new dissipated energy and partial regularity for thin film equations abstract we study the thin film equation h t h n h x x x x and prove that the energy h p h x 2 d x dissipates for suitable weak solutions when n p lies in the whole domain e defined by laugesen which extends partially the corresponding results of laugesen chugunova and taranets and then improve some previous results of partial regularity by chou and the first author

parallel dynamics of disordered ising spin systems on finitely connected directed random graphs with arbitrary degree distributions we study the stochastic parallel dynamics of ising spin systems defined on finitely connected directed random graphs with arbitrary degree distributions using generating functional analysis for fully asymmetric graphs the dynamics of the system can be completely solved due to the asymptotic absence of loops for arbitrary graph symmetry we solve the dynamics exactly for the first few time steps and we construct approximate stationary solutions

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

remarks on the singular set of suitable weak solutions for the three dimensional navier stokes equations abstract in this study s denotes the possible interior singular set of suitable weak solutions for the three dimensional navier stokes equations we improve the known upper box counting dimension of this set from 360 277 1 300 given by 24 to 975 758 1 286 we also show that s r log e r 0 0 27 113 which extends the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor given by choe and lewis 3 j funct anal 175 348 369 2000 and by choe and yang 4 comm math phys 336 171 198 2015 the proof is inspired by a new e regularity criterion which was proved by guevara and phuc 7 calc var 56 68 2017

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

the minkowski dimension of interior singular points in the incompressible navier stokes equations abstract we study the possible interior singular points of suitable weak solutions to the three dimensional incompressible navier stokes equations we present an improved parabolic upper minkowski dimension of the possible singular set which is bounded by 95 63 the result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty

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

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

existence of suitable weak solutions to the navier stokes equations in time varying domains abstract we consider suitable weak solutions to the navier stokes equations in time varying domains we develop schauder theory for the approximate stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary existence of suitable weak solutions in time varying domains follows from compactness in lebesgue and sobolev spaces

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

regularity of solutions to axisymmetric navier stokes equations with a slightly supercritical condition abstract consider an axisymmetric suitable weak solution of 3d incompressible navier stokes equations with nontrivial swirl v v r e r v e v z e z let z denote the axis of symmetry and r be the distance to the z axis if the solution satisfies a slightly supercritical assumption that is v c ln ln r r for 0 0 028 when r is small then we prove that v is regular this extends the results in 6 16 18 where regularities under critical assumptions such as v c r were proven as a useful tool in the proof of our main result an upper bound estimate to the fundamental solution of the parabolic equation with a critical drift term will be given in the last part of this paper

a regularity criterion at one scale without pressure for suitable weak solutions to the navier stokes equations abstract in this paper we continue our work in 15 to derive e regularity criteria at one scale without pressure for suitable weak solutions to the navier stokes equations we establish an e regularity criterion below of suitable weak solutions for any 0 q 1 u 5 2 d x d t e as an application we extend the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor

partial regularity for the navier stokes equations with a force in a morrey space abstract in the paper we address the partial regularity of solutions of the navier stokes system earlier we have proved that the one dimensional parabolic hausdorff measure of the singular set is zero under the assumption that the force f belongs locally to l 5 3 here we prove the same statement under a more general assumption that the morrey norm in l 5 7 10 7 of the force is sufficiently small we do so by establishing a fractional integration theorem using the morrey spaces and by a suitable iteration using a localized version of the morrey norm

hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations we derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations combining biot savart law interpolation inequalities and the local energy inequality we prove a partial regularity theorem for suitable weak solutions furthermore we obtain an improved estimate for the logarithmic hausdorff dimension of the singular set of suitable weak solutions

global strong solutions to incompressible ericksen leslie system in r3 abstract in this paper we consider the cauchy problem to the ericksen leslie system of liquid crystals in r 3 global well posedness of strong solutions are obtained under the condition that the product of u 0 2 d 0 2 and u 0 2 2 d 0 2 is suitably small this result can be viewed as a supplement to the local existence and blow up criteria recently established in hong et al 2014

a criterion for uniqueness of lagrangian trajectories for weak solutions of the 3d navier stokes equations foias guillope temam showed in 1985 that for a given weak solution u in l infty 0 t l 2 cap l 2 0 t h 1 of the three dimensional navier stokes equations on a domain one can define a trajectory mapping phi omega times 0 t rightarrow omega that gives a consistent choice of trajectory through each initial condition a in omega xi a t phi a t and that respects the volume preserving property one would expect for smooth flows the uniqueness of this mapping is guaranteed by the theory of renormalised solutions of non smooth odes due to diperna lions

optimization problems and replica symmetry breaking in finite connectivity spin glasses a formalism capable of handling the first step of hierarchical replica symmetry breaking rsb in finite connectivity models is introduced the emerging order parameter is claimed to be a probability distribution over the space of field distributions or equivalently magnetization distributions inside the cluster of states the approach is shown to coincide with previous works in the replica symmetric case and in the two limiting cases m 0 and 1 where m is parisi s break point as an application to the study of optimization problems the gs properties of the random 3 satisfiability problem are investigated and we present a first rsb solution improving replica symmetric results

boundary regularity criteria for the 6d steady navier stokes and mhd equations abstract it is shown in this paper that suitable weak solutions to the 6d steady incompressible navier stokes and mhd equations are holder continuous near boundary provided that either r 3 b r u x 3 d x or r 2 b r u x 2 d x is sufficiently small which implies that the 2d hausdorff measure of the set of singular points near the boundary is zero this generalizes recent interior regularity results by dong strain 5

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

a tamed 3d navier stokes equation in uniform c2 domains abstract in this paper we analyze a tamed 3d navier stokes equation in uniform c 2 domains not necessarily bounded which obeys the scaling invariance principle and prove the existence and uniqueness of strong solutions to this tamed equation in particular if there exists a bounded solution to the classical 3d navier stokes equation then this solution satisfies our tamed equation moreover the existence of a global attractor for the tamed equation in bounded domains is also proved as a simple application we obtain that the set of all initial values for which the classical navier stokes equation admits a bounded strong solution is open in h 2

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

navier stokes equations with lower bounds on the pressure we prove that weak solutions of the three dimensional incompressible navier stokes equations are smooth if the negative part of the pressure is controlled or if the positive part of the quantity v 2 2p is controlled

analytic solution of attractor neural networks on scale free graphs we study the influence of network topology on retrieval properties of recurrent neural networks using replica techniques for dilute systems the theory is presented for a network with an arbitrary degree distribution p k and applied to power law distributions p k k i e to neural networks on scale free graphs a bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase using a population dynamics algorithm the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram it is shown that there is an enhancement of the retrieval properties compared with a poissonian random graph we compare our findings with simulations

local regularity of the navier stokes equations near the curved boundary abstract we present some regularity conditions for suitable weak solutions of the navier stokes equations near the curved boundary of a sufficiently smooth domain our extend the work that was results established near a flat boundary by gustafson kang and tsai 2006 6

finite connectivity attractor neural networks we study a family of diluted attractor neural networks with a finite average number of symmetric connections per neuron as in finite connectivity spin glasses their equilibrium properties are described by order parameter functions for which we derive an integral equation in replica symmetric approximation a bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin glass transition lines in the phase diagram the line separating the retrieval phase from the spin glass phase is calculated at zero temperature all phase transitions are found to be continuous

partial regularity of solutions to the four dimensional navier stokes equations at the first blow up time the solutions of incompressible navier stokes equations in four spatial dimensions are considered we prove that the two dimensional hausdorff measure of the set of singular points at the first blow up time is equal to zero

the brown comenetz dual of the k 2 local sphere at the prime 3 abstract we calculate the homotopy type of the brown comenetz dual i 2 of the k 2 local sphere at the prime 3 and show that there is an equivalence in the k 2 local category between i 2 and a smash product of the determinant twisted sphere and an exotic element p in the picard group we give a characterization of p as well a secondary aim of the paper is to extend our library of calculations in the k 2 local category

a resolution of the k 2 local sphere at the prime 3 we develop a framework for displaying the stable homotopy theory of the sphere at least after localization at the second morava k theory k 2 at the prime 3 we write the spectrum lk 2 s 0 as the inverse limit of a tower of fibrations with four layers the successive fibers are of the form e hf where f is a finite subgroup of the morava stabilizer group and e2 is the second morava or lubin tate homology theory we give explicit calculation of the homotopy groups of these fibers the case n 2a tp 3 represents the edge of our current knowledge n 1 is classical and at n 2 the prime 3 is the largest prime where the morava stabilizer group has a p torsion subgroup so that the homotopy theory is not entirely algebraic the problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics a period of calculation beginning with serre s computation of the cohomology of eilenberg maclane spaces and the advent of the adams spectral sequence culminated in the late 1970s with the work of miller ravenel and wilson on periodic phenomena in the homotopy groups of spheres and ravenel s nilpotence conjectures the solutions to most of these conjectures by devinatz hopkins and smith in the middle 1980s established the primacy of the chromatic point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory computations passed from center stage to some extent although there has been steady work in the wings most notably by shimomura and his coworkers and ravenel and more lately by hopkins and

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

on the number of singular points of weak solutions to the navier stokes equations we consider a suitable weak solution to the three dimensional navier stokes equations in the space time cylinder 0 t let be the set of singular points for this solution and t x t for a given open subset and for a given moment of time t 0 t we obtain an upper bound for the number of points of the set t 2001 john wiley sons inc

metastable configurations of spin models on random graphs one flip stable configurations of an ising model on a random graph with fluctuating connectivity are examined in order to perform the quenched average of the number of stable configurations we introduce a global order parameter function with two arguments the analytical results are compared with numerical simulations

on partial regularity problem for 3d boussinesq equations abstract in this paper we study the partial regularity of the solutions for the three dimensional boussinesq equations we first prove a criterion of local holder continuous of the suitable weak solutions of the boussinesq equations and show that one dimensional hausdorff measure of the singular point set is zero secondly we present a local uniform gradient estimate on the suitable weak solutions and assert that the local behavior of the solution can be dominated by some scaled quantities such as the scaled local l 3 norm of the velocity besides when the initial data v 0 and 0 decay sufficiently rapidly at the distribution of the regular point set of the suitable weak solutions is also considered based on it one can find that mhd equations are more similar to navier stokes equations than boussinesq equations finally we give a local regularity criterion of the suitable weak solutions near the boundary

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

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

slowly evolving random graphs ii adaptive geometry in finite connectivity hopfield models we present an analytically solvable random graph model in which the connections between the nodes can evolve in time adiabatically slowly compared to the dynamics of the nodes we apply the formalism to finite connectivity attractor neural network hopfield models and show that due to the minimization of the frustration effects the retrieval region of the phase diagram can be significantly enlarged moreover the fraction of misaligned spins is reduced by this effect and is smaller than that in the infinite connectivity regime the main cause of this difference is found to be the non zero fraction of sites with vanishing local field when the connectivity is finite

a liouville theorem for the axially symmetric navier stokes equations let v x t vrer v e vzez be a solution to the three dimensional incompressible axially symmetric navier stokes equations denote by b vrer vzez the radial axial vector field under a general scaling invariant condition on b we prove that the quantity rv is holder continuous at r 0 t 0 as an application we prove that the ancient weak solutions of axi symmetric navier stokes equations must be zero which was raised by koch nadirashvili seregin and sverak 2009 in 15 and seregin and sverak 2009 in 26 as a conjecture under the condition that b l 0 t bmo 1 as another application we prove that if b l 0 t bmo 1 then v is regular

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

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

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

the parallel dynamics of a dilute symmetric hebb rule network the authors find that the dynamics of a dilute network of symmetric hebb rule synapses with parallel evolution the model considered by patrick and zagrebnov 1990 are considerably more complicated than that paper predicted the number of parameters is not constant but equal to the square of the number of time steps taken and they are generated by exact but increasingly complicated relations a replica symmetric calculation of long term behaviour gives an estimate of alpha c approximately 1 26 the calculations also describe parallel evolution of fully connected spin glasses

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

centralizers of elementary abelian p subgroups and mod p cohomology of profinite groups 1 1 let g be a profinite group and p be a fixed prime in this paper we will be concerned with h c g fp the continuous cohomology of g with coefficients in the trivial module fp we will abbreviate h c g fp by h g fp or simply by h g if p is understood from the context we recall that if g is the inverse limit of finite groups gi then h g colim h gi throughout this paper we will assume that h g is finitely generated as fp algebra by work of lazard la it is known that this holds for many interesting groups for example for profinite p analytic groups like gl n zp the general linear groups over the p adic integers in case h g is finitely generated as fp algebra quillen has shown q1 that there are only finitely many conjugacy classes of elementary abelian p subgroups of g i e groups isomorphic to z p for some natural number n in other words the following category a g is equivalent to a finite category objects of a g are all elementary abelian p subgroups of g and if e1 and e2 are elementary abelian p subgroups of g then the set of morphisms from e1 to e2 in a g consists precisely of those homomorphisms e1 e2 of abelian groups for which there exists an element g g with e geg 1 e e1 the category a g plays an important role both in quillen s results and in the work presented here this category entered into quillen s work as follows the assignment e 7 h e extends to a functor from the opposite category a g op to graded fp algebras and the restriction homomorphisms h g h e for e running through the elementary abelian p subgroups of g induce a canonical map of algebras q h g lima g op h e

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

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

boundary partial regularity for the high dimensional navier stokes equations abstract we consider suitable weak solutions of the incompressible navier stokes equations in two cases the 4d time dependent case and the 6d stationary case we prove that up to the boundary the two dimensional hausdorff measure of the set of singular points is equal to zero in both cases

limiting case for the regularity criterion to the 3 d magneto hydrodynamics equations abstract we prove that any weak solution u b of three dimensional incompressible magneto hydrodynamics equations is regular if u l 0 t l 3 r 3 and b l 0 t vmo 1 r 3 the proof is based on the blow up analysis and backward uniqueness for the parabolic operator developed by escauriaza seregin sverak

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

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

regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space abstract we study three dimensional incompressible magnetohydrodynamic equations in bounded domains or a half space we present new regularity criteria of weak solutions a pair of weak solutions u b become regular if u satisfies serrin s type conditions when we consider no slip or slip boundary conditions for the velocity field u and slip boundary conditions for the magnetic field b in either bounded domains or a half space in addition in the case of a half space with no slip boundary conditions for u and slip boundary conditions for b we demonstrate that if tangential components of u and normal component of b satisfy serrin s type conditions then a pair of weak solutions u b become regular

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

a refinement of the local serrin type regularity criterion for a suitable weak solution to the navier stokes equations we formulate a new criterion for regularity of a suitable weak solution v to the navier stokes equations at the space time point x0 t0 the criterion imposes a serrin type integrability condition on v only in a backward neighbourhood of x0 t0 intersected with the exterior of a certain space time paraboloid with vertex at point x0 t0 we make no special assumptions on the solution in the interior of the paraboloid

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

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

on partial regularity for weak solutions to the navier stokes equations in this paper we study the partial regularity of the general weak solution u l 0 t l2 l2 0 t h1 to the navier stokes equations which include the well known leray hopf weak solutions it is shown that there is a absolute constant e such that for the weak solution u if either the scaled local lq 1 q 2 norm of the gradient of the solution or the scaled local lq 1 q 103 norm of u is less than e then u is locally bounded this implies that the one dimensional hausdorff measure is zero for the possible singular point set which extends the corresponding result due to caffarelli et al comm pure appl math 35 1982 717 to more general weak solution

on finite time singularity and global regularity of an axisymmetric model for the 3d euler equations in comm pure appl math 62 4 502 564 2009 hou and lei proposed a 3d model for the axisymmetric incompressible euler and navier stokes equations with swirl this model shares a number of properties of the 3d incompressible euler and navier stokes equations in this paper we prove that the 3d inviscid model with an appropriate neumann robin or dirichlet robin boundary condition will develop a finite time singularity in an axisymmetric domain we also provide numerical confirmation for our finite time blowup results we further demonstrate that the energy of the blowup solution is bounded up to the singularity time and the blowup mechanism for the mixed dirichlet robin boundary condition is essentially the same as that for the energy conserving homogeneous dirichlet boundary condition finally we prove that the 3d inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition both the analysis and the results we obtain here improve the previous work in a rectangular domain by hou et al adv math 230 607 641 2012 in several respects

boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations abstract we present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three we prove that suitable weak solutions are holder continuous near boundary provided that either the scaled l x t p q norm of the velocity with 3 p 2 q 2 2 q or the scaled l x t p q norm of the vorticity with 3 p 2 q 3 2 q are sufficiently small near the boundary

new results of general n dimensional incompressible navier stokes equations let u u x t u0 represent the global strong weak solutions of the cauchy problems for the general n dimensional incompressible navier stokes equations ut e 2nut u u u p u 0 u 0 u x 0 u0 x u0 0 where the spatial dimension n 2 0 e 1 is a constant and 1 2 n t rn is a constant vector note that if e 0 and 0 then the problem reduces to the traditional navier stokes equations let the scalar functions ij c2 rn l1 rn ij xj l1 rn h2n rn i j 1 2 n define the real vector valued functions i i1 i2 in t let the initial data u0 x j 1n 1j xj x j 1n 2j xj x j 1n nj xj x t satisfy i 1n j 1n 2 ij xi xj x 0 then limt 1 t 1 n 2 rn u x t 2 e nu x t 2 dx 1 2 n 2 n 214 k 1n rn k x dx 2 for any integer m 1 we will establish the following limit limt 1 t 2m 1 n 2 rn mu x t 2 e m nu x t 2 dx 1 2 n 2 n 2 14 2m 1 l 12m 2l n k 1n rn k x dx 2 this kind of exact limit will have great influence on the hausdorff dimension of the global attractor of the model equations

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

partial regularity of suitable weak solutions to the fractional navier stokes equations in this paper we study the partial regularity of fractional navier stokes equations in mathbb r 3 times 0 infty with 3 4 s 1 we show that the suitable weak solution is regular away from a relatively closed singular set whose 5 4s dimentional hausdorff measure is zero the result is a generalization of the partial regularity for the classical navier stokes system in caffarelli et al commun pure appl math 35 771 831 1982

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

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

analogue neural networks on correlated random graphs we consider a generalization of the hopfield model where the entries of patterns are gaussian and diluted we focus on the high storage regime and we investigate analytically the topological properties of the emergent network as well as the thermodynamic properties of the model we find that by properly tuning the dilution in the pattern entries the network can recover different topological regimes characterized by peculiar scalings of the average coordination number with respect to the system size the structure is also shown to exhibit a large degree of cliquishness even when very sparse moreover we obtain explicitly the replica symmetric free energy and the self consistency equations for the overlaps order parameters of the theory which turn out to be classical weighted sums of sub overlaps defined on all possible sub graphs finally a study of criticality is performed through a small overlap expansion of the self consistencies and through a whole fluctuation theory developed for their rescaled correlations both approaches show that the net effect of dilution in pattern entries is to rescale the critical noise level at which ergodicity breaks down

on the partial regularity of a 3d model of the navier stokes equations we study the partial regularity of a 3d model of the incompressible navier stokes equations which was recently introduced by the authors in 11 this model is derived for axisymmetric flows with swirl using a set of new variables it preserves almost all the properties of the full 3d euler or navier stokes equations except for the convection term which is neglected in the model if we add the convection term back to our model we would recover the full navier stokes equations in 11 we presented numerical evidence which seems to support that the 3d model develops finite time singularities while the corresponding solution of the 3d navier stokes equations remains smooth this suggests that the convection term play an essential role in stabilizing the nonlinear vortex stretching term in this paper we prove that for any suitable weak solution of the 3d model in an open set in space time the one dimensional hausdorff measure of the associated singular set is zero the partial regularity result of this paper is an analogue of the caffarelli kohn nirenberg theory for the 3d navier stokes equations

finitely connected vector spin systems with random matrix interactions we use finite connectivity equilibrium replica theory to solve models of finitely connected unit length vectorial spins with random pair interactions which are of the orthogonal matrix type finitely connected spin models although still of a mean field nature can be regarded as a convenient level of description in between fully connected and finite dimensional ones since the spins are continuous and the connectivity c remains finite in the thermodynamic limit the replica symmetric order parameter is a functional the general theory is developed for arbitrary values of the dimension d of the spins and arbitrary choices of the ensemble of random orthogonal matrices we calculate phase diagrams and the values of moments of the order parameter explicitly for d 2 finitely connected xy spins with random chiral interactions and for d 3 finitely connected classical heisenberg spins with random chiral interactions numerical simulations are shown to support our predictions quite satisfactorily

dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl in this paper we study the dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl to this purpose we propose a new one dimensional model that approximates the navier stokes equations along the symmetry axis an important property of this one dimensional model is that one can construct from its solutions a family of exact solutions of the three dimensionafinal navier stokes equations the nonlinear structure of the one dimensional model has some very interesting properties on one hand it can lead to tremendous dynamic growth of the solution within a short time on the other hand it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time by exploiting this special nonlinear structure we prove the global regularity of the three dimensional navier stokes equations for a family of initial data whose solutions can lead to large dynamic growth but yet have global smooth solutions