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

the role of the pressure in the partial regularity theory for weak solutions of the navier stokes equations we study the role of the pressure in the partial regularity theory for weak solutions of the navier stokes equations by introducing the notion of dissipative solutions due to duchon robert we will provide a generalization of the caffarelli kohn and nirenberg theory our approach gives a new enlightenment of the role of the pressure in this theory in connection to serrin s local regularity criterion

hausdorff measure of boundary singular points in the magnetohydrodynamic equations abstract we study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations by using the generalised hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors

weak solution to the steady compressible flow of nematic liquid crystals abstract in this study three dimensional equations for the steady compressible flow of nematic liquid crystals are considered in a bounded domain for the adiabatic exponent 1 we establish the existence of a weak solution by three level approximation and adapting the weak convergence method

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

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

on a particular analytical solution to the 3d navier stokes equations and its peculiarity for high reynolds numbers a special class of axially symmetric nonstationary flows of incompressible viscous fluids is examined for it the 3d navier stokes equations are reduced to a nonlinear partial differential equation of the third order and a linear partial differential equation of the second order these equations are studied and their particular analytical solutions are found the obtained particular solution to the navier stokes equations could be used to describe some types of turbulent flows of viscous fluids in the case of high reynolds numbers

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

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

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

global regularity for generalized anisotropic navier stokes equations we consider the global cauchy problem for the three dimensional generalized anisotropic navier stokes system with the viscosity term hu m2u where m is a fourier multiplier whose symbol m r r is non negative the case m 3 3 is essentially the navier stokes system while the case m 0 is the anisotropic navier stokes system we obtain the global regularity in the case that m 3 3 32 g 3 for all sufficiently large 3 where g r r is a nondecreasing function such that 1 dsg2 s slns then using the method of descent we can obtain the classical global regularity result of the two dimensional navier stokes system

a free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension abstract in this paper we consider the free boundary problem for a simplified version of ericksen leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one we obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum

a recursive maximum likelihood algorithm for arma line enhancement optimal filtering of narrow band signals corrupted by noise requires an infinite impulse response iir filter a recursive maximum likelihood algorithm is proposed for adaptively adjusting the coefficients of an iir line enhancer the performance of this adaptive iir filter is analyzed and compared to that of a finite impulse response filter preliminary simulation results are presented to illustrate the behavior of the proposed algorithm under various signal to noise conditions

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

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

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

generic structure of two dimensional dimages under gaussian blurring for medical imaging problems computer scientists pizer and eberly introduced the core as the analogue for grayscale images of the medial axis it is defined for two dimensional images as the ridge of a medial function defined on 2 1 dimensional scale space in this paper we extend methods from singularity theory to analyze the generic properties of ridges and cores for two dimensional images including the generic transitions which occur for sequences of images varying with a parameter such as time we do so by placing the ridge construction into a larger relative critical set structure which captures properties undetected by the disjoint ridge core segments the genericity properties are established using transversality conditions for associated mappings we introduce a method for establishing transversality conditions for medial functions obtained by gaussian blurring by using a basis of local solutions to the heat equation the method applies to many other properties besides those we consid

global weak solutions and attractors of the three dimensional maxwell bloch two level laser systems the three dimensional maxwell bloch system governs the multi longitudinal and transverse mode dynamics of two level wide aperture lasers in an optical ring cavity the system is hyperbolic in the propagation direction and dispersive in the transverse directions due to diffraction effects a rich variety of optical patterns and chaos are present in the dynamics we show the global existence of weak solutions inlp 2 p spaces of the maxwell bloch system under both absorbing and periodic boundary conditions the weak solutions are unique within the class of solutions provided by our regularization procedure and approach a universal attractor which has only partial smoothing instead of thec smoothing property found in early works for the longitudinal one dimensional and transverse two dimensional cases the idea of the proof makes essential use of both the hyperbolicity and dispersivity of the system in the case of periodic boundary condition our result depends on a conjectural strichartz inequality

local kinetic energy and singularities of the incompressible navier stokes equations abstract we study the partial regularity problem of the incompressible navier stokes equations a reverse holder inequality of velocity gradient with increasing support is obtained under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded as an application we give a new bound for the hausdorff dimension and the minkowski dimension of singular set when weak solutions v belong to l 0 t l 3 w r 3 where l 3 w r 3 denotes the standard weak lebesgue space

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

on the partial regularity theory for the mhd equations abstract we generalize here the celebrated partial regularity theory of caffarelli kohn and nirenberg to the mhd equations in the framework of parabolic morrey spaces this type of parabolic generalization using morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory for the classical navier stokes equations as well as for the mhd equations

existence of suitable weak solutions of complex ginzburg landau equations and properties of the set of singular points in this paper we consider the supercritical complex ginzburg landau equation we discuss the existence of suitable weak solution in where is a bounded domain in rn or the whole space we also discuss the properties of the set of the singular points of the suitable weak solution in rn which means that the possible singular points are located in a bounded ball for any given time and there is no singular point on the whole space after limited time

navier stokes equations with regularity in one direction we consider sufficient conditions for the regularity of leray hopf solutions of the navier stokes equations we prove that if the third derivative of the velocity u x3 belongs to the space lts0lxr0 where 2 s0 3 r0 2 and 9 4 r0 3 then the solution is regular this extends a result of beirao da veiga chin ann math ser b 16 407 412 1995 c r acad sci ser i math 321 405 408 1995 by making a requirement only on one direction of the velocity instead of on the full gradient the derivative u x3 can be substituted with any directional derivative of u

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

a priori bound on the velocity in axially symmetric navier stokes equations let v be the velocity of leray hopf solutions to the axially symmetric three dimensional navier stokes equations under suitable conditions for initial values we prove the following a priori bound v x t le frac c ln r 1 2 r 2 qquad 0 r le 1 2 where r is the distance from x to the z axis and c is a constant depending only on the initial value this provides a pointwise upper bound worst case scenario for possible singularities while the recent papers chiun chuan et al commun pde 34 1 3 203 232 2009 koch et al acta math 203 1 83 105 2009 gave a lower bound the gap is polynomial order 1 modulo a half log term

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

a new order determination technique for arma processes it is generally recognized that when performing spectral estimation via the modeling approach autoregressive moving average arma models can provide estimates of higher resolution than auto gressive models alone 1 2 however there is lacking a systematic method for determining the orders of an arma process from its output of the very few order determination techniques in existence most are trial and error based the akaike information criterion for arma processes is one such example this paper presents a new method that determines the arma orders in one pass it is a modification of the method of chow 6 and uses the gram schmidt orthonormalization procedure to determine linear dependency of the columns in an order determination the theoretical development is verified by simulation which also includes results on order determination from output sequences alone

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

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

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

a sufficient integral condition for local regularity of solutions to the surface growth model abstract the surface growth model u t u x x x x x x u x 2 0 is a one dimensional fourth order equation which shares a number of striking similarities with the three dimensional incompressible navier stokes equations including the results regarding existence and uniqueness of solutions and the partial regularity theory here we show that a weak solution of this equation is smooth on a space time cylinder q if the serrin condition u x l q l q q is satisfied where q q 1 are such that either 1 q 4 q 1 or 1 q 4 q 1 q

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

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

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

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

arma order estimation via matrix perturbation theory using matrix perturbation theory we obtain the statistical distribution of the smallest eigenvalue of the hankel matrix built upon the estimated covariances under the hypothesis that the corresponding exact hankel matrix possesses one single zero eigenvalue this allows us to develop and justify a new order determination scheme whose performance appears similar to those based on likelihood maximization but with a much lower computational complexity

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

remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations abstract in this paper we are concerned with the partial regularity for suitable weak solutions of the tri dimensional magnetohydrodynamic equations with the help of the de giorgi iteration method we obtain the results proved by he and xin c he z xin partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations j funct anal 227 2005 113 152 namely the one dimensional parabolic hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero

on leray s self similar solutions of the navier stokes equations satisfying local energy estimates this paper proves that leray s self similar solutions of the three dimensional navier stokes equations must be trivial under very general assumptions for example if they satisfy local energy estimates

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

a profile decomposition approach to the l infty _t l 3 _x navier stokes regularity criterion in this paper we continue to develop an alternative viewpoint on recent studies of navier stokes regularity in critical spaces a program which was started in the recent work by kenig and koch ann inst h poincare anal non lineaire 28 2 159 187 2011 specifically we prove that strong solutions which remain bounded in the space l 3 mathbb r 3 do not become singular in finite time a known result established by escauriaza et al uspekhi mat nauk 58 2 350 3 44 2003 in the context of suitable weak solutions here we use the method of critical elements which was recently developed by kenig and merle to treat critical dispersive equations our main tool is a profile decomposition for the navier stokes equations in critical besov spaces which we develop here as a byproduct of this tool assuming a singularity producing initial datum for navier stokes exists in a critical lebesgue or besov space we show there is one with minimal norm generalizing a result of rusin and sverak j funct anal 260 3 879 891 2011

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

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

pressure representation and boundary regularity of the navier stokes equations with slip boundary condition abstract we first represent the pressure in terms of the velocity in r 3 using this representation we prove that a solution to the navier stokes equations is in l r 3 0 under the critical assumption that u l loc r r 3 r 2 r 1 with r 3 while for r 3 the smallness is required in h j choe boundary regularity of weak solutions of the navier stokes equations j differential equations 149 2 1998 211 247 a boundary l estimate for the solution is derived if the pressure on the boundary is bounded in our work we remove the boundedness assumption of the pressure here our estimate is local indeed employing moser type iteration and the reverse holder inequality we find an integral estimate for l norm of u

hypersurfaces of extremal slope the hypersurface of extremal slope consists of extrema of the gradient magnitude along integral curves of the gradient vector field of a smooth function we study the singularities and other geometric features of hypersurfaces of extremal slope associated with generic functions and with one parameter families of functions representing generic solutions to the heat equation given a smooth function g on an open subset r n let denote the locus of points in which the squared gradient magnitude along the integral curves of the gradient vector field associated with g attains an extremum for generic functions g is a hypersurface in with isolated singular points we are interested in the following questions what types of singular points and other geometric features can have for generically chosen functions g what types of transitions can arise on for generic one parameter families of functions and for one parameter families representing generic solutions to the heat equation here generic means for an open and dense subset of the space of smooth functions or one parameter families of smooth functions or solutions to the heat equation the present study extends the work in 12 on curves of extremal slope in the plane to higher dimensions the original motivation for the work in 12 was to study the possible evolutions of canny edges 4 in scale space given a generic two dimensional signal g for example the intensity function of a computer image it is time for some definitions given a smooth function

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

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