backward error analysis of a full discretization scheme for a class of semilinear parabolic partial differential equations a numerical method for reaction diffusion equations with analytic nonlinearity is presented for which a rigorous backward error analysis is possible we construct a modified equation which describes the behavior of the full discretization scheme up to exponentially small errors in the step size in the construction the numerical scheme is first exactly embedded into a nonautonomous equation this equation is then averaged with only exponentially small remainder terms the long time behavior near hyperbolic equilibria the persistence of homoclinic orbits and regularity properties are analyzed

global well posedness and analyticity of solutions to three dimensional hall mhd equations abstract in this paper suppose that the initial data is sufficiently small we study the global well posedness and analyticity of mild solutions to the three dimensional incompressible hall mhd equations

existence of solutions and gevrey class regularity for leray alpha equations abstract in this paper we consider some equations similar to navier stokes equations the three dimensional leray alpha equations with space periodic boundary conditions we establish the regularity of the equations by using the classical faedo galerkin method our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the leray alpha equations furthermore the strong solutions are analytic in time with values in the gevrey class of functions for the space variable the relations between the leray alpha equations and the navier stokes equations are also considered

on smooth dependence on initial conditions for dissipative pdes an ode type approach abstract using elementary ode tools we show that for a class of dissipative pdes including navier stokes equations on the plane with periodic boundary conditions galerkin projections are c 1 convergent i e the partial derivatives with respect to initial conditions for galerkin projections converge giving rise to smooth dependence on initial conditions for full pde

natural coordinate system approach to coupled n phase fluid flow in curved domains a natural coordinate system is introduced in this work to handle the flow of a multiphase fluid in domains possessing curved boundaries a multiple transformation that is capable of mapping a single physical domain onto a single computational domain is derived this work furnishes the necessary theoretical foundation for the concept of natural coordinate systems for multiphase flows

well posedness and decay of solutions for three dimensional generalized navier stokes equations abstract in this paper we study the generalized incompressible navier stokes equations in r 3 based on the energy estimates and regularization of the initial data with the heat semigroup we prove the well posedness of solutions in h 5 4 2 r 3 provided that the h 5 4 2 r 3 norm of initial data is sufficiently small in addition in contrast to the generalized heat equation the upper bound of the time decay rate of solutions to the generalized navier stokes equations is also established

local existence and gevrey regularity of 3 d navier stokes equations with l p initial data abstract we obtain local existence and gevrey regularity of 3 d periodic navier stokes equations in case the sequence of fourier coefficients of the initial data is in l p p 3 2 the l p norm of the sequence of fourier coefficients of the solution and its analogous gevrey norm remains bounded on a time interval whose length depends only on the size of the body force and the l p norm of the fourier coefficient sequence of the initial data the control on the gevrey norm produces explicit estimates on the analyticity radius of the solution as in foias and temam j funct anal 87 1989 359 369 the results provide an alternate approach in estimating the space analyticity radius of solutions to navier stokes equations than the one presented by grujic and kukavica j funct anal 152 1998 447 466

high order compact computation and nonuniform grids for streamfunction vorticity equations abstract fourth order compact difference schemes for steady streamfunction vorticity formulation of 2d incompressible navier stokes equations on nonuniform grids are derived using maple software package especially we deduce fourth order compact difference scheme of the first partial derivative terms in order to resolve boundary layers grid transformation techniques are used which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme a krylov subspace iterative method with an ilut preconditioning technique is employed to solve the resulting linear system the proposed high accuracy computation method is applied to two model problems computational results are compared with results computed by other schemes in the literature

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

on the helicity in 3d periodic navier stokes equations i the non statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

on the helicity in 3d periodic navier stokes equations ii the statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

a nonstiff euler discretization of the complex ginzburg landau equation in one space dimension a nonstiff discretization method is applied to the complex ginzburg landau cgl equation with periodic boundary conditions begin eqnarray partial t u 1 i nu partial xx 2 u ru 1 i mu u 2 u u t geq 0 end eqnarray this parabolic equation is discretized first in space by a truncated fourier series that is a finite fourier sum and then in time by an explicit euler method the exponential decay of fourier modes in both the original cgl equation and the resulting equation for the truncated fourier series independently from the number of fourier modes in the truncation is used in an essential way to prove the nonstiffness also dissipativity of the discretized equation is established and numerical results are discussed

multiscale fractional step schemes for the numerical simulation of the rotating shallow water flows with complex periodic topography in this paper we study several multiscale fractional step schemes for the numerical solution of the rotating shallow water equations with complex topography we consider the case of periodic boundary conditions f plane model spatial discretization is obtained using a fourier spectral galerkin method for the schemes presented in this paper we consider two approaches the first approach multiscale schemes is based on topography scale separation and the numerical time integration is function of the scales the second approach is based on a splitting of the operators and the time integration method is function of the operator considered fractional step schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme with these multiscale fractional step schemes the objective is to propose new schemes giving numerical results similar to those obtained using only one uniform fine grid n nn n and a time step t but with a cpu time near the cpu time needed when using only one coarse grid n1 n1n1 n1 n1 t t t

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

oscillations of solutions of the kuramoto sivashinsky equation abstract oscillation properties of graphs of solutions of the kuramoto sivashinsky equation are studied it is shown that every solution either converges to the stationary solution 0 with an exponential rate or after some time the cardinalities of the sets of zeros in the space variable of all the spatial derivatives remain bounded by certain constants independent of the particular solution

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

exponential averaging for hamiltonian evolution equations we derive estimates on the magnitude of non adiabatic interaction between a hamiltonian partial differential equation and a high frequency nonlinear oscillator assuming spatial analyticity of the initial conditions we show that the dynamics can be transformed to the uncoupled dynamics of an infinite dimensional hamiltonian system and an anharmonic oscillator up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator the result is derived from an abstract averaging theorem for infinite dimensional analytic evolution equations in gevrey spaces refining upon a similar result by neishtadt for analytic ordinary differential equations the temporal estimate crucially depends on the spatial regularity of the initial condition the result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data an application is provided by a system of nonlinear schrodinger equations coupled to a rapidly forcing single mode representing small scale oscillations we provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations

a new boundary problem for the two dimensional navier stokes system we formulate a new boundary value problem for the 2d navier stokes system on the unit square under some suitable assumptions on the initial velocity we obtain quantitative decay estimates of the fourier modes of both the vorticity and the velocity it is found that in one direction the fourier modes decay exponentially and along the other direction their decay is only power like

computational complexities and streamfunction coordinates in this work we consider some of the computational aspects of viscous fluid flow through two dimensional curvilinear domains associated with the use of the von mises transformation in mapping a physical plane into a computational plane are some complexities that arise in the main part due to the infinite nature of the jacobian of the transformation some recommendations are made in this work to overcome some of the arising computational difficulties

a rigorous treatment of experimental observations for the two dimensional navier stokes equations within a fully rigorous framework this paper shows that a finite number of point observations distributed in both space and time serve to determine the fully developed flow of a two dimensional fluid throughout the whole flow domain with the measurements taken at one fixed time the results imply that if the flow is resolved on a sufficiently small scale then its dynamics are entirely determined applied to experimental observations taken randomly from a small neighbourhood of one point in space but at staggered times they imply a version of taken s time delay embedding theorem

a note on gevrey class regularity for the solutions of the navier stokes equations abstract in this note we present results on gevrey class regularity of the solutions of the navier stokes equations with space periodic boundary conditions we give a generalization of results of c foias and r temam j funct anal 87 359 369 1988

on the lagrangian and eulerian analyticity for the euler equations abstract we address preservation of the lagrangian analyticity radius of solutions to the euler equations belonging to natural analytic space based on the size of taylor or gevrey coefficients we prove that if the solution belongs to such space then the solution also belongs to it for a positive amount of time we also prove the local analog of this result for a sufficiently large gevrey parameter however we show that the preservation holds independently of the size of the radius finally we construct a solution which shows that the first result does not hold in the eulerian setting

on the regularity and convergence of solutions to the 3d navier stokes voigt equations abstract in this paper we consider the 3d navier stokes voigt equations with periodic boundary conditions we first prove the higher order global regularity including both sobolev and gevrey regularity of solutions to the navier stokes voigt equations then we show the convergence of solutions of the 3d navier stokes voigt equations to the corresponding strong solution of the limit 3d navier stokes equations on the interval of existence of the latter as the parameter tends to zero

controllability of probabilistic boolean control networks based on transition probability matrices in this paper we propose a new approach to investigate the controllability and reachability of probabilistic boolean control networks pbcns with forbidden states we first give a simple algebraic formula for the transition probability between two states in a given number of time step while avoiding a set of forbidden states then we construct the controllability matrix based on a new operator and some necessary and sufficient conditions are obtained for the controllability and reachability of pbcns a numerical example is given to illustrate the efficiency of the obtained results

the sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension abstract we construct local fourth order finite difference approximations of first and second derivatives on nonuniform grids in one dimension the approximations are required to satisfy symmetry relationships that come from the analogous higher dimensional fundamental operators the divergence the gradient and the laplacian for example we require that the discrete divergence and gradient be negative adjoint of each other div grad and the discrete laplacian is defined as lap divgrad the adjointness requirement on the divergence and gradient guarantees that the laplacian is a symmetric negative operator the discrete approximations we derive are fourth order on smooth grids but the approach can be extended to create approximations of arbitrarily high order we analyze the loss of accuracy in the approximations when the grid is not smooth and include a numerical example demonstrating the effectiveness of the higher order methods on nonuniform grids

fractional derivative estimates in gevrey spaces global regularity and decay for solutions to semilinear equations in rn abstract we propose a unified functional analytic approach to study the uniform analytic gevrey regularity and the decay of solutions to semilinear elliptic equations on r n first we develop a fractional calculus for nonlinear maps in banach spaces of l p based gevrey functions 1 h p s r n regularity with s scr 0 depending on the nonlinearity next we investigate the type of decay polynomial or exponential of the derivatives of solutions to semilinear elliptic equations provided they decay a priori slowly as o x x for some small 0 the restrictions involved in our results are optimal in particular given a hyperplane l we construct 2d 2 strongly singular solutions locally in h p s r n for s

first order accurate finite difference schemes for boundary vorticity approximations in curvilinear geometries in this work we develop first order accurate forward finite difference schemes for the first derivative on both a uniform and a non uniform grid the schemes are applied to the calculation of vorticity on a solid wall of a curvilinear two dimensional channel the von mises coordinates are used to transform the governing equations and map the irregular domain onto a rectangular computational domain vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain and the derived finite difference schemes are used to calculate the vorticity at the computational boundary grid points using combinations of up to five computational domain grid points this work extends previous work awartani et al 2005 3 in which higher order schemes were obtained for the first derivative using up to four computational domain grid points the aim here is to shed further light onto the use of first order accurate non uniform finite difference schemes that are essential when the von mises transformation is used results show that the best schemes are those that use a natural sequence of non uniform grid points it is further shown that for non uniform grid with clustering near the boundary solution deteriorates with increasing number of grid points used by contrast when a uniform grid is used solution improves with increasing number of grid points used

stability of solutions of a class of nonlinear fractional laplacian parabolic problems abstract in this paper we study the forward and the backward in time problems for a class of nonlinear partial differential equations of fractional laplace operator with nonlinear source function the equations are generalized from lots of well known equations such as the burger equation the cahn hilliard equation herein we investigate the stability of the solution of the forward problem depended on the fractional order of the operator and the instability of the backward problem and then we establish a regularization method some numerical experiments are performed to verify efficiency and accuracy of our method

on the ensemble average in the study of approximate inertial manifolds ii abstract in this paper we present new estimates on the ensemble averages of the fluid dynamical quantities that have significances in the study of approximate inertial manifolds of the 2 d navier stokes equations some immediate implications of these estimates on the physics of 2 d turbulent flows are discussed

long time stability of high order multistep numerical schemes for two dimensional incompressible navier stokes equations the long time stability properties of a few multistep numerical schemes for the two dimensional incompressible navier stokes equations formulated in vorticity stream function are investigated in this article these semi implicit numerical schemes use a combination of explicit adams bashforth extrapolation for the nonlinear convection term and implicit adams moulton interpolation for the viscous diffusion term up to fourth order accuracy in time as a result only two poisson solvers are needed at each time step to achieve the desired temporal accuracy the fully discrete schemes with fourier pseudospectral approximation in space are analyzed in detail with the help of a priori analysis and aliasing error control techniques we prove uniform in time bounds for these high order schemes in both l 2 and h m norms for m ge 1 provided that the time step is bounded by a given constant such a long time stability is also demonstrated by the numerical experiments

space time analyticity of weak solutions to linear parabolic systems with variable coefficients analytic smoothing properties of a general strongly parabolic linear cauchy problem of second order in rn 0 t with analytic coefficients in space and time variables are investigated they are expressed in terms of holomorphic continuation of global weak l2 type solutions to the system given 0 0 are constants depending upon t the proof uses the extension of a solution to an l2 type solution in a domain in cn c such that this extension satisfies the cauchy riemann equations the holomorphic extension is thus obtained in a hardy space h2 applications include market completion by european options in finance

well posedness and regularity of the generalized burgers equation in periodic gevrey spaces we consider the generalized burgers equation in a class of gevrey functions g s t we show that the generalized burgers equation is well posed in this space furthermore we show that the solution is gevrey in the spacial variable and gevrey 2 in the time variable

space analyticity for the navier stokes and related equations with initial data inlp abstract we introduce a method of estimating the space analyticity radius of solutions for the navier stokes and related equations in terms of l p and l norms of the initial data the method enables us to express the space analyticity radius for 3d navier stokes equations in terms of the reynolds number of the flow also for the kuramoto sivashinsky equation we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period

time averaging under fast periodic forcing of parabolic partial differential equations exponential estimates the phases of a large class of parabolic partial differential equations with rapid time periodic forcing can be separated up to exponential small errors the originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing this is a counterpart for partial differential equations of the theorem by a neishtadt 1984 j appl math mech 48 134 139 for ordinary differential equations in our case the exponential rate depends on time t and the estimates have the form h exp c t h 1 3

on maximum enstrophy growth in a hydrodynamic system abstract enstrophy e plays an important role in the study of regularity of solutions to the three dimensional navier stokes equation the best estimates for its growth available to date do not rule out the possibility of enstrophy becoming unbounded in finite time which would indicate loss of regularity of solutions it is therefore interesting to investigate sharpness of such finite time bounds for enstrophy growth we consider this question in the context of burgers equation which is used as a toy model the problem of saturation of finite time estimates for the enstrophy growth is stated as a pde constrained optimization problem max e t e 0 subject to e 0 e 0 where the control variable represents the initial condition which is solved numerically using an adjoint based gradient method for a wide range of time windows t and initial enstrophies e 0 we show that this optimization problem admits a discrete family of maximizers parameterized by the wavenumber m whose members are rescaled copies of the fundamental maximizer corresponding to m 1 it is found that the maximum enstrophy growth in finite time scales with the initial enstrophy as e 0 where 3 2 the exponent is smaller than 3 predicted by analytic means therefore suggesting possible lack of sharpness of analytical estimates

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

on the exponential type explosion of navier stokes equations abstract the classical results on the explosion of the maximal solution of incompressible navier stokes equations are of type c t t 0 for some 0 0 inspired by the works benameur and selmi 2012 15 chemin 2004 16 we use the sobolev gevrey spaces to get better explosion results precisely if e a d 1 u 0 h s r 3 then e a d 1 u t h s is at least of the order t t 1 exp c t t 2 for some 1 0 and 2 0 fourier analysis and standard techniques are used

analytic well posedness of periodic gkdv abstract in the periodic case it is proved that the cauchy problem for the generalized korteweg de vries equation gkdv is locally well posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x axis thus the uniform analyticity radius of the solution does not change as time progresses also information about the regularity of the solution in the time variable is provided

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

the algebraic approximation of attractors the finite dimensional case abstract in this article we present a constructive method for approximating attractors the approximate sets are parts of algebraic sets i e the zeros of high degree polynomial functions and they can approximate the attractor at an arbitrary high level of accuracy

invariant subspaces of the periodic navier stokes and magnetohydrodynamics equations symmetries and inverse cascades it is shown that when the initial condition and the forcing term of the periodic navier stokes or magnetohydrodynamics equations have fourier coefficients which vanish outside a certain semigroup of frequencies the same happens to the solutions for all time subgroups of frequencies correspond to solutions possessing certain symmetries by taking as a semigroup the frequencies whose fourier components are non negative integers we get a class of solutions for which the higher modes do not influence the evolution of the lower ones therefore the phenomenon of inverse cascading cannot occur for them

on the rate of convergence of the nonlinear galerkin methods in this paper we provide estimates to the rate of convergence of the nonlinear galerkin approximation method in particular and by means of an illustrative example we show that the nonlinear galerkin method converges faster than the usual galerkin method

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

attractors and inertial manifolds for finite difference approximations of the complex ginzburg landau equation a semidiscrete spatial finite difference approximation and two fully discrete finite difference approximations to the complex ginzburg landau equation are considered in this paper the existence of an inertial manifold is proved inside a discrete h1 absorbing ball for the semi discrete approximation by showing that certain spectral properties hold for the linear operator and certain lipschitz properties hold for the nonlinear term convergence of the true and discrete inertial manifolds is then shown by considering the contraction mappings used to construct the inertial manifolds a fully implicit scheme is shown to form a continuous semigroup in a discrete l2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete l2 and h1 spaces are constructed existence of a global attractor follows furthermore the existence of an inertial manifold is proved by showing that certain spectral and lipschitz properties hold for a semi implicit scheme existence of absorbing balls in the discrete space l2 of radii independent of the spatial and temporal mesh is proved existence of a global attractor follows

on depletion of the vortex stretching term in the 3d navier stokes equations certain new cancellation properties in the vortex stretching term are detected leading to new geometric criteria for preventing finite time blow up in the 3d navier stokes equations

postprocessing the galerkin method a novel approach to approximate inertial manifolds a postprocess of the standard galerkin method for the discretization of dissipative equations is presented the postprocessed galerkin method uses the same approximate inertial manifold phi app to approximate the high wave number modes of the solution as in the nonlinear galerkin nlg method however in this postprocessed galerkin method the value of phi app is calculated only once after the time integration of the standard galerkin method is completed contrary to the nlg in which phi app evolves with time and affects the time evolution of the lower wave number modes the postprocessed galerkin method which is much cheaper to implement computationally than the nlg is shown in the case of fourier modes to possess the same rate of convergence accuracy as the simplest version of the nlg which is based on either the foias manley temam approximate inertial manifold or the euler galerkin approximate inertial manifold this is proved for some problems in one and two spatial dimensions including the navier stokes equations under periodic boundary conditions the advantages of postprocessing that we present here apply not only to the standard galerkin method but also to the computationally more efficient pseudospectral method

on approximate inertial manifolds for two and three dimensional turbulent flows abstract in this paper we present some results concerning finite dimensional analytic manifolds that approximate the global attractor and the structure of the small eddies for the navier stokes equations under periodic boundary conditions we show that the smoother the external force the better the approximation

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

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

an algebraic reduction of the scaling gap in the navier stokes regularity problem it is shown within a mathematical framework based on the suitably defined scale of sparseness of the super level sets of the positive and negative parts of the vorticity components and in the context of a blow up type argument that the ever resisting scaling gap that is the scaling distance between a regularity criterion and a corresponding a priori bound shortly a measure of the super criticality of the three dimensional navier stokes regularity problem can be reduced by an algebraic factor since independent fundamental works of ladyzhenskaya prodi and serrin as well as kato and fujita in 1960s all the reductions have been logarithmic in nature regardless of the functional set up utilized more precisely it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy level bound while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria the mathematics presented was inspired by morphology of the regions of intense vorticity velocity gradients observed in computational simulations of turbulent flows as well as by the physics of turbulent cascades and turbulent dissipation

kraichnan turbulence via finite time averages relations central to kraichnan s theory of fully developed two dimensional turbulence are rigorously established for finite time averages in particular we prove that if the ratio of the averages of palinstrophy to enstrophy is large then a large inertial range displaying an enstrophy cascade exists moreover if this ratio is comparable up to a logarithm to the dissipation wave number a necessary condition for turbulence then the power law for the energy spectrum until now derived only heuristically is rigorously shown to provide up to a logarithm an upper bound for the energy spectrum finally we show that deep in the dissipation range the palinstrophy contributed by eddies smaller than a specified length decays exponentially in the corresponding wave number the averaging times needed for these relations are bounded in terms of the generalized grashof number independent of the solution for which the time averages are taken solutions are not assumed to be on the global attractor merely within the absorbing ball an easily verified condition

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

exponential mixing for the 3d stochastic navier stokes equations we study the navier stokes equations in dimension 3 ns3d driven by a noise which is white in time we establish that if the noise is at the same time sufficiently smooth and non degenerate in space then the weak solutions converge exponentially fast to equilibrium

gevrey regularity of mild solutions to the parabolic elliptic system of drift diffusion type in critical besov spaces abstract this paper is devoted to studying analyticity of mild solutions to the parabolic elliptic system of drift diffusion type with small initial data in critical besov spaces more precisely by using multilinear singular integrals and fourier localization argument we show that global in time mild solutions are gevrey regular i e e t 1 v e t 1 w l t b p r 2 n p r n l t 1 b p r n p r n for all 1 p 2 n and 1 r where 1 is the fourier multiplier whose symbol is given by 1 i 1 n i as a corollary we obtain decay estimates of mild solutions in critical besov spaces without using cumbersome recursive estimation of higher order derivatives

analyticity of solutions to the barotropic compressible navier stokes equations abstract in this paper we establish analyticity of solutions to the barotropic compressible navier stokes equations describing the motion of the density and the velocity field u in r 3 we assume that 0 is a small perturbation of 1 and 1 1 0 u 0 are analytic in besov spaces with analyticity radius 0 we show that the corresponding solutions are analytic globally in time when 1 1 0 u 0 are sufficiently small to do this we introduce the exponential operator e t d acting on 1 1 u where d is the differential operator whose fourier symbol is given by 1 1 2 3 and t is chosen to satisfy t globally in time

space analyticity for the nonlinear heat equation in a bounded domain 1 introductionin fot foias and temam introduced a method for estimating thespace analyticity radius of solutions of the navier stokes equation withperiodic boundary conditions for the numerous applications seee g cees crt dth ft gr k lo in our previous paper gk we did not use the fourier series and were thus able to treat the navier stokes and other semilinear parabolic equations with singular l

the dissipative scale of the stochastics navier stokes equation regularization and analyticity we prove spatial analyticity for solutions of the stochastically forced navier stokes equation provided that the forcing is sufficiently smooth spatially we also give estimates which extend to the stationary regime providing strong control of both of the expected rate of dissipation and fluctuations about this mean surprisingly we could not obtain non random estimates of the exponential decay rate of the spatial fourier spectra

on the hydrostatic approximation of the navier stokes equations in a thin strip abstract in this paper we first prove the global well posedness of a scaled anisotropic navier stokes system and the hydrostatic navier stokes system in a 2 d striped domain with small analytic data in the tangential variable then we justify the limit from the anisotropic navier stokes system to the hydrostatic navier stokes system with analytic data

gevrey regularity for a class of water wave models abstract local well posedness for a class of higher order nonlinear dispersive partial differential equations is obtained in spaces of functions analytic on a strip around the real axis the proof relies on estimates in space time norms adapted to the linear part of the equation the class of equations in view contains a number of equations arising in the modeling of waves in fluids and also in other applications