on stabilized finite element methods based on the scott zhang projector circumventing the inf sup condition for the stokes problem abstract in this work we propose a stabilized finite element method that permits us to circumvent discrete inf sup conditions e g allowing equal order interpolation the type of method we propose belongs to the family of symmetric stabilization techniques which are based on the introduction of additional terms that penalize the difference between some quantities i e the pressure gradient in the stokes problem and their finite element projections the key feature of the formulation we propose is the definition of the projection to be used a non standard scott zhang projector that is well defined for l 1 functions the resulting method has some appealing features the projector is local and nested meshes or enriched spaces are not required

monomial modular representations and construction of the held group abstract monomial representations of familiar finite groups over finite fields are used to construct infinite semi direct products of free products of cyclic groups by groups of monomial automorphisms finite homomorphic images of these progenitors in which the actions on the group of automorphisms and on the cyclic components are faithful are sought the smallest non trivial images of this type are often sporadic simple groups the technique is demonstrated by three examples over the fields z 3 z 5 and z 7 which produce the mathieu group m 11 the unitary group u 3 5 2 and the held group respectively

generalized surface quasi geostrophic equations with singular velocities this paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals the first family is a generalized surface quasigeostrophic sqg equation with the velocity field u related to the scalar by where and is the zygmund operator the borderline case 1 corresponds to the sqg equation and the situation is more singular for 1 we obtain the local existence and uniqueness of classical solutions the global existence of weak solutions and the local existence of patch type solutions the second family is a dissipative active scalar equation with which is at least logarithmically more singular than the velocity in the first family we prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data this result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by k ohkitani 2012 wiley periodicals inc

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on mixed finite element methods for linear elastodynamics we construct and analyze finite element methods for approximating the equations of linear elastodynamics using mixed elements for the discretization of the spatial variables we consider two different mixed formulations for the problem and analyze semidiscrete and up to fourth order in time fully discrete approximations l 2 optimal order error estimates are proved for the approximations of displacement and stress

on global regularity of 2d generalized magnetohydrodynamic equations abstract in this article we study the global regularity of 2d generalized magnetohydrodynamic equations 2d gmhd in which the dissipation terms are u and b we show that smooth solutions are global in the following three cases 1 2 1 0 1 2 2 2 2 0 we also show that in the inviscid case 0 if 1 then smooth solutions are global as long as the direction of the magnetic field remains smooth enough

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

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

a space time finite element method for the wave equation a numerical method is introduced which uses finite elements in time and space simultaneously to solve the wave equation an error analysis for this scheme is presented and the results of several computations this scheme is accurate for time slabs of arbitrary thickness without introducing extra least squares or stabilization terms

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a new family of mixed finite elements for the linear elastodynamic problem we construct and analyze a new family of quadrangular in two dimensions or cubic in three dimensions mixed finite elements for the approximation of elastic wave equations our elements lead to explicit schemes via mass lumping after time discretization including in the case of anisotropic media error estimates are given for these new elements

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

existence theorem and blow up criterion of the strong solutions to the two fluid mhd equation in r3 abstract we first give the local well posedness of strong solutions to the cauchy problem of the 3d two fluid mhd equations and then study the blow up criterion of the strong solutions by means of the fourier frequency localization and bony s paraproduct decomposition it is proved that the strong solution u b can be extended after t t if either u l t q b p 0 with 2 q 3 p 1 and b l t 1 b 0 or j l t q b p 0 with 2 q 3 p 2 where t u denotes the vorticity of the velocity and j b stands for the current density

energy and cross helicity conservation for the three dimensional ideal mhd equations in a bounded domain abstract in this paper we prove the energy and cross helicity conservation of weak solutions to the three dimensional ideal mhd equations in a bounded domain under the interior besov regularity conditions which are exactly same as the three dimensional periodic domain case in 3 and the boundedness and the besov type continuity conditions for both the velocity and magnetic fields near the boundary which seem crucial for the bounded domain case due to the boundary effect note that our besov type continuity conditions near the boundary are consistent with the interior besov regularity conditions

graphs related to held s simple group abstract we analyze the permutation representations of low degree of held s simple group he we also determine its primitive multiplicity free permutation representations and show that there is no graph on which it or its automorphism group acts as a distance transitive group of automorphisms in doing so we supply a computer free construction of he

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

the sharp corner formation in 2d euler dynamics of patches infinite double exponential rate of merging for the 2d euler dynamics of patches we investigate the convergence to the singular stationary solution in the presence of a regular strain it is proved that the rate of merging can be double exponential infinitely in time and the estimates we obtain are sharp

a computer free construction of j4 define a finite simple group j to be of j4 type or simply j4 provided that j contains an involution z withcjz 21 12 3aut mat22 the purpose of this paper is to give the first computer free construction of a group of j4 type in addition we achieve yet another uniqueness proof for groups of j4 type via the simple connectedness of the 2 local geometry of such a group

onsager s conjecture in bounded domains for the conservation of entropy and other companion laws we show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy and other respective companion laws if they possess a certain fractional differentiability of order one third in the interior of the domain and if the normal component of the corresponding fluxes tend to zero as one approaches the boundary this extends various recent results of the authors

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

energy and helicity preserving schemes for hydro and magnetohydro dynamics flows with symmetry we propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry by introducing a generalized vorticity stream formulation the divergence free constraints are automatically satisfied in addition with explicit treatment of the nonlinear terms and local vorticity boundary condition the navier stokes mhd respectively equation essentially decouples into 2 4 respectively scalar equation and thus the scheme is very efficient moreover with proper discretization of the nonlinear terms the scheme preserves both energy and helicity identities numerically this is achieved by recasting the nonlinear terms convection vorticity stretching geometric source lorentz force and electro motive force in terms of jacobians this conservative property is valid even in the presence of the pole singularity for axisymmetric flows the exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity numerical examples have demonstrated both accuracy and efficiency of the scheme finally local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost

an analysis of new mixed finite elements for the approximation of wave propagation problems we construct and analyze a new family of rectangular two dimensional or cubic three dimensional mixed finite elements for the approximation of the acoustic wave equations the main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium nonclassical error estimates are given for this new element

on the well posedness of the ideal mhd equations in the triebel lizorkin spaces in this paper we prove the local well posedness for the ideal mhd equations in the triebel lizorkin spaces and obtain a blow up criterion of smooth solutions specifically we fill a gap in a step of the proof of the local well posedness part for the incompressible euler equation in chae comm pure appl math 55 654 678 2002

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

existence for the α patch model and the qg sharp front in sobolev spaces abstract we consider a family of contour dynamics equations depending on a parameter with 0 1 the vortex patch problem of the 2 d euler equation is obtained taking 0 and the case 1 corresponds to a sharp front of the qg equation we prove local in time existence for the family of equations in sobolev spaces

remarks on global regularity of 2d generalized mhd equations abstract in this paper we investigate the global regularity of 2d generalized mhd equations in which the dissipation term and magnetic diffusion term are u and b respectively let u 0 b 0 h s with s 2 it is showed that the smooth solution u x t b x t is globally regular for the case 0 1 2 3 2

stability convergence and accuracy of stabilized finite element methods for the wave equation in mixed form in this paper we propose two stabilized finite element methods for different functional frameworks of the wave equation in mixed form these stabilized finite element methods are stable for any pair of interpolation spaces of the unknowns the variational forms corresponding to different functional settings are treated in a unified manner through the introduction of length scales related to the unknowns stability and convergence analysis is performed together with numerical experiments it is shown that modifying the length scales allows one to mimic at the discrete level the different functional settings of the continuous problem and influence the stability and accuracy of the resulting methods

finite element approximation of the hyperbolic wave equation in mixed form the purpose of this paper is to present a finite element approximation of the scalar hyperbolic wave equation written in mixed form that is introducing an auxiliary vector field to transform the problem into a first order problem in space and time we explain why the standard galerkin method is inappropriate to solve this problem and propose as alternative a stabilized finite element method that can be cast in the variational multiscale framework the unknown is split into its finite element component and a remainder referred to as subscale as original features of our approach we consider the possibility of letting the subscales to be time dependent and orthogonal to the finite element space the formulation depends on algorithmic parameters whose expression is proposed from a heuristic fourier analysis

nonexistence of simple hyperbolic blow up for the quasi geostrophic equation the problem we are concerned with is whether singularities form in finite time in incompressible fluid flows it is well known that the answer is no in the case of euler and navier stokes equations in dimension two in dimension three it is still an open problem for these equations in this paper we focus on a two dimensional active scalar model for the 3d euler vorticity equation constantin majda and tabak 7 suggested by studying rigorous theorems and detailed numerical experiments a general principle if the level set topology in the temperature field for the 2d quasi geostrophic active scalar in the region of strong scalar gradients does not contain a hyperbolic saddle then no finite time singularity is possible numerical simulations showed evidence of singular behavior when the geometry of the level sets of the active scalar contain a hyperbolic saddle there is a naturally associated notion of simple hyperbolic saddle breakdown the main theorem we present in this paper shows that such breakdown cannot occur in finite time we also show that the angle of the saddle cannot close in finite time and it cannot be faster than a double exponential in time using the same techniques we see that analogous results hold for incompressible 2d and 3d euler these results were announced in 9 but with a slight difference in the definition of a simple hyperbolic saddle the definition given in section 4 generalizes the one given in the announcement see also constantin 4 discussed in section 7 remark 5 below the whole work described in this paper is basically part of the author s thesis i am particularly grateful to my thesis advisor charles fefferman for his attention support guidance and advice i am indebted to d christodoulou and p constantin for helpful corrections and suggestions i wish to thank a majda for suggesting the subject and e tabak for discussions and com

remarks on singularities dimension and energy dissipation for ideal hydrodynamics and mhd for weak solutions of the incompressible euler equations there is energy conservation if the velocity is in the besov space b 3 s with s greater than 1 3 b 3 s consists of functions that are lip s i e holder continuous with exponent s measured in the l p norm here this result is applied to a velocity field that is lip 0 except on a set of co dimension on which it is lip agr 1 with uniformity that will be made precise below we show that the frisch parisi multifractal formalism is valid at least in one direction for such a function and that there is energy conservation if analogous conservation results are derived for the equations of incompressible ideal mhd i e zero viscosity and resistivity for both energy and helicity in addition a necessary condition is derived for singularity development in ideal mhd generalizing the beale kato majda condition for ideal hydrodynamics

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

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

convergence of a continuous galerkin method with mesh modification for nonlinear wave equations we consider space time continuous galerkin methods with mesh modification in time for semilinear second order hyperbolic equations we show a priori estimates in the energy norm without mesh conditions under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates estimates of the jump in the riesz projection in two successive time steps are also derived

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

regularity criteria in terms of the pressure for the three dimensional mhd equations in this paper we consider the cauchy problem for the 3d incompressible mhd equations a regularity criterion by providing sufficient condition in terms of the pressure is established more precisely we show that if p l 0 t l r 3 with 3 2 7 4 and 12 5 4 then any weak solutions u b of the 3d mhd equations is regular on 0 t

a priori error estimates for mixed finite element approximations of the acoustic wave equation in this paper we derive optimal a priori l infty l 2 error estimates for mixed finite element displacement formulations of the acoustic wave equation the computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure however the displacement formulations with the physical variables of interest displacement and pressure requires less regularity on the displacement

absence of splash singularities for surface quasi geostrophic sharp fronts and the muskat problem in this paper for both the sharp front surface quasi geostrophic equation and the muskat problem we rule out the splash singularity blow up scenario in other words we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth splash singularities have been shown to hold for the free boundary incompressible euler equation in the form of the water waves contour evolution problem our result confirms the numerical simulations in earlier work in which it was shown that the curvature blows up because the contours collapse at a point here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out in this case a positive volume of fluid between the contours cannot be ejected in finite time

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

a computer free construction of the third group of janko abstract we give a new constructive computer free existence proof of the third group of janko j 3 using the 9 dimensional unitary representation of 3 j 3 over gf 4 thereby we embed the c 2 c geometries for 3 j 3 and j 3 into the 9 dimensional unitary space and the 9 dimensional projective space respectively

on approximate solutions of the equations of incompressible magnetohydrodynamics abstract inspired by an approach proposed previously for the incompressible navier stokes ns equations we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics mhd on a torus of arbitrary dimension d this setting involves a sobolev space of infinite order made of c vector fields with vanishing divergence and mean on the torus given any approximate solution of the mhd cauchy problem its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution and to bound from above the sobolev distance of any order between the exact and the approximate solution in certain cases the above mentioned lower bound on the time of existence is found to be infinite so one infers the global existence of the exact mhd solution we present some applications of this general scheme the most sophisticated one lives in dimension d 3 with the abc flow perturbed magnetically as an initial datum and uses for the cauchy problem a galerkin approximate solution in 124 fourier modes we illustrate the conclusions arising in this case from the a posteriori analysis of the galerkin approximant these include the derivation of global existence of the exact mhd solution with the abc datum when the dimensionless viscosity and resistivity are equal and stay above a critical value

nonexistence of self similar singularities in the viscous magnetohydrodynamics with zero resistivity abstract we are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in r n n 2 3 namely the mhd with positive viscosity and zero resistivity in the special case of zero magnetic field the system reduces to the navier stokes equations in r n in this paper we exclude the scenario of finite time singularity in the form of self similarity under suitable integrability conditions on the velocity and the magnetic field we also prove the nonexistence of asymptotically self similar singularity this provides us information on the behavior of solutions near possible singularity of general type as described in corollary 1 1

analysis of a stabilized finite element approximation of the transient convection diffusion equation using an ale framework in this paper we analyze a stabilized finite element method to approximate the convection diffusion equation on moving domains using an arbitrary lagrangian eulerian ale framework as basic numerical strategy we discretize the equation in time using first and second order backward differencing bdf schemes whereas space is discretized using a stabilized finite element method the orthogonal subgrid scale formulation to deal with convection dominated flows the semidiscrete problem continuous in space is first analyzed in this situation it is easy to identify the error introduced by the ale approach after that the fully discrete method is considered we obtain optimal error estimates in both space and time in a mesh dependent norm the analysis reveals that the ale approach introduces an upper bound for the time step size for the results to hold the results obtained for the fully discretized second order scheme in time are associated to a weaker norm than the one used for the first order method nevertheless optimal convergence results have been proved for fixed domains we recover stability and convergence results with the strong norm for the second order scheme stressing the aspects that make the analysis of this method much more involved

on some time marching schemes for the stabilized finite element approximation of the mixed wave equation abstract in this paper we analyze time marching schemes for the wave equation in mixed form the problem is discretized in space using stabilized finite elements on the one hand stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings on the other hand we use fourier techniques also known as von neumann analysis in order to analyze stability dispersion and dissipation numerical convergence tests are presented for various time integration schemes polynomial interpolations for the spatial discretization stabilization methods and variational forms to analyze the behavior of the different schemes considered a 1d wave propagation problem is solved

a remark on global regularity of 2d generalized magnetohydrodynamic equations abstract in this paper we study the global regularity of the following 2d two dimensional generalized magnetohydrodynamic equations u t u u p b b u b t u b b u b and get global regular solutions when 0 1 2 1 3 2 3 which improves the results in 5 in particular we obtain the global regularity of the 2d generalized mhd when 0 and 3 2

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

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

a robust finite element method for darcy stokes flow finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed the system is approximately a linear stokes problem when the perturbation parameter is large while it degenerates to a mixed formulation of poisson s equation as the perturbation parameter tends to zero it is established basically by numerical experiments that most of the proposed finite element methods for stokes problem or the mixed poisson s system are not well behaved uniformly in the perturbation parameter this is used as the motivation for introducing a new robust finite element which exhibits this property

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

regularity criteria for the 3d generalized mhd equations in terms of vorticity abstract we prove regularity criteria for the 3d generalized mhd equations these criteria impose assumptions on the vorticity only in addition we also prove a result of global existence for smooth solution under some special conditions

higher order commutator estimates and local existence for the non resistive mhd equations and related models this paper establishes the local in time existence and uniqueness of strong solutions in hs for s n 2 to the viscous non resistive magnetohydrodynamics mhd equations in rn n 2 3 as well as for a related model where the advection terms are removed from the velocity equation the uniform bounds required for proving existence are established by means of a new estimate which is a partial generalisation of the commutator estimate of kato ponce comm pure appl math 41 7 891 907 1988

global cauchy problem of 2d generalized magnetohydrodynamic equations abstract in this paper we prove the global in time existence of smooth solutions of the 2d generalized magnetohydrodynamic equations with zero viscosity when the power of the laplacian operator in the dissipation term of the induction equation is between 1 and 2 we also show a regularity criterion on the direction of the magnetic field b 0 b b when 1

remarks on the blow up criteria for three dimensional ideal magnetohydrodynamics equations in this paper we establish some improved regularity conditions for the three dimensional 3d magnetohydrodynamics equations we proved the logarithmically improved caflisch klapper steele s criteria remarks on sigularities dimension and energy dissipation for ideal hydrodynamics and mhd commun math phys 184 443 1997 as well as the logarithmically improved he xin s on the regularity of weak solutions to the magnetohydrodynamic equations j differ equ 23 235 2005 and wu s bounds and new approach for the 3d mhd equations j nonlinear sci 12 395 2002 results for 3d ideal mhd equations

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

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

scalars convected by a two dimensional incompressible flow we provide a test for numerical simulations for several two dimensional incompressible flows that appear to develop sharp fronts we show that in order to have a front the velocity has to have uncontrolled velocity growth 2001 john wiley sons inc

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence

growth of solutions for qg and 2d euler equations the work of constantin majda tabak 1 developed an analogy between the quasi geostrophic and 3d euler equations constantin majda and tabak proposed a candidate for a singularity for the quasi geostrophic equation their numerics showed evidence of a blow up for a particular initial data where the level sets of the temperature contain a hyperbolic saddle the arms of the saddle tend to close in finite time producing a sharp front numerics studies done later by ohkitani yamada 8 and constantin nie schorghofer 2 with the same initial data suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time the study of collapse on a curve was first studied in 1 for the quasi geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology at the time of collapse the scalar is discontinuous across the curve x2 f x1 with different limiting values for the temperature on each side of the front they show that under this topology the directional field remains smooth up to the collapse which contradicts the following theorem proven in 1