the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

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

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

weak solutions to the navier stokes inequality with arbitrary energy profiles in a recent paper buckmaster and vicol ann math 2 189 1 101 144 2019 used the method of convex integration to construct weak solutions u to the 3d incompressible navier stokes equations such that vert u t vert l 2 e t for a given non negative and smooth energy profile e 0 t rightarrow mathbb r however it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions that is weak solutions satisfying the strong energy inequality sei and the local energy inequality lei leray hopf weak solutions that is weak solutions satisfying the sei or at least to exclude energy profiles that are not nonincreasing in this paper we are concerned with weak solutions to the navier stokes inequality on mathbb r 3 that is vector fields that satisfy both the sei and the lei but not necessarily solve the navier stokes equations given t 0 and a nonincreasing energy profile e 0 t rightarrow 0 infty we construct weak solution to the navier stokes inequality that are localised in space and whose energy profile vert u t vert l 2 mathbb r 3 stays arbitrarily close to e t for all t in 0 t our method applies only to nonincreasing energy profiles the relevance of such solutions is that despite not satisfying the navier stokes equations they satisfy the partial regularity theory of caffarelli et al commun pure appl math 35 6 771 831 1982 in fact scheffer s constructions of weak solutions to the navier stokes inequality with blow ups commun math phys 101 1 47 85 1985 commun math phys 110 4 525 551 1987 show that the caffarelli kohn nirenberg s theory is sharp for such solutions our approach gives an indication of a number of ideas used by scheffer moreover it can be used to obtain a stronger result than scheffer s namely we obtain weak solutions to the navier stokes inequality with both blow up and a prescribed energy profile

on non equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case it is shown that for a set of full measure with respect to any translation invariant probability distribution on the space of initial configurarations of classical particle systems on d with interaction given by a smooth superstable potential of finite range there is a solution to the newtonian equations of motion provided that the specific energy and the particle density of the initial configuration exist a s

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

algorithms to enumerate codewords for dc sup 2 constrained channels we introduce a new method for enumerating codewords that can be applied to dc sup 2 constrained channels based on this method two efficient algorithms for evaluating the number of codewords with specified characteristics are developed computer calculation results show that these algorithms are significantly more computationally efficient than other techniques developed to date

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

stochastic three dimensional rotating navier stokes equations averaging convergence and regularity we consider stochastic three dimensional rotating navier stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems

existence of densities for the 3d navier stokes equations driven by gaussian noise we prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic navier stokes equations in dimension 3 in particular under very mild assumptions on the noise we prove that finite dimensional projections of the solutions have densities with respect to the lebesgue measure which have some smoothness when measured in a besov space this is proved thanks to a new argument inspired by an idea introduced in fournier and printems 2010

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

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

a contribution to scattering calculation for small wavelengths the high frequency panel method numerical treatment of electromagnetic scattering in the case of small wavelengths too small for reasonable matrix sizes when using discretization methods and too large for traditional asymptotic methods is considered an approach which takes advantage of the user friendly nature of matrix discretization methods while avoiding the explosive increase in computing effort as the wavelength decreases is suggested it appears that no sensible increase of matrix size is required as the wavelength decreases since this is an untried technique applied in an inaccessible computational domain the preliminary work described is confined to two dimensional conducting circular cylinders where the answers are well known and well understood the overall idea however appears to be quite general

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

design of some new efficient balanced codes a balanced code with r check bits and k information bits is a binary code of length k r and cardinality 2 sup k such that each codeword is balanced that is it has k r 2 1 s and k r 2 0 s this paper contains new methods to construct efficient balanced codes to design a balanced code an information word with a low number of 1 s or 0 s is compressed and then balanced using the saved space on the other hand an information word having almost the same number of 1 s and 0 s is encoded using the single maps defined by knuth s 1986 complementation method three different constructions are presented balanced codes with r check bits and k information bits with k spl les 2 sup r 1 2 k spl les 3 spl times 2 sup r 8 and k spl les 5 spl times 2 sup r 10r c r c r spl isin 15 10 5 0 5 are given improving the constructions found in the literature in some cases the first two constructions have a parallel coding scheme

nearly one dimensional singularities of solutions to the navier stokes inequality there exists a functionu satisfying 1 u is a weak solution to the navier stokes equations of incompressible fluid flow in three space with an external force that reduces the speed at every point 2 the internal singularities ofu have hausdorff dimension close to 1

sampling method using prefiltered band limited green s functions for the solution of electromagnetic integral equations in the context of far field scattering the spatial spectrum of fields or currents on a scatterer may be approximated as band limited on this premise one can greatly simplify the computation of radiation integrals by eliminating in advance those spectral components of the green s function not expected to be present in the fields this approach enables one to dispense with the notion of basis functions and instead represent fields and currents as well as the impedance matrix representing the green s function entirely in terms of a sparse set of samples thereby obviating the need for integration moreover by taking full advantage of the band limited character of the solution this sampling theoretical approach yields an efficient representation with a minimal number of degrees of freedom the method was explored for scattering from two dimensional perfect electric conductors including ellipsoidal cylinders and flat strips in both polarizations using uniform as well as nonuniform sampling and yielded surprisingly good results

efficient encoding algorithm for third order spectral null codes an efficient algorithm is presented for encoding unconstrained information sequences into a third order spectral null code of length n and redundancy 9log sub 2 n o log log n the encoding can be implemented using o n integer additions and o nlog n counter increments

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

irreducibility of the 3 d stochastic navier stokes equation abstract a 3 dimensional navier stokes equation with random force is investigated a form of irreducibility of interest in ergodic theory is proved under a full noise assumption the basic tool is the fact that even if the equation is a priori non well posed the solutions depend continuously on the noise around regular solutions

on the use of fourier series fft s as global basis functions in the solution of boundary integral equations for em scattering current methods for the solution of the surface integral equations of interest to practical electromagnetic scattering appear to have reached an impasse plagued by disputes concerning the suitability and performance of various competing choices for piecewise basis functions and facing horribly complicated piecewise quadratures for the construction of influence matrix elements it has become quite commonplace for these methods to be matrix build time limited rather than matrix solve time limited an extremely simple alternative to all this is described below and demonstrated in an introductory sense on some 2 d problems in this procedure the fft method the basis functions are all global fourier series in a suitably chosen parameter and one attempts to estimate the coefficients describing the unknown surface current directly from the integral equation this format circumvents all of the shortcomings discussed above there are no singular integrals to evaluate furthermore since the fft algorithm is used throughout the computation is extremely efficient the results shown below suggest that so far this approach is very effective

a solution to the navier stokes inequality with an internal singularity we consider weak solutions to the time dependent navier stokes equations of incompressible fluid flow in three dimensional space with an external force that always acts against the direction of the flow we show that there exists a solution with an internal singularity the speed of the flow reaches infinity at this singular point in addition the solution has finite kinetic energy

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

partial regularity for the stochastic navier stokes equations the effects of random forces on the emergence of singularities in the navier stokes equations are investigated in spite of the presence of white noise the paths of a martingale suitable weak solution have a set of singular points of one dimensional hausdorff measure zero furthermore statistically stationary solutions with finite mean dissipation rate are analysed for these stationary solutions it is proved that at any time t the set of singular points is empty the same result holds true for every martingale solution starting from 0 a e initial condition u 0 where 0 is the law at time zero of a stationary solution finally the previous result is non trivial when the noise is sufficiently non degenerate since for any stationary solution the measure 0 is supported on the whole space h of initial conditions

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

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

an inverse scattering method based on contour deformations by means of a level set method using frequency hopping technique a shape reconstruction method for electromagnetic tomography based on contour deformations by means of a level set method is proposed the level set method adds quite valuable features to the inverse procedure such as natural regularization of the contours and automatic breaking and merging capability the frequency hopping technique turns out to be a key point of the inverse scattering procedure the numerical results clearly indicate that the use of this inversion algorithm yields to accurate reconstructions of one or several objects from one initial guess object even with noise contaminated data and limited coverage of the fields

local martingale and pathwise solutions for an abstract fluids model we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system the primary application of this abstract framework is to infer the local existence of strong pathwise solutions to the 3d primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise instead of developing our results specifically for the 3d primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations atmosphere oceans coupled atmosphere ocean on the sphere on the beta plane approximation etc and the incompressible navier stokes equations in applications all of the details are given for the beta plane approximation of the oceans equations

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

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

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

efficient encoding algorithm for second order spectral null codes using cyclic bit shift some efficient second order spectral null codes encoded an index of the random walk function recursively and ended with the short base second order spectral null codes all of these codes used the tallini bose random walk function that exchanges two consecutive bits in this paper we propose a new random walk function based on cyclic bit shift on which the redundancy can be improved moreover the bit shift can be implemented efficiently by either software or hardware

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

theory of unidirectional error correcting detecting codes in this paper we present some basic theory on unidirectional error correcting detecting codes we define symmetric asymmetric and unidirectional error classes and proceed to derive the necessary and sufficient conditions for a binary code to be unidirectional error correcting detecting

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

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

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

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

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

stationary solutions to the compressible navier stokes system driven by stochastic forces we study the long time behavior of solutions to a stochastically driven navier stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation the existence of stationary solutions is established in the framework of lebesgue sobolev spaces pertinent to the class of weak martingale solutions the methods are based on new global in time estimates and a combination of deterministic and stochastic compactness arguments an essential tool in order to obtain the global in time estimate is the stationarity of solutions on each approximation level which provides a certain regularizing effect in contrast with the deterministic case where related results were obtained only under rather restrictive constitutive assumptions for the pressure the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory due to the underlying martingale structure of the noise

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

irreducibility of the three and two and a half dimensional hall magnetohydrodynamics system abstract the hall magnetohydrodynamics system has been considered by many to play an important role in astrophysical plasmas star formation and magnetic re connection the hall term mathematically elevates the magnetohydrodynamics system from a semi linear to a quasi linear type and standard results which are well known for the navier stokes equations and the magnetohydrodynamics system such as the local well posedness in the non diffusive case as well as the uniqueness of the weak solution in case the solution depends only on two spatial variables are completely open for the hall magnetohydrodynamics system because the hall magnetohydrodynamics system is not known to be globally well posed even if the solution depends only on two spatial variables a typical method to prove ergodicity for the two dimensional navier stokes equations such as hairer and mattingly 2006 seems to have no chance in this manuscript we take advantage of the structure of the hall term follow the work of flandoli 1997 and prove its irreducibility property in order to elaborate on the effect of the hall term we also prove analogous results for related systems such as the magnetohydrodynamics system benard problem magnetic benard problem micropolar fluid system as well as the magneto micropolar fluid system at the time of writing this manuscript due to a technical difficulty the strong feller property of the hall magnetohydrodynamics system is an open problem

exponential mixing for the 3d stochastic navier stokes equations we study the navier stokes equations in dimension 3 ns3d driven by a noise which is white in time we establish that if the noise is at the same time sufficiently smooth and non degenerate in space then the weak solutions converge exponentially fast to equilibrium

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

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

convergence results for a class of spectrally hyperviscous models of 3 d turbulent flow abstract we consider the spectrally hyperviscous navier stokes equations shnse which add hyperviscosity to the nse but only to the higher frequencies past a cutoff wavenumber m 0 in guermond and prudhomme 2003 18 subsequence convergence of shnse galerkin solutions to dissipative solutions of the nse was achieved in a specific spectral vanishing viscosity setting our goal is to obtain similar results in a more general setting and to obtain convergence to the stronger class of leray solutions in particular we obtain subsequence convergence of shnse strong solutions to leray solutions of the nse by fixing the hyperviscosity coefficient while the spectral hyperviscosity cutoff m 0 goes to infinity this formulation presents new technical challenges and we discuss how its motivation can be derived from computational experiments e g those in borue and orszag 1996 1998 3 4 we also obtain weak subsequence convergence to leray weak solutions under the general assumption that the hyperviscous coefficient goes to zero with no constraints imposed on the spectral cutoff in both of our main results the aubin compactness theorem provides the underlying framework for the convergence to leray solutions