classical solutions for a generalized euler equation in two dimensions abstract it is well known that the euler equations in two spatial dimensions have global classical solutions we provide a new proof which is analytic rather than geometric it is set in an abstract framework that applies to the so called lake and the great lake equations describing weakly non hydrostatic effects of bottom topography on the motion of shallow water the key ingredient is a new lpestimate on the nonlinear term the estimate is used to develop a global hmtheory for bounded domains in r 2which is similar in spirit to a 1975 paper by r temam it also provides explicit bounds on the hmnorm which grow like exp exp t

global well posedness for the 3d primitive equations in anisotropic framework abstract this paper is concerned with the initial boundary value problem of the 3d primitive equations with only horizontal viscosities and horizontal diffusion we first establish the local well posedness result of this system with the initial data belongs to the anisotropic sobolev pace h 0 s s 1 2 then under the basic and high order energy estimates one can extend the local solution to the global one provided that s 1 the restrictions on the initial data is more weaker than the previous results which are obtained by cao et al 2016 8 and 2017 9 in which the initial data belongs to h 2 and h 1 l respectively

theory and methodologyminimax regret solution to linear programming problems with an interval objective function this paper deals with a linear programming problem with interval objective function coefficients a new treatment of an interval objective function is presented by introducing the minimax regret criterion as used in decision theory the properties of minimax regret solution and the relations with possibly and necessarily optimal solutions are investigated next the minimax regret criterion is applied to the final determination of the solution when a reference solution set is given a method of solution by a relaxation procedure is proposed the solution is obtained by repetitional use of the simplex method the minimax regret solution is obtained by the proposed solution method when the reference solution set is the set of possibly optimal solutions in order to illustrate the proposed solution method a numerical example is given

minimax regret solution to linear programming problems with an interval objective function in this paper a linear programming problem with an interval objective function is treated first the previous approaches to this problem are reviewed and the drawbacks are pointed out to improve the drawbacks a new approach to this problem is proposed by introducing the minimax regret criterion as used in decision theory the properties of minimax regret solution are investigated in order to obtain the minimax regret solution a method of solution by a relaxation procedure is proposed it is shown that the solution is obtained by repetitional use of the simplex method a numerical example is given to illustrate the proposed solution method

uniqueness of the global weak solutions to 2d compressible primitive equations abstract in this paper we investigate an initial boundary value problem for 2d compressible primitive equations with some specific viscosities we will prove the uniqueness of the weak solutions obtained by b v gatapov and a v kazhikhov 12

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

pullback attractors for closed cocycles abstract in this article using a result of pata and zelik 2007 45 we derive a general result on the existence of pullback attractors for closed cocycles acting on a banach space where the strong continuity is replaced by a much weaker requirement that the cocycle be a closed map as application we prove the existence of the pullback attractor of a cocycle associated with the z weak solutions of a non autonomous two dimensional primitive equations of the ocean

strong solutions to the 3d primitive equations with only horizontal dissipation near h 1 initial data abstract in this paper we consider the initial boundary value problem of the three dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity we establish the local in time well posedness of strong solutions for any initial data v 0 t 0 h 1 by using the local in space type energy estimate we also establish the global well posedness of strong solutions for this system with any initial data v 0 t 0 h 1 l such that z v 0 l m for some m 2 by using the logarithmic type anisotropic sobolev inequality and a logarithmic type gronwall inequality this paper improves the previous results obtained in cao et al 2016 10 where the initial data v 0 t 0 was assumed to have h 2 regularity

violation analysis on two step method for interval linear programming in many real world problems system parameters or model coefficients may be bounded between lower and upper bounds due to a variety of uncertainties over the past decades intensive research efforts have focused on interval linear programming ilp to tackle such uncertainties as one of the most popular methods for solving ilp problems two step method tsm proposed by huang et al in 1995 allows uncertain information to be directly communicated into the optimization process and resulting solutions such that decision alternatives could be generated through the interpretation of interval solutions however part of optimum solution points obtained through tsm may go beyond the decision space in some cases this phenomenon referred to as solution violation may mislead decision makers to unreasonable policies plans or strategies which play important roles in the social and economic development therefore this study first investigates and identifies the essential cause for the solution violation of tsm following that an improved solution method namely itsm is proposed to avoid resulting violation by introducing extra constraints in the solving process a numeric example is then presented to demonstrate the effectiveness of itsm in handling solution violation

random attractor for the 3d viscous primitive equations driven by fractional noises abstract we develop a new and general method to prove the existence of the random attractor strong attractor for the primitive equations pes of large scale ocean and atmosphere dynamics under non periodic boundary conditions and driven by infinite dimensional additive fractional wiener processes in contrast to our new method the common method compact sobolev embedding theorem is to obtain the time uniform a priori estimates in some sobolev space whose regularity is higher than the solution space but this method can not be applied to the 3d stochastic pes with the non periodic boundary conditions therefore the existence of universal attractor weak attractor was established in previous works see 15 16 the main idea of our method is that we first derive that p almost surely the solution operator of stochastic pes is compact then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set we should point out that our method has some advantages over the common method of using compact sobolev embedding theorem i e using our method we only need to obtain time uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system while the common method need to establish time uniform a priori estimates in a more regular functional space than the solution space take the stochastic pes for example as the unique strong solution to the stochastic pes belongs to c 0 t h 1 3 in view of our method we only need to obtain the time uniform a priori estimates in the solution space h 1 3 to prove the existence of random attractor for this stochastic system while the common method need to establish time uniform a priori estimates for the solution in the functional space h 2 3 however time uniform a priori estimates in h 2 3 for the solution to stochastic pes are too difficult to be established the present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by wiener noises fractional noises and even jump noises in a forth coming paper using this new method we 46 prove the existence of random attractor for the stochastic nematic liquid crystals equations this is the first result about the long time behavior of stochastic nematic liquid crystals equations

on the solutions of the 3d steady and unsteady primitive equations of the ocean abstract in this article the existence and uniqueness of the solution for the 3d steady viscous primitive equations pes for brevity of the ocean under the uniqueness condition are analyzed also the h 2 regularity results of the solution for the 3d steady viscous pes of the ocean under the stability condition is provided finally the convergence of the solution for the 3d unsteady viscous pes of the ocean to the solution for the 3d steady viscous pes of the ocean when t towards to infinity under the uniqueness condition is studied

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

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

on the superconformal index of argyres douglas theories we conjecture a closed form expression for the schur limit of the superconformal index of two infinite series of argyres douglas ad superconformal field theories scfts the a1 a2n 3 and the a1 d2n theories while these scfts can be realized at special points on the coulomb branch of certain n 2 gauge theories their superconformal r symmetries are emergent and hence their indices cannot be evaluated by localization instead we construct the a1 a2n 3 and a1 d2n indices by using a relation to two dimensional qdeformed yang mills theory and data from the class s construction our results generalize the indices derived from the torus partition functions of the two dimensional chiral algebras associated with the a1 a3 and a1 d4 scfts as checks of our conjectures we study the consistency of our results with an s duality recently discussed by us in collaboration with giacomelli and papageorgakis we reproduce known higgs branch relations we check consistency with a series of renormalization group flows and we verify that the small s 1 limits of our indices reproduce expected cardy like behavior we will discuss the s 1

global strong lp well posedness of the 3d primitive equations with heat and salinity diffusion abstract consider the full primitive equations i e the three dimensional primitive equations coupled to the equation for temperature and salinity and subject to outer forces it is shown that this set of equations is globally strongly well posed for arbitrary large initial data lying in certain interpolation spaces which are explicitly characterized as subspaces of h 2 p p 1 p satisfying certain boundary conditions in particular global well posedness of the full primitive equations is obtained for initial data having less differentiability properties than h 1 hereby generalizing a result by cao and titi 2007 5 to the case of non smooth data in addition it is shown that the solutions are exponentially decaying provided the outer forces possess this property

global strong solutions of navier stokes equations with interface boundary in three dimensional thin domains abstract in this article we study the spectrum of the stokes operator in a 3d two layer domain with interface obtain the asymptotic estimates on the spectrum of the stokes operator as thickness e goes to zero based on the spectral decomposition of the stokes operator a new average like operator is introduced and applied to the study of navier stokes equation in the two layer thin domains under interface boundary condition we prove the global existence of strong solutions to the 3d navier stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small this article is a continuation of our study on the stokes operator under navier friction boundary condition due to the viscosity distinction between the two layers the stokes operator displays radically different spectral structure from that under navier friction boundary condition then causes great difficulty to the analysis

torus n point functions for mathbb r graded vertex operator superalgebras and continuous fermion orbifolds we consider genus one n point functions for a vertex operator superalgebra with a real grading we compute all n point functions for rank one and rank two fermion vertex operator superalgebras in the rank two fermion case we obtain all orbifold n point functions for a twisted module associated with a continuous automorphism generated by a heisenberg bosonic state the modular properties of these orbifold n point functions are given and we describe a generalization of fay s trisecant identity for elliptic functions

existence of the universal attractor for the 3 d viscous primitive equations of large scale moist atmosphere abstract in this paper we consider the initial boundary value problem for the three dimensional viscous primitive equations of large scale moist atmosphere which are used to describe the turbulent behavior of long term weather prediction and climate changes by obtaining the existence and uniqueness of global strong solutions for the problem and studying the long time behavior of strong solutions we prove the existence of the universal attractor for the dynamical system generated by the primitive equations of large scale moist atmosphere

gauge theories and macdonald polynomials we study the mathcal n 2 four dimensional superconformal index in various interesting limits such that only states annihilated by more than one supercharge contribute extrapolating from the su 2 generalized quivers which have a lagrangian description we conjecture explicit formulae for all a type quivers of class mathcal s which in general do not have one we test our proposals against several expected dualities the index can always be interpreted as a correlator in a two dimensional topological theory which we identify in each limit as a certain deformation of two dimensional yang mills theory the structure constants of the topological algebra are diagonal in the basis of macdonald polynomials of the holonomies

existence of weak solutions and trajectory attractors for the moist atmospheric equations in geophysics in this paper we consider the initial boundary value problem for the primitive equations of moist atmospheric dynamics that are used to describe the turbulent behavior of long term weather prediction and climate changes by the faedo galerkin method we obtain the existence of global weak solutions to the problem in a large scale atmosphere by studying the long time behavior of solutions we obtain trajectory and global attractors for the primitive equations of moist atmosphere

global well posedness of strong solutions to the 3d primitive equations with horizontal eddy diffusivity in this paper we consider the initial boundary value problem of the 3d primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation global well posedness of strong solutions are established with h2 initial data 2014 elsevier inc all rights reserved

zero mach number limit of the compressible primitive equations well prepared initial data author s liu xin titi edriss s abstract this work concerns the zero mach number limit of the compressible primitive equations the primitive equations with the incompressibility condition are identified as the limiting equations the convergence with well prepared initial data i e initial data without acoustic oscillations is rigorously justified and the convergence rate is shown to be of order mathcal o varepsilon as varepsilon rightarrow 0 where varepsilon represents the mach number as a byproduct we construct a class of global solutions to the compressible primitive equations which are close to the incompressible flows

on the long time stability of a temporal discretization scheme for the three dimensional viscous primitive equations in this article a semi discretized euler scheme to solve the three dimensional viscous primitive equations is studied based on suitable assumptions on the initial data and forcing terms the long time stability of the proposed scheme is proven by showing that the h 1 norm in space variables of the solutions is bounded at each time step when the time step satisfies certain smallness condition

existence of a strong solution and trajectory attractor for a climate dynamics model with topography effects abstract the primitive three dimensional viscous equations for large scale atmosphere dynamics are commonly used in weather and climate predictions and multiple theoretical analyses have been performed on them however few studies have considered topographic effects which have a remarkable influence on climate factors e g atmospheric temperature and wind velocity in this study a climate dynamics model with topography and non stationary external force effects based on the navier stokes equations and a temperature equation is analyzed the existence and uniqueness of a global strong solution for this system is demonstrated based on the initial data assumptions in addition the existence of a universal attractor in the dynamic system is confirmed

on the complexity of minmax regret linear programming abstract we consider linear programming problems with uncertain objective function coefficients for each coefficient of the objective function an interval of uncertainty is known and it is assumed that any coefficient can take on any value from the corresponding interval of uncertainty regardless of the values taken by other coefficients it is required to find a minmax regret solution this problem received considerable attention in the recent literature but its computational complexity status remained unknown we prove that the problem is strongly np hard this gives the first known example of a minmax regret optimization problem that is np hard in the case of interval data representation of uncertainty but is polynomially solvable in the case of discrete scenario representation of uncertainty

robust regret for uncertain linear programs with application to co production models abstract this paper considers the regret optimization criterion for linear programming problems with uncertainty in the data inputs the problems of study are more challenging than those considered in previous works that address only interval objective coefficients and furthermore the uncertainties are allowed to arise from arbitrarily specified polyhedral sets to this end a safe approximation of the regret function is developed so that the maximum regret can be evaluated reasonably efficiently by leveraging on previous established results and solution algorithms the proposed approach is then applied to a two stage co production newsvendor problem that contains uncertainties in both supplies and demands computational experiments demonstrate that the proposed regret approximation is reasonably accurate and the corresponding regret optimization model performs competitively well against other optimization approaches such as worst case and sample average optimization across different performance measures

voas and rank two instanton scfts we analyze the mathcal n 2 superconformal field theories that arise when a pair of d3 branes probe an f theory singularity from the perspective of the associated vertex operator algebra we identify these vertex operator algebras for all cases we find that they have a completely uniform description parameterized by the dual coxeter number of the corresponding global symmetry group we further present free field realizations for these algebras in the style of recent work by three of the authors these realizations transparently reflect the algebraic structure of the higgs branches of these theories we find fourth order linear modular differential equations for the vacuum characters schur indices of these theories which are again uniform across the full family of theories and parameterized by the dual coxeter number we comment briefly on expectations for the still higher rank cases

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

infinite chiral symmetry in four dimensions we describe a new correspondence between four dimensional conformal field theories with extended supersymmetry and two dimensional chiral algebras the meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four dimensional theory infinite chiral symmetry has far reaching consequences for the spectral data correlation functions and central charges of any four dimensional theory with mathcal n 2 superconformal symmetry

large deviation principles for 3d stochastic primitive equations abstract in this paper we establish the freidlin wentzell s large deviations for 3d stochastic primitive equations with small noise perturbation the weak convergence approach plays an important role

the existence of a pullback attractor for the three dimensional non autonomous planetary geostrophic viscous equations of large scale ocean circulation abstract in this paper we prove the existence of a pullback attractor in h 2 for the process u t t associated with the three dimensional non autonomous planetary geostrophic viscous equations of large scale ocean circulation by asymptotic a priori estimates

optimal control of the primitive equations of the ocean with state constraints abstract we investigate in this article pontryagin s maximum principle for a class of control problems associated with the primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in wang 2002 7 for the three dimensional navier stokes ns equations the main difference between this work and wang 2002 7 is that the nonlinearity in the pes is stronger than in the three dimensional ns systems

the exponential behavior of a stochastic globally modified cahn hilliard navier stokes model with multiplicative noise abstract in this article we study the stability of weak solutions to a stochastic version of a globally modified coupled cahn hilliard navier stokes model with multiplicative noise the model consists of the globally modified navier stokes equations for the velocity coupled with an cahn hilliard model for the order phase parameter we prove that under some conditions on the forcing terms the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions we also prove a result related to the stabilization of these equations

a non autonomous two phase flow model with oscillating external force and its global attractor abstract in this article we consider a non autonomous diffuse interface model for an isothermal incompressible two phase flow in a two dimensional bounded domain assuming that the external force is singularly oscillating and depends on a small parameter we prove the existence of the uniform global attractor a furthermore using the method similar to that of chepyzhov and vishik 2007 22 in the case of the two dimensional navier stokes systems we study the convergence of a as goes to zero let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two dimensional navier stokes system studied in chepyzhov and vishik 2007 22

finite time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics in an earlier work we have shown the global for all initial data and all time well posedness of strong solutions to the three dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics in this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid non viscous primitive equations if they exist they blow up in finite time specifically we consider the three dimensional inviscid primitive equations in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no normal flow boundary conditions on the solid top and bottom boundaries for certain class of initial data we reduce this system into the two dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions and then we show that for specific sub class of initial data the corresponding smooth solutions of the reduced inviscid two dimensional system develop singularities in finite time

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

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

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

robust multi market newsvendor models with interval demand data we present a robust model for determining the optimal order quantity and market selection for short life cycle products in a single period newsvendor setting due to limited information about demand distribution in particular for short life cycle products stochastic modeling approaches may not be suitable we propose the minimax regret multi market newsvendor model where the demands are only known to be bounded within some given interval in the basic version of the problem a linear time solution method is developed for the capacitated case we establish some structural results to reduce the problem size and then propose an approximation solution algorithm based on integer programming finally we compare the performance of the proposed minimax regret model against the typical average case and worst case models our test results demonstrate that the proposed minimax regret model outperformed the average case and worst case models in terms of risk related criteria and mean profit respectively

pontryagin s maximum principle for optimal control of the stationary primitive equations of the ocean abstract we study in this article pontryagin s maximum principle for a class of control problems associated with the stationary primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in g wang pontryagin s maximum principle for optimal control of the stationary navier stokes equations nonlinear anal 52 2003 1853 1866 in the case of the three dimensional navier stokes ns equations the main difference between this work and the one cited above is that the nonlinearity in pes is stronger than that in the three dimensional ns equations

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

global well posedness for the lake equations abstract we prove global well posedness for the lake equations these equations arise to leading order in a low aspect ratio low froude number i e low wave speed and very small wave amplitude expansion of the three dimensional incompressible euler equations in a basin with a free upper surface and a spatially varying bottom topography well posedness means that there exists a solution that it is unique and that it depends continuously on the data i e the initial condition and the bottom topography our approach follows the works of yudovitch and bardos in constructing the solutions as the inviscid limit of solutions to a system with artificial viscosity which is the analog of the navier stokes with respect to the euler equations one of the main assumptions in the present work is a nondegenerate bottom topography

averaging of the planetary 3d geostrophic equations with oscillating external forces abstract in this article we consider a non autonomous three dimensional planetary geostrophic model of the ocean with a singularly oscillating external force depending on a small parameter we prove the existence of the uniform global attractor a furthermore using the method of 11 in the case of the two dimensional navier stokes systems we study the convergence of a as goes to zero

first order decoupled method of the primitive equations of the ocean i time discretization abstract in this article we study the time discretization of the first order decoupled semi implicit scheme of the 3d primitive equations of the ocean in the case of the non slip and non heat flux boundary conditions on the side the almost unconditional stability of the scheme and the optimal error estimates of the time discrete velocity pressure and density are provided

a second order decoupled implicit explicit method of the 3d primitive equations of ocean ii finite element spatial discretization summary a fully discrete second order decoupled implicit explicit method is proposed for solving 3d primitive equations of ocean in the case of dirichlet boundary conditions on the side where a second order decoupled implicit explicit scheme is used for time discretization and a finite element method based on the p1 p1 p1 p1 p1 elements for velocity pressure and density is used for spatial discretization of these primitive equations optimal h1 l2 h1 error estimates for numerical solution uhn phn hn and an optimal l2 error estimate for uhn hn are established under the convergence condition of 0 h 1 0 2 and 3h for some positive constants 1 2 and 3 furthermore numerical computations show that the h1 l2 h1 convergence rate for numerical solution uhn phn hn is of o h 2 and an l2 convergence rate for uhn hn is o h2 2 with the assumed convergence condition where h is a mesh size and is a time step size more practical calculations are performed as a further validation of the numerical method copyright 2016 john wiley sons ltd

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

a fourth order numerical method for the planetary geostrophic equations with inviscid geostrophic balance the planetary geostrophic equations with inviscid balance equation are reformulated in an alternate form and a fourth order finite difference numerical method of solution is proposed and analyzed in this article in the reformulation there is only one prognostic equation for the temperature field and the velocity field is statically determined by the planetary geostrophic balance combined with the incompressibility condition the key observation is that all the velocity profiles can be explicitly determined by the temperature gradient by utilizing the special form of the coriolis parameter this brings convenience and efficiency in the numerical study in the fourth order scheme the temperature is dynamically updated at the regular numerical grid by long stencil approximation along with a one sided extrapolation near the boundary the velocity variables are recovered by special solvers on the 3 d staggered grid furthermore it is shown that the numerical velocity field is divergence free at the discrete level in a suitable sense fourth order convergence is proven under mild regularity requirements

a heuristic to minimax absolute regret for linear programs with interval objective function coefficients decision makers faced with uncertain information often experience regret upon learning that an alternative action would have been preferable to the one actually selected models that minimize the maximum regret can be useful in such situations especially when decisions are subject to ex post review of particular interest are those decision problems that can be modeled as linear programs with interval objective function coefficients the minimax regret solution for these formulations can be found using an algorithm that at each iteration solves first a linear program to obtain a candidate solution and then a mixed integer program mip to maximize the corresponding regret the exact solution of the mip is computationally expensive and becomes impractical as the problem size increases in this paper we develop a heuristic for the mip and investigate its performance both alone and in combination with exact procedures the heuristic is shown to be effective for problems that are significantly larger than those previously reported in the literature

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

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

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 robust lot sizing problem with ill known demands the paper deals with a lot sizing problem with ill known demands modeled by fuzzy intervals whose membership functions are possibility distributions for the values of the uncertain demands optimization criteria in the setting of possibility theory that lead to choose robust production plans under fuzzy demands are given some algorithms for determining optimal robust production plans with respect to the proposed criteria and for evaluating production plans are provided some computational experiments are presented

on a constant factor approximation for minmax regret problems using a symmetry point scenario abstract in order to find a robust solution under an unknown linear cost function it will be considered the minmax regret criterion it is assumed the vector of costs can take on values from a given uncertainty set the resulting optimization model has been extensively analyzed in the literature when the uncertain costs are modeled by closed intervals unfortunately except for rare applications this problem has np hard complexity which has led to the appearance of approximated methods seeking for good solutions in a short computational time the most recent approaches solve the deterministic version of the optimization problem under the midpoint scenario that realization in which each cost takes the midpoint of the closed interval proceeding in this way it can be obtained a 2 approximation for a wide class of minmax regret optimization models in this paper it is shown how this approach can be extended to a new class of minmax regret problems including more general uncertainty sets

discontinuous galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems abstract we formulate a coupled surface subsurface flow model that relies on hydrostatic equations with free surface in the free flow domain and on darcy s law in the subsurface the model is discretized using the local discontinuous galerkin method and a statement of discrete energy stability is proved for the fully non linear coupled system the performance of the proposed scheme is illustrated by a numerical convergence study and a realistic example

pullback attractors for a non autonomous homogeneous two phase flow model abstract this article studies the pullback asymptotic behavior of solutions for a non autonomous homogeneous two phase flow model in a two dimensional domain we prove the existence of pullback attractors a v in v the velocity has the h 1 regularity and a y in y the velocity has the l 2 regularity then we verify the regularity of the pullback attractors by proving that a v a y which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data the method used in this article is similar to the one used in zhao and zhou 2007 42 in the case of the non autonomous incompressible non newtonian fluid in a two dimensional domain let us mention that the nonlinearity involved in the model considered in this article is stronger than the one in the two dimensional non newtonian flow studied in zhao and zhou 2007 42

estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems we show that the long time behavior of the projection of the exact solutions to the navier stokes equations and other dissipative evolution equations on the finite dimensional space of interpolant polynomials determines the long time behavior of the solution itself provided that the spatial mesh is fine enough we also provide an explicit estimate on the size of the mesh moreover we show that if the evolution equation has an inertial manifold then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite dimensional space spanned by the approximating polynomials our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation

global well posedness of the three dimensional primitive equations with only horizontal viscosity and diffusion 2015 wiley periodicals inc in this paper we consider the initial boundary value problem of the three dimen sional primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal momentum equations and only horizontal diffusion in the temperature equation global well posedness of the strong solution is established for any h2 initial data an n dimensional logarithmic sobolev embedding inequality which bounds the l norm in terms of the lq norms up to a logarithm of the lp norm for p n of the first order derivatives and a system version of the classic gronwall inequality are exploited to establish the required a priori h2 estimates for global regularity