small global solutions is proved in spaces with the right homogeneity to allow self similar solutions moreover we analyze the long time behavior of the solutions and obtain a class of asymptotically self similar ones

a multilevel nonlinear method a multilevel nonlinear method mnm for the numerical solution of nonlinear partial differential equations is developed the mnm algorithm is motivated and analyzed using a simplified model which retains the essential features of the new approach it is thereby shown to combine the advantages of the two classical multigrid approaches to nonlinear problems the analysis is supported by numerical tests for nonlinear differential equations in one and two dimensions

diffusive expansion for solutions of the boltzmann equation in the whole space this paper is concerned with the diffusive expansion for solutions of the rescaled boltzmann equation in the whole space 0 1 tf v xf 1 q f f x rn v rn t 0 with prescribed initial data 0 2 f t x v t 0 f0 x v x rn v rn our main purpose is to justify the global validity of the diffusive expansion 0 3 f t x v f1 t x v 2f2 t x v n 1fn 1 t x v nfn t x v for a solution f t x v of the rescaled boltzmann equation 0 1 in the whole space rn for all t 0 with initial data f0 x v satisfying the initial expansion 0 4 f0 x v f1 0 x v 2f2 0 x v n 1fn 1 0 x v nfn 0 x v x v r2n here v 2 n2exp v 22 is a normalized global maxwellian under the assumption that the fluid components of the coefficients fm 0 x v 1 m n of the initial expansion f0 x v have divergence free velocity fields um0 x as well as temperature fields m0 x if we assume further that the velocity temperature fields u10 x 10 x of f1 0 x v have small amplitude in hs rn s 2 n n 2 we can determine these coefficients fm t x v 1 m n in the diffusive expansion 0 3 uniquely by an iteration method and energy method the hydrodynamic component of these coefficients fm t x v 1 m n satisfies the incompressible condition the boussinesq relations and or the navier stokes fourier system respectively while the microscopic component of these coefficients is determined by a recursive formula compared with the corresponding problem inside a periodic box studied in y guo 2006 18 the main difficulty here is due to the fact that poincare s inequality is not valid in the whole space rn and this difficulty is overcome by using the lp lq estimate on the riesz potential moreover by exploiting the energy method we can also deduce certain the space time energy estimates on these coefficients fm t x v 1 m n once the coefficients fm t x v 1 m n in the diffusive expansion 0 3 are uniquely determined and some delicate estimates have been obtained the uniform estimates with respect to on the remainders fn t x v are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion 0 3 in the large provided that 0 5 n 2n 2 notice that for m 2 um t x is no longer a divergence free vector and it is worth to pointing out that for m 3 it was in deducing certain estimates on pm t x by the lp lq estimate on the riesz potential that we need to require that n 2n 2

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

long time decay to the lei lin solution of 3d navier stokes equations abstract in this paper we prove if u c 0 x 1 r 3 is a global solution of 3d navier stokes equations then u t x 1 decays to zero as time goes to infinity fourier analysis and standard techniques are used

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

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

analysis of velocity flux least squares principles for the navier stokes equations part ii this paper continues the development of the least squares methodology for the solution of the incompressible navier stokes equations started in part i here we again use a velocity flux first order navier stokes system but our focus now is on a practical algorithm based on a discrete negative norm

sobolev multipliers maximal functions and parabolic equations with a quadratic nonlinearity abstract we develop a general framework to describe global mild solutions to a cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity this class includes the navier stokes equations the subcritical dissipative quasi geostrophic equation and the parabolic elliptic keller segel system

unigrid for multigrid simulation this paper develops unigrid which is an algorithm that results from interpreting multigrid as it directly affects the fine grid approximation unigrid is theoretically equivalent to multigrid under certain assumptions yet has different computational characteristics though it is generally less efficient unigrid is very useful as a software tool for testing the feasibility of applying multigrid to a given problem this is illustrated with several numerical examples 1 introduction the subject of this paper is unigrid a method designed primarily for solving the equations arising from discretization of partial differential boundary value problems the method is closely related and under certain conditions theoretically equivalent to multigrid although all of the unigrid computations are performed on a single grid as the name implies unigrid is generally more computa tionally costly than multigrid yet its simplicity is reflected in its relative ease of code implementation and use with existing applications software this enables unigrid to act effectively as a software tool to simulate the multigrid approach that requires a

global solutions of the navier stokes equations in thin three dimensional domains this paper is concerned with the spatially periodic navier stokes equations in a thin three dimensional domain it is shown that the classical solution exists globally when the initial velocityu0and the external forcefare chosen in the set significantly larger than those presented in earlier works what is more it is shown that these global solutions which seem large in a function space are small in nature

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

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

decay estimates of gradient of a generalized oseen evolution operator arising from time dependent rigid motions in exterior domains let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions where the translational and angular velocities of the body are prescribed but time dependent in a reference frame attached to the body we have the navier stokes system with the drift and one half of the coriolis terms in a fixed exterior domain the existence of the evolution operator t t s in the space l q generated by the linearized non autonomous system was proved by hansel and rhandi j reine angew math 694 1 26 2014 and the large time behavior of t t s f in l r for t s rightarrow infty was then developed by hishida math ann 372 915 949 2018 when f is taken from l q with q leqq r the contribution of the present paper concerns such l q l r decay estimates of nabla t t s with optimal rates which must be useful for the study of stability attainability of the navier stokes flow in several physically relevant situations our main theorem completely recovers the l q l r estimates for the autonomous case stokes and oseen semigroups those semigroups with rotating effect in three dimensional exterior domains which were established by hishida and shibata arch ration mech anal 193 339 421 2009 iwashita math ann 285 265 288 1989 kobayashi and shibata math ann 310 1 45 1998 maremonti and solonnikov ann sc norm super pisa 24 395 449 1997 and shibata in amann arendt hieber neubrander nicaise von below eds functional analysis and evolution equations the gunter lumer volume birkhauser basel pp 595 611 2008

a line search multigrid method for large scale nonlinear optimization we present a line search multigrid method for solving discretized versions of general unconstrained infinite dimensional optimization problems at each iteration on each level the algorithm computes either a direct search direction on the current level or a recursive search direction from coarser level models introducing a new condition that must be satisfied by a backtracking line search procedure the recursive search direction is guaranteed to be a descent direction global convergence is proved under fairly minimal requirements on the minimization method used at all grid levels using a limited memory bfgs quasi newton method to produce the direct search direction preliminary numerical experiments show that our line search multigrid approach is promising

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

on multilevel iterative methods for optimization problems this paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces the construction of auxiliary problems as well as applications to elasto plastic model and linear programming are described the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates equivalents of this condition are presented in special cases

besov morrey spaces function space theory and applications to non linear pde this paper is devoted to the analysis of function spaces modeled on besov spaces and their applications to non linear partial differential equations with emphasis on the incompressible isotropic navier stokes system and semilinear heat equations specifically we consider the class introduced by hideo kozono and masao yamazaki of besov spaces based on morrey spaces which we call besov morrey or bm spaces we obtain equivalent representations in terms of the weierstrass semigroup and wavelets and various embeddings in classical spaces we then establish pseudo differential and para differential estimates our results cover non regular and exotic symbols although the heat semigroup is not strongly continuous on morrey spaces we show that its action defines an equivalent norm in particular homogeneous bm spaces belong to a larger class constructed by grzegorz karch to analyze scaling in parabolic equations we compare karch s results with those of kozono and yamazaki and generalize them by obtaining short time existence and uniqueness of solutions for arbitrary data with subcritical regularity we exploit pseudo differential calculus to extend the analysis to compact smooth boundaryless riemannian manifolds bm spaces are defined by means of partitions of unity and coordinate patches and intrinsically in terms of functions of the laplace operator

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

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

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

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces this paper considers the existence and large time behavior of solutions to the convection diffusion equation ut u b x u u q 1 f x t in n 0 where f x t is slowly decaying and q 1 1 n or in some particular cases q 1 the initial condition u0 is supposed to be in an appropriate lp space uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term

a zero sum game with incomplete information and compact action spaces we make use of distributional strategies to describe player s choices of actions in a zero sum game with incomplete information and compact action spaces under very mild assumptions we obtain the existence of equilibria

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

projection multilevel methods for quasilinear elliptic partial differential equations theoretical results the goal of this paper is to introduce a new multilevel solver for two dimensional elliptic systems of nonlinear partial differential equations pdes where the nonlinearity is of the type u partial v the incompressible navier stokes equations are an important representative of this class and are the target of this study using a first order system least squares fosls approach and introducing a new variable for partial v for this class of pdes we obtain a formulation in which the nonlinearity appears as a product of two different dependent variables the result is a system that is linear within each variable but nonlinear in the cross terms in this paper we introduce a new multilevel method that treats the nonlinearities directly this approach is based on a projection multilevel pml method s f mccormick multilevel projection methods for partial differential equations siam philadelphia 1992 applied to the fosls functional the implementation of the discretization process relaxation coarse grid correction and cycling strategies is discussed and optimal performance is established numerically a companion paper t a manteuffel s f mccormick and o rohrle siam j numer anal 44 2006 pp 139 152 establishes a two level convergence proof for this new multilevel method

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach

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