the global well posedness and global attractor for the solutions to the 2d boussinesq system with variable viscosity and thermal diffusivity abstract global well posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2d boussinesq system in a periodic channel with non homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved

initial boundary value problem for two dimensional viscous boussinesq equations we study the initial boundary value problem of two dimensional viscous boussinesq equations over a bounded domain with smooth boundary we show that the equations have a unique classical solution for h3 initial data and the no slip boundary condition in addition we show that the kinetic energy is uniformly bounded in time

full factors and co amenable inclusions we show that if m is a full factor and n subset m is a co amenable subfactor with expectation then n is also full this answers a question of popa from 1986 we also generalize a theorem of tomatsu by showing that if m is a full factor and sigma g curvearrowright m is an outer action of a compact group g then sigma is automatically minimal and m g is a full factor which has w spectral gap in m finally in the appendix we give a proof of the fact that several natural notions of co amenability for an inclusion n subset m of von neumann algebras are equivalent thus closing the cycle of implications given in anantharaman delaroche s paper in 1995

infinite prandtl number limit of rayleigh benard convection we rigorously justify the infinite prandtl number model of convection as the limit of the boussinesq approximation to the rayleigh benard convection as the prandtl number approaches infinity this is a singular limit problem involving an initial layer 2003 wiley periodicals inc

parabolic equations with vmo coefficients in sobolev spaces with mixed norms an lq lp theory of divergence and non divergence form parabolic equations is presented the main coefficients are supposed to belong to the class vmox which in particular contains all measurable functions depending only on t the method of proving simplifies the methods previously used in the case p q

on actions of amenable groups on ii1 factors abstract given a ii 1 factor m with separable predual and a free action of a countable amenable discrete group g on m we show that the crossed product m g has property resp is mcduff when m itself has property resp is mcduff

stationary statistical properties of rayleigh benard convection at large prandtl number this is the third in a series of our study of rayleigh benard convection at large prandtl number here we investigate whether stationary statistical properties of the boussinesq system for rayleigh benard convection at large prandtl number are related to those of the infinite prandtl number model for convection that is formally derived from the boussinesq system via setting the prandtl number to infinity we study asymptotic behavior of stationary statistical solutions or invariant measures to the boussinesq system for rayleigh benard convection at large prandtl number in particular we show that the invariant measures of the boussinesq system for rayleigh benard convection converge to those of the infinite prandtl number model for convection as the prandtl number approaches infinity we also show that the nusselt number for the boussinesq system a specific statistical property of the system is asymptotically bounded by the nusselt number of the infinite prandtl number model for convection at large prandtl number we discover that the nusselt numbers are saturated by ergodic invariant measures moreover we derive a new upper bound on the nusselt number for the boussinesq system at large prandtl number of the form

global regularity results for the 2d boussinesq equations with vertical dissipation abstract the two dimensional 2d incompressible boussinesq equations model geophysical fluids and play an important role in the study of the raleigh bernard convection mathematically this 2d system retains some key features of the 3d navier stokes and euler equations such as the vortex stretching mechanism the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation or no dissipation at all this paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases these results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2d boussinesq equations

global well posedness for the euler boussinesq system with axisymmetric data abstract in this paper we prove the global well posedness for the three dimensional euler boussinesq system with axisymmetric initial data without swirl this system couples the euler equation with a transport diffusion equation governing the temperature

global regularity for the initial boundary value problem of the 2 d boussinesq system with variable viscosity and thermal diffusivity abstract we prove the global existence of strong solution to the initial boundary value problem of the 2 d boussinesq system and 3 d infinite prandtl number model with viscosity and thermal conductivity depending on the temperature

duality in parabolic set up for questions in kazhdan lusztig theory in this short article we expand on the ideas developed in de21 viz the so called parabolic analogue of kazhdan lusztig polynomials in that paper we considered two hecke algebra modules corresponding to the two roots 1 and q of the equation u q q 1 which give rise to two sets of polynomials the question of the relationship between those two modules was not considered there since it seemed somewhat of a digression in that context it was noted however that these two sets of polynomials have different recursive relations and so they are different in nature special attention was given to the module corresponding to 1 and it was shown that the coefficients of the polynomials are given by the dimensions of stalks of intersection cohomology sheaves of certain generalized schubert varieties in crystallographic cases recently the author has been working on a combinatorial set up which is used to obtain a closed formula for the kazhdan lusztig polynomials de3 while dealing with the parabolic case one can not use the map j kl sect 11 which gives a dual basis viz ci in case j d see remark 2 5 below at this stage the author came across a preprint by j m douglass do where concrete models for the two hecke algebra modulus are constructed for the parabolic case in a finite coxeter group it is further shown that these two are related to each other via a certain map defined by g lusztig l unfortunately those methods use the finiteness assumption in an essential way the author realized that one may use alternate arguments which do not assume the finiteness as a matter of fact one can now tie up the two hecke algebra modules by a natural duality see theorem 2 1 below which then relates various other

the vanishing diffusivity limit for the 2 d boussinesq equations with boundary effect abstract in this paper we tackle the issue of the vanishing diffusivity limit of the 2 d incompressible boussinesq equations in the half plane our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter goes to zero under the homogeneous dirichlet boundary condition of velocity and the nonhomogeneous dirichlet boundary condition in the x direction for temperature we show that the boundary layer thickness is of the order o with 0 2 3 in contrast with jiang et al 2011 the bl thickness we got is thinner than that in jiang et al 2011 moreover we prove that as diffusivity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion equations we also obtain the convergence rates of the vanishing diffusivity limit

on the blowup criteria and global regularity for the non diffusive boussinesq equations with temperature dependent viscosity coefficient abstract in this paper we consider the boussinesq equations with temperature dependent viscosity and zero thermal diffusivity a blowup criterion of classical solution depending only on l q 0 t l p with p 2 and 1 p 1 q 1 2 is obtained for the two dimensional equations this is in particular consistent with the results in chae 2006 hou and li 2005 lai et al 2011 where the global regularity of the 2d non diffusive boussinesq equations with constant viscosity coefficient i e const was proved a serrin s type blowup criterion depending on u l s 0 t l r l q 0 t l p with r p 3 3 r 2 s 1 and 3 p 2 q 1 is obtained for the three dimensional case we also study the global regularity with large data for the two dimensional equations in a non divergence form

global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity abstract we prove the global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity

global well posedness for 2 d boussinesq system with the temperature dependent viscosity and supercritical dissipation abstract the present paper is dedicated to the global well posedness issue for the boussinesq system with the temperature dependent viscosity in r 2 motivated by the work abidi and zhang adv math 305 2017 1202 1249 on the critical dissipation d with 1 in the temperature equation of the system we consider the extension of their work to the supercritical dissipation for 2 3 1

a combinatorial formula for the character of the diagonal coinvariants author s haglund j haiman m loehr n remmel j b ulyanov a abstract let rn be the ring of coinvariants for the diagonal action of the symmetric group sn it is known that the character of rn as a doubly graded s module can be expressed using the frobenius characteristic map as nabla en where en is the n th elementary symmetric function and nabla is an operator from the theory of macdonald polynomials we conjecture a combinatorial formula for nabla en and prove that it has many desirable properties that support our conjecture in particular we prove that our formula is a symmetric function which is not obvious and that it is schur positive these results make use of the theory of ribbon tableau generating functions of lascoux leclerc and thibon we also show that a variety of earlier conjectures and theorems on nabla en are special cases of our conjecture finally we extend our conjectures on nabla en and several on the results supporting them to higher powers nablam en

initial boundary value problem to 2d boussinesq equations for mhd convection with stratification effects abstract this paper is concerned with the initial boundary value problem to 2d magnetohydrodynamics boussinesq system with the temperature dependent viscosity thermal diffusivity and electrical conductivity first we establish the global weak solutions under the minimal initial assumption then by imposing higher regularity assumption on the initial data we obtain the global strong solution with uniqueness moreover the exponential decay rates of weak solutions and strong solution are obtained respectively

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

the 2d boussinesq equations with vertical viscosity and vertical diffusivity this paper aims at the global regularity of classical solutions to the 2d boussinesq equations with vertical dissipation and vertical thermal diffusion we prove that the l r norm of the vertical velocity v for any 1 0 would guarantee the global regularity of classical solutions

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

parabolic kazhdan lusztig r polynomials for tight quotients of the symmetric groups abstract we give explicit closed combinatorial formulas for the parabolic kazhdan lusztig r polynomials of the tight quotients of the symmetric groups we give two formulations of our result one in terms of permutations and one in terms of motzkin paths as an application of our results we obtain explicit closed combinatorial formulas for certain sums and alternating sums of ordinary kazhdan lusztig r polynomials