generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

global well posedness of an inviscid three dimensional pseudo hasegawa mima model the three dimensional inviscid hasegawa mima model is one of the fundamental models that describe plasma turbulence the model also appears as a simplified reduced rayleigh benard convection model the mathematical analysis of the hasegawa mima equation is challenging due to the absence of any smoothing viscous terms as well as to the presence of an analogue of the vortex stretching terms in this paper we introduce and study a model which is inspired by the inviscid hasegawa mima model which we call a pseudo hasegawa mima model the introduced model is easier to investigate analytically than the original inviscid hasegawa mima model as it has a nicer mathematical structure the resemblance between this model and the euler equations of inviscid incompressible fluids inspired us to adapt the techniques and ideas introduced for the two dimensional and the three dimensional euler equations to prove the global existence and uniqueness of solutions for our model this is in addition to proving and implementing a new technical logarithmic inequality generalizing the brezis gallouet and the brezis wainger inequalities moreover we prove the continuous dependence on initial data of solutions for the pseudo hasegawa mima model these are the first results on existence and uniqueness of solutions for a model that is related to the three dimensional inviscid hasegawa mima equations

the decomposition of level 1 irreducible highest weight modules with respect to the level 0 actions of the quantum affine algebra the level 1 irreducible highest weight modules of the quantum affine algebra uq bn are decomposed into irreducible components with respect to the level 0 u 0 q bn action previously defined in 13 the components of the decomposition are found to be the so called tame representations of u 0 q bn parametrized by the skew young diagrams of the border strip type this result verifies a recent conjecture due to kirillov et al

global regularity for the 2d magnetic benard fluid system with mixed partial viscosity abstract inspired by the previous works the main goal of this paper is to establish the global well posedness and conditional regularity of the incompressible magnetic benard fluid system with mixed partial viscosity in two dimensions as a result our results generalize and extend the previous results

existence and asymptotic behavior for a convection problem abstract we prove global existence and exponential decay of solutions for a system which arise in thermal convection flow for sufficiently small initial data these results improve previous ones in funkcial ekvac 34 1991 449 further we investigate the behavior of solutions for arbitrarily large initial data in particular we show that the length of the interval on which we have existence and exponential decay is inverse proportional to the size of the initial data

skew young diagram method in spectral decomposition of integrable lattice models the spectral decomposition of the path space of the vertex model associated to the vector representation of the quantized affine algebra is studied we give a one to one correspondence between the spin configurations and the semistandard tableaux of skew young diagrams as a result we obtain a formula of the characters for the degeneracy of the spectrum in terms of skew schur functions we conjecture that our result describes the module contents of the yangian module structures of the level 1 integrable modules of the affine lie algebra an analogous result is obtained also for a vertex model associated to the quantized twisted affine algebra where characters appear for the degeneracy of the spectrum the relations to the spectrum of the haldane shastry and the polychronakos models are also discussed

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

a nonlinear analysis of the stabilizing effect of rotation in the benard problem this paper uses a novel generalized energy to study the stabilizing effect of rotation in the benard problem the nonlinear stability boundary we find is in very close agreement with the experiments of rossby rossby h t j fluid mech 36 309 335 1969 who predicts sub critical instabilities for high taylor numbers for fluids with prandtl number greater than or equal to 1 such as water

thermosolutal convection at infinite prandtl number with or without rotation bifurcation and stability in physical space we examine the nature of the thermosolutal convection with or without rotation in the infinite prandtl number regime which is applicable to magma chambers the onset of bifurcation and the structure of the bifurcated solutions in this double diffusion problem are analyzed the stress free boundary condition is imposed at the top and bottom plates confining the fluid for the rotation free case two dimensional boussinesq equations are considered and we prove that there are bifurcating solutions from the basic solution and that the bifurcated solutions consist of only one cycle of steady state solutions that are homeomorphic to s1 by thoroughly investigating the structure and transitions of the solutions of the thermosolutal convection problem in physical space we confirm that the bifurcated solutions are indeed structurally stable in the presence of rotation we consider three dimensional boussinesq equations and we can get similar results as of the rotation free case we also see how intensively the

global well posedness of the 3d primitive equations with partial vertical turbulence mixing heat diffusion the three dimensional incompressible viscous boussinesq equations under the assumption of hydrostatic balance govern the large scale dynamics of atmospheric and oceanic motion and are commonly called the primitive equations to overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection or density stratification equation in this paper we prove the global regularity of strong solutions to this model in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no penetration and stress free boundary conditions on the solid top and bottom boundaries specifically we show that short time strong solutions to the above problem exist globally in time and that they depend continuously on the initial data

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

some backward uniqueness results soit 7b u un operateur non borne lineaire ou non lineaire sur un espace de hilbert h tel que 7b u 0 0 soit une fonction solution de l equation d evolution d dt 7b u 0 et telle que t 0 pour un t 0 on etudie la propriete d unicite retrograde sur 0 t pour des inegalites d t dt a t t h n t t d a 1 2 t ou a t t 0 est une famille d operateurs lineaires non bornes autoadjoint sur h et u l 2 0 t

stability and bifurcation of the taylor problem the main objective of this paper is to address the stability and bifurcation of the couette flow between two concentric rotating cylinders and to verify rigorously taylor s observation in his experiments 13 a nonlinear theory is obtained for the taylor problem leading in particular to rigorous justifications of the linear theory used by physicists and the taylor vortex structure the main technical tools are the dynamic bifurcation theory and the geometric theory for incompressible flows both developed recently by ma wang 10 12

skew quasisymmetric schur functions and noncommutative schur functions abstract recently a new basis for the hopf algebra of quasisymmetric functions qsym called quasisymmetric schur functions has been introduced by haglund luoto mason van willigenburg in this paper we extend the definition of quasisymmetric schur functions to introduce skew quasisymmetric schur functions these functions include both classical skew schur functions and quasisymmetric schur functions as examples and give rise to a new poset l c that is analogous to young s lattice we also introduce a new basis for the hopf algebra of noncommutative symmetric functions nsym this basis of nsym is dual to the basis of quasisymmetric schur functions and its elements are the pre image of the schur functions under the forgetful map nsym sym we prove that the multiplicative structure constants of the noncommutative schur functions equivalently the coefficients of the skew quasisymmetric schur functions when expanded in the quasisymmetric schur basis are nonnegative integers satisfying a littlewood richardson rule analogue that reduces to the classical littlewood richardson rule under as an application we show that the morphism of algebras from the algebra of poirier reutenauer to sym factors through nsym we also extend the definition of schur functions in noncommuting variables of rosas sagan in the algebra ncsym to define quasisymmetric schur functions in the algebra ncqsym we prove these latter functions refine the former and their properties and project onto quasisymmetric schur functions under the forgetful map lastly we show that by suitably labeling l c skew quasisymmetric schur functions arise in the theory of pieri operators on posets

blow up and global existence for heat flows of harmonic maps in this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small

structure of bifurcated solutions of two dimensional infinite prandtl number convection with no slip boundary conditions abstract we consider the two dimensional infinite prandtl number convection problem with no slip boundary conditions the existence of a bifurcation from the trivial solution to an attractor r was proved by park 13 the no stress case has been examined in 14 we prove in this paper that the bifurcated attractor r consists of only one cycle of steady state solutions and it is homeomorphic to s 1 by thoroughly investigating the structure and transitions the solutions of the infinite prandtl number convection problem in physical space we confirm that the bifurcated solutions are indeed structurally stable we show what the asymptotic structure of the bifurcated solutions looks like this bifurcation analysis is based on a new notion of bifurcation called attractor bifurcation and structural stability is derived using geometric theories of incompressible flows

on heat convection equations in a half space with non decaying data and stokes semi group on besov spaces based on l abstract the local existence theorem for the boussinesq equations in the half space r n 1 x x x n r n 1 x r n x n 1 0 with non decaying data is shown to this end the stokes operator is shown to generate an analytic semi group on the besov spaces of negative order based on l

topics in quantum integrable systems recent developments in the theory of quantum integrable particle systems in one dimension with inverse square interactions are reviewed first the yangian symmetry is introduced and the energy spectra of the related spin models are discussed the character of the su n 1 wznw theory is shown to be closely related with the rogers szego polynomial second the infinite dimensional representation for solutions of the yang baxter equation and the reflection equation is given based on the representation the dunkl operators associated with the classical root systems are constructed the macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case finally some results on short range interacting systems are mentioned

modelling control of crystal growth processes abstract the goal of this paper is to describe an efficient cost effective mathematical modelfor controlling the czochralski crystal growth process we first consider the governing equations for a class of mhd problems we then choose the use of an externally applied magnetic field as a control mechanism we examine theoretical and numerical results applicable to the optimal control of evolutionary magnetically conducted flows the appropriate state equations are derived first order necessary conditions are examined next we develop numerical methods for solving the necessary conditions of optimality and alternative cost efficient algorithms are discussed finally these results are used to design a control strategy for the czochralski crystal growth process

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

spectral decomposition of path space in solvable lattice model we give thespectral decomposition of the path space of the u q widehat sl 2 vertex model with respect to the local energy functions the result suggests the hidden yangian module structure on the widehat sl 2 levell integrable modules which is consistent with the earlier work 1 in the level one case also we prove the fermionic character formula of the widehat sl 2 levell integrable representations in consequence

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

chemically reacting fluid flows weak solutions and global attractors abstract in this paper we present a model for incompressible chemically reacting flows where reactants enter the domain react and then leave the domain in the interior of the domain we have the navier stokes equations for the fluid flow coupled with reaction diffusion equations for the chemistry and temperature on the boundary we use mixed neumann and inhomogeneous robin boundary conditions for the chemistry and temperature equations which fix the amount of chemicals and heat flowing into the system for three dimensional domains we then show the existence of weak solutions which are physically reasonable i e which satisfy certain maximum principle properties these solutions are sufficient to provide the basis for a dynamical systems study of these reacting flows we do not show that the solutions are unique however we are able to show the existence of global attractors for the system

bifurcation of infinite prandtl number rotating convection abstract we consider infinite prandtl number convection with rotation which is the basic model in geophysical fluid dynamics for the rotation free case the rigorous analysis has been provided by park 2005 2007 revised for publication 5 6 25 under various boundary conditions by thoroughly investigating we prove in this paper that the solutions bifurcate from the trivial solution u 0 to an attractor r which consists of only one cycle of steady state solutions and is homeomorphic to s 1 we also see how intensively the rotation inhibits the onset of convective motion this bifurcation analysis is based on a new notion of bifurcation called attractor bifurcation which was developed by ma and wang 2005 see 15

inertial manifold and large deviations approach to reduced pde dynamics in this paper a certain type of reaction diffusion equation similar to the allen cahn equation is the starting point for setting up a genuine thermodynamic reduction i e involving a finite number of parameters or collective variables of the initial system we firstly operate a finite lyapunov schmidt reduction of the cited reaction diffusion equation when reformulated as a variational problem in this way we gain a finite dimensional ode description of the initial system which preserves the gradient structure of the original one and that is exact for the static case and only approximate for the dynamic case our main concern is how to deal with this approximate reduced description of the initial pde to start with we note that our approximate reduced ode is similar to the approximate inertial manifold introduced by temam and coworkers for navier stokes equations as a second approach we take into account the uncertainty loss of information introduced with the above mentioned approximate reduction by considering the stochastic version of the ode we study this reduced stochastic system using classical tools from large deviations viscosity solutions and weak kam hamilton jacobi theory in the last part we suggest a possible use of a result of our approach in the comprehensive treatment non equilibrium thermodynamics given by macroscopic fluctuation theory

decomposable compositions symmetric quasisymmetric functions and equality of ribbon schur functions abstract we define an equivalence relation on integer compositions and show that two ribbon schur functions are identical if and only if their defining compositions are equivalent in this sense this equivalence is completely determined by means of a factorization for compositions equivalent compositions have factorizations that differ only by reversing some of the terms as an application we can derive identities on certain littlewood richardson coefficients finally we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis we show the schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

the regularity of harmonic maps into spheres and applications to bernstein problems we show the regularity of and derive a priori estimates for weakly harmonic maps from a riemannian manifold into a euclidean sphere under the assumption that the image avoids some neighborhood of a half equator the proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems we apply these results to the spherical and euclidean bernstein problems for minimal hypersurfaces obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in euclidean spaces are trivial

approximation of attractors large eddy simulations and multiscale methods recent advances in the mathematical theory of the navier stokes equations have produced new insight in the mathematical theory of turbulence in particular the study of the attractor for the navier stokes equations produced the first connection between two approaches to turbulence that seemed far apart namely the conventional approach of kolmogorov and the dynamical systems theory approach similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows our aim in this article is to survey and describe some of the new results concerning the functional properties of the navier stokes equations and to discuss their relevance to turbulence

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

continuous data assimilation for the 2d benard convection through velocity measurements alone 2015 elsevier b v an algorithm for continuous data assimilation for the two dimensional benard convection problem is introduced and analyzed it is inspired by the data assimilation algorithm developed for the navier stokes equations which allows for the implementation of variety of observables low fourier modes nodal values finite volume averages and finite elements the novelty here is that the observed data is obtained for the velocity field alone i e no temperature measurements are needed for this algorithm we provide conditions on the spatial resolution of the observed data under the assumption that the observed data is free of noise which are sufficient to show that the solution of the algorithm approaches at an exponential rate the unique exact unknown solution of the benard convection problem associated with the observed finite dimensional projection of velocity

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

existence and finite dimensionality of the global attractor for evolution equations on unbounded domains abstract this paper is devoted to the asymptotic behavior for some evolution equations when the underlying euclidean domain is unbounded we present a method that is able to overcome the lack of compactness of the trajectories in such situations and that allows us to obtain some results on the global attractor similar to those on bounded domains this method is based on the utilization of an appropriate weight integration function that depends on x and t and becomes effective in x for large t s