on the pressure regularity criterion of the 3d navier stokes equations abstract in the study of regularity criteria for the weak solutions of the 3d navier stokes equations an improved regularity criterion is obtained more precisely it is proved that if the pressure satisfies the critical growth condition x t l 2 2 r 0 t b r r 3 for 1 r 1 then the weak solution u x t is regular on 0 t the finding is mainly based on the innovative function decomposition methods together with besov space techniques

interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations we present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three a suitable weak solution is regular near an interior point z if the scaled lx tp q norm of the velocity with 1 3 p 2 q 2 1 q is sufficiently small near z and if the scaled lx tl m norm of the magnetic field with 1 3 l 2 m 2 1 m is bounded near z similar results are also obtained for the vorticity and for the gradient of the vorticity furthermore with the aid of the regularity criteria we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations

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

p systems with minimal insertion and deletion in this paper we consider insertion deletion p systems with priority of deletion over insertion we show that such systems with one symbol context free insertion and deletion rules are able to generate parikh sets of all recursively enumerable languages psre if a one symbol one sided context is added to the insertion or deletion rules then all recursively enumerable languages can be generated the same result holds if a deletion of two symbols is permitted we also show that the priority relation is very important and in its absence the corresponding class of p systems is strictly included in the family of matrix languages mat

the importance of being discrete life always wins on the surface many systems in chemistry biology finance and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components in the past the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions it turns out that to the contrary taking into account the actual individual discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior in fact we find that in conditions in which the continuum approach would predict the extinction of all of the population respectively the vanishing of the invested capital or the concentration of a chemical substance etc the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development in particular it is found that in two dimensions life the localized proliferating phase always prevails

discrete opinion dynamics on networks based on social influence a model of opinion dynamics based on social influence on networks was studied the opinion of each agent can have integer values i 1 2 i and opinion exchanges are restricted to connected agents it was found that for any i 2 and self confidence parameter 0 u 1 when u is a degree independent constant the weighted proportion qi of the population that hold a given opinion i is a martingale and the fraction qi of opinion i will gradually converge to qi the tendency can slow down with the increase of degree assortativity of networks when u is degree dependent qi does not possess the martingale property however qi still converges to it in both cases for a finite network the states of all agents will finally reach consensus further if there exist stubborn persons in the population whose opinions do not change over time it was found that for degree independent constant u both qi and qi will converge to fixed proportions which only depend on the distribution of initial obstinate persons and naturally the final equilibrium state will be the coexistence of diverse opinions held by the stubborn people the analytical results were verified by numerical simulations on barabasi albert ba networks the model highlights the influence of high degree agents on the final consensus or coexistence state and captures some realistic features of the diffusion of opinions in social networks

killer geometries in competing species dynamics we discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species the model implements local fights among small group of individuals and a synchronous random walk on a 2d lattice the faith of the system i e the spreading or disappearance of the species is determined by their initial density and fight frequency the initial density of the emergent species has to be higher than a critical threshold for total spreading but this value depends in a non trivial way on the fight frequency below the threshold any better adapted species disappears showing that a qualitative advantage is not enough for a minority to win no strategy is involved but spatial organization turns out to be crucial for instance at minority densities of zero measure some very rare local geometries which occur by chance are found to be killer geometries once set they lead with high probability to the total destruction of the preexisting majority species the occurrence rate of these killer geometries is a function of the system size this model may apply to a large spectrum of competing groups like smoker non smoker opinion forming diffusion of innovation setting of industrial standards species evolution epidemic spreading and cancer growth

dynamical analysis of a iwsr rumor spreading model with considering the self growth mechanism and indiscernible degree abstract propagation of rumor as one of the most important aspects of spreading dynamics on networks which is often accompanied by the self growth of individuals and indiscernible degree of events and the analysis of this phenomenon can help us to better understand the information dissemination in the social network here with the wiseman supplemented to the classical ignorant spreader stifler i s r rumor spreading model a modified ignorant wiseman spreader stifler i w s r model is established with self growth mechanism by considering indiscernible degree of events by mean field equations of i w s r model the spreading threshold c is calculated to describe the theory of steady state dynamics of the rumor spreading numerical simulation is performed to assess the influence of self growth of individual and indiscernible degree of events on dynamic spreading process the results show that when the spreading rate is smaller than the threshold c the rumor will not spread in the population the stronger the self growth ability of the individual is the smaller the range of rumor will be the higher the indiscernible degree of events is the greater influence of the rumor these obtained findings in the present study could not only elaborate our understandings of spreading dynamics in network but also provide us with an insight to develop effective strategies of inhibiting rumor propagation

effects of degree biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks we introduce a generalized rumor spreading model and analytically investigate the spreading of rumors on scale free sf networks in the standard rumor spreading model each node has an infectivity equal to its degree and connectivity is uniform across all links to generalize this model we introduce an infectivity function that determines the number of simultaneous contacts that a given node individual may establish with its connected neighbors and a connectivity strength function csf for the direct link between two connected nodes these lead to a degree biased propagation of rumors for nonlinear functions this generalization is reflected in the infectivity s exponent ensuremath alpha and the csf s exponent ensuremath beta we show that by adjusting exponents ensuremath alpha and ensuremath beta the epidemic threshold can be controlled this feature is absent in the standard rumor spreading model in addition we obtain a critical threshold we show that the critical threshold for our generalized model is greater than that of the standard model on a finite sf network theoretically we show that ensuremath beta ensuremath 1 leads to a maximum spreading of rumors and computation results on different networks verify our theoretical prediction also we show that a smaller ensuremath alpha leads to a larger spreading of rumors our results are interesting since we obtain these results regardless of the network topology and configuration

infrequent social interaction can accelerate the spread of a persuasive idea we study the spread of a persuasive new idea through a population of continuous time random walkers in one dimension the idea spreads via social gatherings involving groups of nearby walkers who act according to a biased majority rule after each gathering the group takes on the new idea if more than a critical fraction 1 2 1 2 of them already hold it otherwise they all reject it the boundary of a domain where the new idea has taken hold expands as a traveling wave in the density of new idea holders our walkers move by levy motion and we compute the wave velocity analytically as a function of the frequency of social gatherings and the exponent of the jump distribution when this distribution is sufficiently heavy tailed then counter to intuition the idea can propagate faster if social gatherings are held less frequently when jumps are truncated a critical gathering frequency can emerge which maximizes propagation velocity we explore our model by simulation confirming our analytical results

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

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 criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor in this paper we provide a sufficient condition in terms of only one of the nine entries of the gradient tensor that is the jacobian matrix of the velocity vector field for the global regularity of strong solutions to the three dimensional navier stokes equations in the whole space as well as for the case of periodic boundary conditions

a proof of onsager s conjecture for any 1 3 we construct weak solutions to the 3d incompressible euler equations in the class c tc x that have nonempty compact support in time on r t 3 and therefore fail to conserve the total kinetic energy this result together with the proof of energy conservation for 1 3 due to eyink and constantin e titi solves onsager s conjecture that the exponent 1 3 marks the threshold for conservation of energy for weak solutions in the class l t c x the previous best results were solutions in the classc tc x for 1 5 due to isett and in the class l t 1c x for 1 3 due to buckmaster de lellis szekelyhidi both based on the method of convex integration developed for the incompressible euler equations by de lellis szekelyhidi the present proof combines the method of convex integration and a new gluing approximation technique the convex integration part of the proof relies on the mikado flows introduced by daneri szekelyhidi and the framework of estimates developed in the author s previous work

a generalized regularity criterion for 3d navier stokes equations in terms of one velocity component abstract in this article we study the global regularity of the 3d navier stokes equations in terms of one velocity component in particular we establish a new version of regularity criterion in the framework of the anisotropic lebesgue space it is a generalization of the notable work of c cao and e s titi 2011 3

who is left who is right abstract we propose a model to simulate the process of opinion formation in a society where the traditional division into leftists and rightists is not enough any longer the model is based on the ashkin teller idea to assign two ising spins to each site agent the two states up and down of the first spin represent two possible opinions on any question in an economic area and the two states of the second spin in a personal area we assume that the mechanisms of opinion formation in both of these areas are different the interaction between them is introduced in the form of a tolerance factor it has been found among others that the formation of consensus between groups of agents which differ only in the economic area is quite simple whereas if they differ in the personal area no consensus is possible

a strong regularity result for parabolic equations we consider a parabolic equation with a drift term u b u ut 0 under the condition divb 0 we prove that solutions possess dramatically better regularity than those provided by standard theory for example we prove continuity of solutions when not even boundedness is expected

social paradoxes of majority rule voting and renormalization group real space renormalization group ideas are used to study a voting problem in political science a model to construct self directed pyramidal structures from bottom up to the top is presented using majority rules it is shown that a minority and even a majority can be systematically self eliminated from top leadership provided the hierarchy has a minimal number of levels in some cases 70 of the population is found to have zero representation after six hierarchical levels results are discussed with respect to the internal operation of political organizations

logarithmically improved regularity criteria for the navier stokes equations in lorentz spaces abstract in this paper we investigate three dimensional incompressible navier stokes equations we establish some logarithmically improved regularity criteria in term of the lorentz spaces to the navier stokes equations

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

spatial evolutionary prisoner s dilemma game with three strategies and external constraints the emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner s dilemma game in which the players are located on the sites of cubic lattices for dimensions d51 2 and 3 each player can choose one of the three following strategies cooperation c defection d or tit for tat t during the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff moreover an external constraint imposes that the players always cooperate with probability p the stationary state phase diagram is computed by both using generalized mean field approximations and monte carlo simulations nonequilibrium second order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class it is shown that externally forcing the collaboration does not always produce the desired result

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

high dimensionality and h principle in pde in this note we present an analyst s point of view on the nash kuiper theorem and in particular highlight the very close connection to turbulence a paradigm example of a high dimensional phenomenon our aim is to explain recent applications of nash s ideas in connection with the incompressible euler equations and onsager s famous conjecture on the energy dissipation in 3d turbulence

modelling rumors the no plane pentagon french hoax case abstract the recent astonishing wide adhesion of french people to the rumor claiming no plane did crash on the pentagon on september 11 is given a generic explanation in terms of a model of minority opinion spreading using a majority rule reaction diffusion dynamics a rumor is shown to invade for sure a social group provided it fulfills simultaneously two criteria first it must initiate with a support beyond some critical threshold which however turns out to be always very low then it has to be consistent with some larger collective social paradigm of the group otherwise it just dies out both conditions were satisfied in the french case with the associated book sold at more than 200 000 copies in just a few days the rumor was stopped by the firm stand of most newspaper editors stating it is nonsense such an incredible social dynamics is shown to result naturally from an open and free public debate among friends and colleagues each one searching for the truth sincerely on a free will basis and without individual biases the polarization process appears also to be very quick in agreement with reality it is a very strong anti democratic reversal of opinion although made quite democratically the model may apply to a large range of rumors

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

regularity conditions of 3d navier stokes flow in terms of large spectral components abstract we develop ladyzhenskaya prodi serrin type spectral regularity criteria for 3d incompressible navier stokes equations in a torus concretely for any n 0 let w n be the sum of all spectral components of the velocity fields whose wave numbers k i n for all i 1 2 3 then we show that for any n 0 the finiteness of the serrin type norm of w n implies the regularity of the flow it implies that if the flow breaks down in a finite time the energy of the velocity fields cascades down to the arbitrarily large spectral components of w n and corresponding energy spectrum in some sense roughly decays slower than 2

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

on the energy of inviscid singular flows abstract it is known that the energy of a weak solution to the euler equation is conserved if it is slightly more regular than the besov space b 3 1 3 when the singular set of the solution is or belongs to a smooth manifold we derive various l p space regularity criteria dimensionally equivalent to the critical one in particular if the singular set is a hypersurface the energy of u is conserved provided the one sided non tangential limits to the surface exist and the non tangential maximal function is l 3 integrable while the maximal function of the pressure is l 3 2 integrable the results directly apply to prove energy conservation of the classical vortex sheets in both 2d and 3d at least in those cases where the energy is finite

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

contextual insertions deletions and computability we investigate two generalizations of insertion and deletion of words that have recently become of interest in the context of molecular computing given a pair of words x y called a context the x y contextual insertion of a wordvinto a worduis performed as follows for each occurrence ofxyas a subword inu we include in the result of the contextual insertion the words obtained by insertingvintou betweenxandy the x y contextual deletion operation is defined in a similar way we study closure properties of the chomsky families under the defined operations contextual ins closed and del closed languages and decidability of existence of solutions to equations involving these operations moreover we prove that every turing machine can be simulated by a system based entirely on contextual insertions and deletions

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

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

a modified galam s model for word of mouth information exchange abstract in this paper we analyze the stochastic model proposed by galam in s galam modelling rumors the no plane pentagon french hoax case physica a 320 2003 571 580 for information spreading in a word of mouth process among agents based on a majority rule using the communications rules among agents defined in the above reference we first perform simulations of the word of mouth process and compare the results with the theoretical values predicted by galam s model some dissimilarities arise in particular when a small number of agents is considered we find motivations for these dissimilarities and suggest some enhancements by introducing a new parameter dependent model we propose a modified galam s scheme which is asymptotically coincident with the original model in the above reference furthermore for relatively small values of the parameter we provide a numerical experience proving that the modified model often outperforms the original one

discrete opinion models as a limit case of the coda model abstract opinion dynamics models can be for most of them divided between discrete and continuous they are used in different circumstances and the relationship between them is not clear here we will explore the relationship between a model where choices are discrete but opinions are a continuous function the continuous opinions and discrete actions coda model and traditional discrete models i will show that when coda is altered to include reasoning about the influence one agent can have on its own neighbors agreement and disagreement no longer have the same importance the limit when an agent considers itself to be more and more influent will be studied and we will see that one recovers discrete dynamics like those of the voter model in that limit

a regularity criterion for the navier stokes equation involving only the middle eigenvalue of the strain tensor this manuscript derives an evolution equation for the symmetric part of the gradient of the velocity the strain tensor in the incompressible navier stokes equation on mathbb r 3 and proves the existence of l 2 mild solutions to this equation we use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor not on the nonlocal interaction of the strain tensor with the vorticity the resulting identity allows us to prove a new family of scale critical necessary and sufficient conditions for the blow up of a solution at some finite time t max infty which depend only on the history of the positive part of the second eigenvalue of the strain matrix since this matrix is trace free this severely restricts the geometry of any finite time blow up this regularity criterion provides analytic evidence of the numerically observed tendency of the vorticity to align with the eigenvector corresponding to the middle eigenvalue of the strain matrix this regularity criterion also allows us to prove as a corollary a new scale critical one component type regularity criterion for a range of exponents for which there were previously no known critical one component type regularity criteria furthermore our analysis permits us to extend the known time of existence of smooth solutions with fixed initial enstrophy e 0 frac 1 2 left nabla otimes u 0 right l 2 2 by a factor of 4920 75 although the previous constant in the literature was not expected to be close to optimal so this improvement is less drastic than it sounds especially compared with numerical results finally we will prove the existence and stability of blow up for a toy model ode for the strain equation

combined update scheme in the sznajd model we analyze the sznajd opinion formation model where a pair of neighboring individuals sharing the same opinion on a square lattice convinces its six neighbors to adopt their opinions when a fraction of the individuals is updated according to the usual random sequential updating rule asynchronous updating and the other fraction the simultaneous updating synchronous updating this combined updating scheme provides that the bigger the synchronous frequency becomes the more difficult the system reaches a consensus moreover in the thermodynamic limit the system needs only a small fraction of individuals following a different kind of updating rules to present a non consensus state as a final state c 2006 elsevier b v all rights reserved

reaching consensus on rumors abstract an important contribution in sociophysics is the galam s model of rumors spreading this model provides an explanation of rumors spreading in a population and explains some interesting social phenomena such as the diffusion of hoaxes in this paper the model has been reformulated as a markov process highlighting the stochastic nature of the phenomena this formalization allows us to derive conditions for consensus to be reached and for the existence of some interesting phenomena such as the emergence of impasses the proposed formulation allows a deeper and more comprehensive analysis of the diffusion of rumors

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 family of models for schelling binary choices abstract we introduce and study a family of discrete time dynamical systems to model binary choices based on the framework proposed by schelling in 1973 the model we propose uses a gradient like adjustment mechanism by means of a family of smooth maps and allows understanding and analytically studying the phenomena qualitatively described by schelling in particular we investigate existence of steady states and their relation to the equilibria of the static model studied by schelling and we analyze local stability linking several examples and considerations provided by schelling with bifurcation theory we provide examples to confirm the theoretical results and to numerically investigate the possible destabilizations as well as the emergence of coexisting attractors we show the existence of chaos for a particular example

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

effect of initial concentration and spatial heterogeneity of active agent distribution on opinion dynamics abstract we analyze the effect of spatial heterogeneity in the initial spin distribution on spin dynamics in a three state square lattice divided into spatial cells districts in the spirit of the statistical mechanics of social impact we introduce a dominant influence rule dir according to which in a single update step a chosen node adopts the state determined by the influence of its discussion group formed by the node itself and its neighbors within one or more coordination spheres in contrast to models based on some form of majority rule mr a system governed by the dir is easily trapped in a stable non consensus state if all nodes of the discussion group have the same weight of influence to ensure that a consensus in the dir system is necessarily reached we need to put a stochastic process in the update rule further the stochastic dir model is used as a starting point for understanding the effect of spatial heterogeneity of active agent non zero spin distribution on the exit probabilities initially the positive and negative spins active agents are assigned to some nodes with non uniform spatial distributions while the rest of the nodes remain in the state with spin zero uncommitted voters by varying the relative means and skewness of the initial spin distributions we observe critical behaviors of exit probabilities in finite size systems the critical exponents are obtained by monte carlo simulations the results of numerical simulations are discussed in the context of social dynamics

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

characterizations of recursively enumerable languages by means of insertion grammars abstract an insertion grammar is based on pure rules of the form uv uxv the string x is inserted in the context u v a strict subfamily of the context sensitive family is obtained incomparable with the family of linear languages we prove here that each recursively enumerable language can be written as the weak coding of the image by an inverse morphism of a language generated by an insertion grammar with the maximal length of strings u v as above equal to seven this result is rather surprising in view of some closure properties established earlier in the literature some consequences of this result are also stated when also erasing rules of the form uxv uv are present the string x is erased from the context u v then a much easier representation of recursively enumerable languages is obtained as the intersection with v of a language generated by an insertion grammar with erased strings having the maximal length of strings u v as above equal to two

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

the energy measure for the euler and navier stokes equations the potential failure of energy equality for a solution u of the euler or navier stokes equations can be quantified using a so called energy measure the weak limit of the measures u t 2 rm d x as t approaches the first possible blowup time we show that membership of u in certain weak or strong l q l p classes gives a uniform lower bound on the lower local dimension of mathcal e more precisely it implies uniform boundedness of a certain upper s density of mathcal e we also define and give lower bounds on the concentration dimension associated to mathcal e which is the hausdorff dimension of the smallest set on which energy can concentrate both the lower local dimension and the concentration dimension of mathcal e measure the departure from energy equality as an application of our estimates we prove that any solution to the 3 dimensional navier stokes equations which is type i in time must satisfy the energy equality at the first blowup time

on the global regularity of a helical decimated version of the 3d navier stokes equations we study the global regularity for all time and all initial data in h 1 2 o f ar e cently introduced decimated version of the incompressible 3d navier stokes dns equa tions the model is based on a projection of the dynamical evolution of navier stokes ns equations into the subspace where helicity the l 2 scalar product of velocity and vorticity is sign definite the presence of a second beside energy sign definite inviscid conserved quadratic quantity which is equivalent to the h 1 2 sobolev norm allows us to demonstrate global existence and uniqueness of space periodic solutions together with continuity with respect to the initial conditions for this decimated 3d model this is achieved thanks to the establishment of two new estimates for this 3d model which show that the h 1 2 and the time average of the square of the h 3 2 norms of the velocity field remain finite such two additional bounds are known in the spirit of the work of h fujita and t kato arch ration mech anal 16 269 315 1964 rend semin mat univ padova 32 243 260 1962 to be sufficient for showing well posedness for the 3d ns equations furthermore they are di rectly linked to the helicity evolution for the dns model and therefore with a clear physical meaning and consequences

the regularity criterion for 3d navier stokes equations involving one velocity gradient component abstract in this article we establish sufficient conditions for the regularity of solutions of navier stokes equations based on one of the nine entries of the gradient tensor we improve the recent results of c s cao e s titi c s cao e s titi global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor arch ration mech anal 202 2011 919 932 and y zhou m pokorn y zhou m pokorn on the regularity of the solutions of the navier stokes equations via one velocity component nonlinearity 23 2010 1097 1107

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

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

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

regularity for the stationary navier stokes equations in bounded domain let u p be a weak solution of the stationary navier stokes equations in a bounded domain n 5 n if u p satisfy the additional conditions begin gathered text a text left frac u text 2 text 2 p right in l text loc q omega text q in left left frac n 2 infty right right hfill hfill text b text int limits omega nabla u nabla u gamma dx leqq text int limits omega left frac u text 2 text 2 p right text u nabla gamma dx int limits omega fu gamma dx text forall gamma in c 0 infty omega text gamma geqq 0 hfill end gathered they become regular moreover it is proved that every weak solution u p satisfying a with q is regular the existence of such solutions for n 5 has been established in a former paper 3

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

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

on the computational completeness of graph controlled insertion deletion systems with binary sizes abstract a graph controlled insertion deletion gcid system is a regulated extension of an insertion deletion system such a system has several components and each component has some insertion deletion rules the transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule the language of the system is the set of all terminal strings collected in the final component the parameters in the size k n i i m j j of a gcid system denote from left to right the maximum number of components the maximal length of the insertion string the maximal length of the left context for insertion the maximal length of the right context for insertion the last three parameters follow a similar representation with respect to deletion in this paper we discuss the computational completeness of the families of gcid systems of size k 1 i i 1 j j with k 3 5 and for nearly all values of i i j j 0 1 all proofs are based on the simulation of type 0 grammars given in special geffert normal form sgnf the novelty in our proof presentation is that the context free and the non context free rules of the given sgnf grammar are simulated by gcid systems of different sizes and finally we combine them by stitching and overlaying to characterize the recursive enumerable languages this proof presentation greatly simplifies and unifies the proof of such characterization results we also connect some of the obtained gcid simulations to the domain of insertion deletion p systems

regularity of weak solutions and the number of singular points to the 3d simplified nematic liquid crystal system abstract in this paper we investigate the regularity of weak solutions to the cauchy problem of the 3d simplified nematic liquid crystal flows with lower bound on pressure and show that if the negative part of the pressure is controlled or if the positive part of quantity u 2 d 2 2 is controlled then the weak solution u d is global in time smooth we also study the singular points of weak solutions and prove that if a weak solution u d on r 3 0 t satisfies u t l p r 3 d t l p r 3 c t t p 3 2 p for some 3 p where c is a positive constant then the number of singular points is finite

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