remarques sur l apparition de singularites dans les ecoulements euleriens compressible we recall some basic facts about the breakdown of smoothness in general hyperbolic systems and then we prove that the breakdown of smoothness in an euclidean flow comes from an accumulation of vorticity divergence or compression

a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

an extension of the beale kato majda criterion for the euler equations the beale kato majda criterion asserts that smooth solutions to the euler equations remain bounded past t as long as is finite ohgr being the vorticity we show how to replace this by a weaker statement on where j is a frequency localization around

the well posedness of the surface quasi geostrophic equations in the besov morrey spaces abstract in this paper we develop the boundedness of riesz transform in morrey spaces which generalize the corresponding result in the usual l p spaces as an application of this boundedness we establish the well posedness and blow up criterion for the surface quasi geostrophic equations in the framework of besov morrey spaces

analytic results related to magneto hydrodynamic turbulence abstract analytic results are provided for topics related to the magneto hydrodynamic turbulence scaling exponents of general structure functions geometric criteria for singularity formation and dissipative solutions

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

a blowup criterion for the compressible nematic liquid crystal flows in dimension two abstract in this paper we establish a blowup criterion for the two dimensional compressible nematic liquid crystal flows the criterion is given in terms of the density and the gradient of direction field where the later satisfies the serrin type blowup criterion for this result we do not need the initial density to be positive

sharp lp estimates for singular transport equations abstract we prove that l p estimates for a singular transport equation are sharp by building what we call a cascading solution the equation we consider studies the combined effect of multiplying by a bounded function and application of the hilbert transform along the way we prove an invariance result for the hilbert transform which could be of independent interest finally we give an example of a bounded and incompressible velocity field u for which the equation t f u f h f develops sharp l p growth the equations we study are relevant as models in the study of fluid equations as well as in general relativity

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

a blow up criterion of strong solutions to a viscous liquid gas two phase flow model with vacuum in 3d abstract in this paper we get a blow up criterion for the strong solutions to the 3d viscous liquid gas two phase flow model with vacuum more precisely we prove that the bound of l t 1 l x norm of the deformation tensor of velocity gradient controls the possible breakdown of the strong solutions

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

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

the three dimensional euler equations where do we stand the three dimensional euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists while much has been discovered the nature of solutions is still largely a mystery this paper surveys some of the issues such as singularity formation that have cost so much effort in the last 25 years in this light we review the beale kato majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted a different line of endeavour focuses on work concerning the pressure hessian and how it may be used and modelled the euler equations are finally discussed in terms of their membership of a class of general lagrangian evolution equations using hamilton s quaternions these are reformulated in an elegant manner to describe the motion and rotation of fluid particles c 2007 elsevier b v all rights reserved

clustering categorical data based on distance vectors we introduce a novel statistical procedure for clustering categorical data based on hamming distance hd vectors the proposed method is conceptually simple and computationally straightforward because it does not require any specific statistical models or any convergence criteria moreover unlike most currently existing algorithms that compute the class membership or membership probability for every data point at each iteration our algorithm sequentially extracts clusters from the given dataset that is at each iteration our algorithm strives to identify only one cluster which will then be deleted from the dataset at the next iteration this procedure repeats until there are no more significant clusters in the remaining data consequently the number of clusters can be determined automatically by the algorithm as for the identification and extraction of a cluster we first locate the cluster center by using a pearson chi squared type statistic on the basis of hd vectors the partition of the datase

axisymmetric solutions to the 3d euler equations abstract here we consider the 3 d incompressible euler equations with axisymmetric velocity without swirl first we will show that if u 0 c s l 2 s 1 and 0 x c x 1 2 x 2 2 then there exists a unique u c 0 c s that solves the equation this conclusion improves on the related results given by majda a l bertozzi a majda vorticity and incompressible flow in cambridge texts in applied mathematics vol 27 2002 a majda vorticity and the mathematical theory of incompressible fluid flow comm pure appl math 39 1986 s187 s220 and by raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 on the other hand if u 0 l 2 0 l and 0 x 1 2 x 2 2 l then there exists a unique quasilipschitzian solution u c 0 c 1 this improves on the corresponding results due to raymond x saint raymond remarks on axisymmetric solutions of the incompressible euler system comm partial differential equations 19 1994 321 334 and to chae and kim d chae n kim axisymmetric weak solutions of the 3 d euler equations for incompressible fluid flows nonlinear anal 29 12 1997 1393 1404

fourier pade approximations and filtering for spectral simulations of an incompressible boussinesq convection problem in this paper we present rational approximations based on fourier series representation for periodic piecewise analytic functions the well known gibbs phenomenon hampers the convergence of the standard fourier method here for a given set of the fourier coefficients from a periodic piecewise analytic function we define fourier pade galerkin and fourier pade collocation methods by expressing the coefficients for the rational approximations using the fourier data we show that those methods converge exponentially in the smooth region and successfully reduce the gibbs oscillations as the degrees of the denominators and the numerators of the pade approximants increase numerical results are demonstrated in several examples the collocation method is applied as a postprocessing step to the standard pseudospectral simulations for the one dimensional inviscid burgers equation and the two dimensional incompressible inviscid boussinesq convection flow

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

blow up criterion for the compressible magnetohydrodynamic equations with vacuum abstract in this paper the 3 d compressible magnetohydrodynamic mhd equations with initial vacuum or infinite electric conductivity is considered we prove that the l norms of the deformation tensor d u and the absolute temperature control the possible blow up see 18 23 of strong solutions especially for the non resistive mhd system when the magnetic diffusion vanishes this conclusion means that if a solution of the compressible mhd equations is initially regular and loses its regularity at some later time then the formation of singularity must be caused by losing the bound of d u or as the critical time approaches the viscosity coefficients are only restricted by the physical conditions our criterion see 1 17 is similar to 17 for 3 d incompressible euler equations and to 12 for 3 d compressible isentropic navier stokes equations

3d euler equations and ideal mhd mapped to regular systems probing the finite time blowup hypothesis abstract we prove by an explicit construction that solutions to incompressible 3d euler equations defined in the periodic cube 0 l 3 can be mapped bijectively to a new system of equations whose solutions are globally regular we establish that the usual beale kato majda criterion for finite time singularity or blowup of a solution to the 3d euler system is equivalent to a condition on the corresponding regular solution of the new system in the hypothetical case of euler finite time singularity we provide an explicit formula for the blowup time in terms of the regular solution of the new system the new system is amenable to being integrated numerically using similar methods as in euler equations we propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite time singularity problem of euler equations based on the conservation of quantities directly related to energy and circulation the method of mapping to a regular system can be extended to any fluid equation that admits a beale kato majda type of theorem e g 3d navier stokes 2d and 3d magnetohydrodynamics and 1d inviscid burgers we discuss briefly the case of 2d ideal magnetohydrodynamics in order to illustrate the usefulness of the mapping we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1d inviscid burgers equation

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

a probabilistic interpretation and stochastic particle approximations of the 3 dimensional navier stokes equations we develop a probabilistic interpretation of local mild solutions of the three dimensional navier stokes equation in the l p spaces when the initial vorticity field is integrable this is done by associating a generalized nonlinear diffusion of the mckean vlasov type with the solution of the corresponding vortex equation we then construct trajectorial chaotic stochastic particle approximations of this nonlinear process these results provide the first complete proof of convergence of a stochastic vortex method for the navier stokes equation in three dimensions and rectify the algorithm conjectured by esposito and pulvirenti in 1989 our techniques rely on a fine regularity study of the vortex equation in the supercritical l p spaces and on an extension of the classic mckean vlasov model which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

a multi fluid compressible system as the limit of weak solutions of the isentropic compressible navier stokes equations this paper mainly concerns the mathematical justification of a viscous compressible multi fluid model linked to the baer nunziato model used by engineers see for instance ishii thermo fluid dynamic theory of two phase flow eyrolles paris 1975 under a stratification assumption more precisely we show that some approximate finite energy weak solutions of the isentropic compressible navier stokes equations converge on a short time interval to the strong solution of this viscous compressible multi fluid model provided the initial density sequence is uniformly bounded with corresponding young measures which are linear convex combinations of m dirac measures to the authors knowledge this provides in the multidimensional in space case a first positive answer to an open question see hillairet j math fluid mech 9 343 376 2007 with a stratification assumption the proof is based on the weak solutions constructed by desjardins commun partial differ equ 22 5 6 977 1008 1997 and on the existence and uniqueness of a local strong solution for the multi fluid model established by hillairet assuming initial density to be far from vacuum in a first step adapting the ideas from hoff and santos arch ration mech anal 188 509 543 2008 we prove that the sequence of weak solutions built by desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between and coupled with the uniform bound of the density property this allows us to use appropriate defect measures and their nice properties introduced and proved by hillairet aspects interactifs de la m ecanique des fluides phd thesis ens lyon 2005 in order to prove that the young measure associated to the weak limit is the convex combination of m dirac measures finally under a non degeneracy assumption of this combination stratification assumption this provides a multi fluid system using a weak strong uniqueness argument we prove that this convex combination is the one corresponding to the strong solution of the multi fluid model built by hillairet if initial data are equal we will briefly discuss this assumption to complete the paper we also present a blow up criterion for this multi fluid system following huang et al in serrin type criterion for the three dimensional viscous compressible flows arxiv 2010

towards a sufficient criterion for collapse in 3d euler equations abstract a sufficient integral criterion for a blow up solution of the hopf equations the euler equations with zero pressure is found this criterion shows that a certain positive integral quantity blows up in a finite time under specific initial conditions blow up of this quantity means that solution of the hopf equation in 3d cannot be continued in the sobolev space h 2 r 3 for infinite time

sharper criteria for the wave collapse abstract sharper criteria for three dimensional wave collapse described by the nonlinear schrodinger equation nlse are derived the collapse threshold corresponds to the ground state soliton which is known to be unstable thus for nonprefocusing distributions this represents the separatrix between collapsing and noncollapsing sectors numerical results support the theoretical results generalizations of the criteria for the nlse with arbitrary power nonlinearity are also presented

on the global well posedness for the axisymmetric euler equations this paper deals with the global well posedness of the 3d axisymmetric euler equations for initial data lying in critical besov spaces b p 1 1 3 p in this case the bkm criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity

on the dual systems to euler and navier stokes equations in r3 in this paper we consider two model systems of equations obtained by taking the hodge duals of the euler and the navier stokes systems respectively the vorticity formulation of the new system becomes a nonlinear non local scalar equation having similarities to the quasi geostrophic equations we prove the local existence of classical solutions and derive the beale kato majda type of blow up criterion for the dual euler system moreover we actually show that the finite time blow up occurs for generic non trivial initial data for this system for the dual navier stokes equations we construct an explicitly global in time smooth solution for general initial data using the hopf cole transform in the appendix we formulate the euler system and the dual euler system in terms of the clifford algebra

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

blow up criterion for the 3d compressible non isentropic navier stokes equations without thermal conductivity abstract in this paper we prove a blow up criterion in terms of the density and the pressure p for the strong solutions with vacuum to cauchy problem of the 3d compressible non isentropic navier stokes equations without thermal conductivity more precisely we show that the strong solution exists globally if the norm p l 0 t r 3 is bounded from above

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

blow up of solutions for a 1d transport equation with nonlocal velocity and supercritical dissipation we study a 1d transport equation with nonlocal velocity and supercritical dissipation we show that for a certain class of initial data the solution blows up in finite time

evolutions of the momentum density deformation tensor and the nonlocal term of the camassa holm equation abstract methods originally developed to study the finite time blow up problem of the regular solutions of the three dimensional incompressible euler equations are used to investigate the regular solutions of the camassa holm equation we obtain results on the relative behaviors of the momentum density the deformation tensor and the nonlocal term along the trajectories in terms of these behaviors we get new types of asymptotic properties of global solutions blow up criterion and blow up time estimate for local solutions more precisely certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions finite time blow up of the accumulated momentum density is necessary and sufficient for the finite time blow up of the solution an upper estimate of the blow up time and a blow up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term

superunitary reduction of the orthosymplectic supertableaux the supertableaux of the orthosymplectic groups osp 2 2p and osp 2 1 2p are decomposed in sums of supertableaux of the superunitary group su p

numerical simulations of possible finite time singularities in the incompressible euler equations comparison of numerical methods abstract the numerical simulation of the 3d incompressible euler equations is analyzed with respect to different integration methods the numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods based on this comparison a kida pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given this estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective

complex space singularities of 2d euler flow in lagrangian coordinates we show that for 2d space periodic incompressible flow the solution can be evaluated numerically in lagrangian coordinates with the same accuracy that is achieved in standard eulerian spectral methods this allows the determination of complex space lagrangian singularities lagrangian singularities are found to be closer to the real domain than eulerian singularities and seem to correspond to fluid particles which escape to complex infinity by the current time various mathematical conjectures regarding eulerian lagrangian singularities are presented

blow up criterions of strong solutions to 3d compressible navier stokes equations with vacuum abstract in the paper we establish a blow up criterion in terms of the integrability of the density for strong solutions to the cauchy problem of compressible isentropic navier stokes equations in r 3 with vacuum under the assumptions on the coefficients of viscosity 29 3 this extends the corresponding results in huang et al 2011 sun et al 2011 20 36 where a blow up criterion in terms of the upper bound of the density was obtained under the condition 7 as a byproduct the restriction 7 in fan et al 2010 sun et al 2011 12 37 is relaxed to 29 3 for the full compressible navier stokes equations by giving a new proof of lemma 3 1 besides we get a blow up criterion in terms of the upper bound of the density and the temperature for strong solutions to the cauchy problem of the full compressible navier stokes equations in r 3 the appearance of vacuum could be allowed this extends the corresponding results in sun et al 2011 37 where a blow up criterion in terms of the upper bound of 1 was obtained without vacuum the effective viscous flux plays a very important role in the proofs

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

relation between the supertableaux of the supergroups osp 2 2 and su 1 2 the classification and the interpretation of the young supertableaux of the orthosymplectic group osp 2 2 are given a comparison is made with the supertableaux of the superunitary group su 1 2 taking advantage of the isomorphism of the corresponding superalgebras

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

remarks on the regularity criteria of generalized mhd and navier stokes systems we study the regularity criteria of the three dimensional generalized magnetohydrodynamics mhd and navier stokes systems in particular we show that the regularity criteria of the generalized mhd system may be reduced to depend only on two diagonal entries of the jacobian matrix of the velocity vector field or one vorticity component and one entry of the jacobian matrix of the velocity vector field

analytical behavior of two dimensional incompressible flow in porous media in this paper we study the analytic structure of a two dimensional mass balance equation of an incompressible fluid in a porous medium given by darcy s law we obtain local existence and uniqueness by the particle trajectory method and we present different global existence criterions these analytical results with numerical simulations are used to indicate nonformation of singularities moreover blow up results are shown in a class of solutions with infinite energy

logarithmically improved bkm s criterion for the 3d nematic liquid crystal flows abstract this paper proves a logarithmically improved beale kato majda s criterion for the 3d nematic liquid crystal flows which says that if 0 t u t b m o ln e u t b m o d t then the solution u d is smooth up to the time t t

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

smooth solutions for the motion of a ball in an incompressible perfect fluid in this paper we investigate the motion of a rigid ball surrounded by an incompressible perfect fluid occupying rn we prove the existence uniqueness and persistence of the regularity for the solutions of this fluid structure interaction problem

nonlinear inviscid incompressible dynamics abstract we describe the mechanisms of geometric depletion of nonlinearity present in the three dimensional incompressible euler equations and in the two dimensional active scalar models

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

local existence and uniqueness of euler equation in weighted sobolev space abstract in this paper we study the existence and uniqueness of euler equation in the weighted sobolev space this is motivated by the study of inviscid limit of navier stokes equation in half plane the use of the weighted calderon zygmund inequality in the estimate of the pressure term is the improvement of our work

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x

incremental learning based multiobjective fuzzy clustering for categorical data the problem of clustering categorical data where attribute values cannot be naturally ordered as numerical values has gained more importance in recent time due to the special properties of categorical attributes the clustering of categorical data seems to be more complicated than that of numerical data although a few clustering algorithms that optimize single clustering objective have been proposed it has found that such single measure may not be appropriate for all kind of datasets hence in this article an incremental learning based multiobjective fuzzy clustering for categorical data is proposed for this purpose a multiobjective modified differential evolution based fuzzy clustering algorithm is developed thereafter it integrates with the well known supervised classifier called random forest using incremental learning to propose the aforementioned technique here the multiobjective algorithm produces a set of optimal clustering solutions known as pareto optimal solutions by optimizing two conflicting objectives simultaneously subsequently through incremental learning using random forest classifier final solution is evolved from the ensemble pareto optimal solutions the results of the proposed method are demonstrated quantitatively and visually in comparison with widely used state of the art methods for six synthetic and four real life datasets finally statistical test is conducted to show the superiority of the results produced by the proposed method

on approximate solutions of the incompressible euler and navier stokes equations abstract we consider the incompressible euler or navier stokes ns equations on a torus t d in the functional setting of the sobolev spaces h 0 n t d of divergence free zero mean vector fields on t d for n d 2 1 we present a general theory of approximate solutions for the euler ns cauchy problem this allows to infer a lower bound t c on the time of existence of the exact solution u analyzing a posteriori any approximate solution u a and also to construct a function n such that u t u a t n n t for all t 0 t c both t c and n are determined solving suitable control inequalities depending on the error of u a the fully quantitative implementation of this scheme depends on some previous estimates of ours on the euler ns quadratic nonlinearity morosi and pizzocchero 2010 in press 7 8 to keep in touch with the existing literature on the subject our results are compared with a setting for approximate euler ns solutions proposed in chernyshenko et al 2007 1 as a first application of the present framework we consider the galerkin approximate solutions of the euler ns cauchy problem with a specific initial datum considered in behr et al 2001 9 in this case our methods allow amongst else to prove global existence for the ns cauchy problem when the viscosity is above an explicitly given bound