attractors for nonautonomous two dimensional space periodic navier stokes equations in this paper we apply a new theorem which was proven by lu et al attractors for non autonomous 2d navier stokes equations with normal external forces discrete contin dyn syst 8 701 2005 to nonautonomous two dimensional space periodic navier stokes equation we obtain the existence of the uniform attractor in d a which is the domain of the stokes operator a with periodic boundary conditions if external forces f t are a normal function in lloc2 r h1 the uniform attractor attracts all bounded subsets of d a in the norm of d a

on the large time galerkin approximation of the navier stokes equations in this article we are interested in the difficult problem of relating the large time behaviour of the navier stokes equations to the large time behaviour of their galerkin approximation we restrict ourselves to the simplest situation dealing with stationary solutions and the question is the following one if the computed approximation of the time dependent equations seems to converge to some limit as t to infty is the same true for the exact problem and are the limits related we give here some sufficient conditions reasonably easy to verify which guarantee a positive answer to this question

flows in a tube structure equation on the graph the steady state navier stokes equations in thin structures lead to some elliptic second order equation for the macroscopic pressure on a graph at the nodes of the graph the pressure satisfies kirchoff type junction conditions in the non steady case the problem for the macroscopic pressure on the graph becomes nonlocal in time in the paper we study the existence and uniqueness of a solution to such one dimensional model on the graph for a pipe wise network we also prove the exponential decay of the solution with respect to the time variable in the case when the data decay exponentially with respect to time

dynamics of convection thermal convection in a fluid layer is an example of a dynamical system governed by partial differential equations as the relevant control parameter the rayleigh number is increased successive bifurcations may lead to chaos and the nature of the transition depends on the spatial structure of the flow numerical experiments with idealized symmetry and boundary conditions make it possible to explore nonlinear behaviour in some detail and to relate bifurcation structures to those found in appropriate low order systems two examples are used to illustrate transitions to chaos in two dimensional thermosolutal convection where the spatial structure is essentially trivial chaos is caused by a heteroclinic bifurcation involving a symmetric pair of saddle foci when convection is driven by internal heating several competing spatial structures are involved and the transition to chaos is more complicated in both two and three dimensional configurations although the first few bifurcations can be isolated a statistical treatment is needed for behaviour at high rayleigh numbers

number of degrees of freedom of two dimensional turbulence we derive upper bounds for the number of degrees of freedom of two dimensional navier stokes turbulence freely decaying from a smooth initial vorticity field omega x y 0 omega0 this number denoted by n is defined as the minimum dimension such that for n or n arbitrary n dimensional balls in phase space centered on the solution trajectory omega x y t for t 0 contract under the dynamics of the system linearized about omega x y t in other words n is the minimum number of greatest lyapunov exponents whose sum becomes negative it is found that n or c1re when the phase space is endowed with the energy norm and n or c2re 1 ln re 1 3 when the phase space is endowed with the enstrophy norm here c1 and c2 are constant and re is the reynolds number defined in terms of omega0 the system length scale and the viscosity nu the linear or nearly linear dependence of n on re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number this result is in a sharp contrast to the forced case for which well known estimates for the hausdorff dimension dh of the global attractor scale highly superlinearly with nu 1 we argue that the extra dependence of dh on nu 1 is not an intrinsic property of the turbulent dynamics rather it is a removable artifact brought about by the use of a time independent forcing as a model for energy and enstrophy injection that drives the turbulence

bifurcations of radially symmetric solutions in a coupled elliptic system with critical growth in rd for d 3 4 abstract we consider a system of coupled elliptic partial differential equations with critical growth in r d for d 3 4 and study bifurcations of three families of radially symmetric bounded solutions we reduce the problems of the three families to those of three symmetric homoclinic orbits in a four dimensional reversible system of ordinary differential equations and show that transcritical or pitchfork bifurcations of the three families occur at infinitely many parameter values numerical computations are also given to demonstrate our theoretical results

optimal liouville type theorems for noncooperative elliptic schrodinger systems and applications we study multi component elliptic schrodinger systems arising in nonlinear optics and bose einstein condensation phenomena we prove new liouville type nonexistence theorems as well as a priori bounds decay and singularity estimates this is shown under an optimal sobolev growth restriction on the nonlinearities thus improving on recent results of dancer et al and of tavares et al these systems are of non cooperative form and hence cannot be tackled by maximum principle methods such as moving planes instead we rely on a delicate combination of rellich pohozaev type identities sobolev and interpolation inequalities on sn 1 and feedback and measure arguments we also extend our results to a rather general class of gradient type systems

extensivity of two dimensional turbulence this study is concerned with how the attractor dimension of the two dimensional navier stokes equations depends on characteristic length scales including the system integral length scale the forcing length scale and the dissipation length scale upper bounds on the attractor dimension derived by constantin foias and temam are analysed it is shown that the optimal attractor dimension estimate grows linearly with the domain area suggestive of extensive chaos for a sufficiently large domain if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed for sufficiently small domain area a slightly super extensive estimate becomes optimal in the extensive regime the attractor dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined on physical grounds in terms of the average rate of shear this dissipation length scale which is not necessarily the scale at which the energy or enstrophy dissipation takes place can be identified with the dimension correlation length scale the square of which is interpreted according to the concept of extensive chaos as the area of a subsystem with one degree of freedom furthermore these length scales can be identified with a minimum length scale of the flow which is rigorously deduced from the concept of determining nodes

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

determining modes for the surface quasi geostrophic equation abstract we introduce a determining wavenumber for the surface quasi geostrophic sqg equation defined for each individual trajectory and then study its dependence on the force while in the subcritical and critical cases this wavenumber has a uniform upper bound it may blow up when the equation is supercritical a bound on the determining wavenumber provides determining modes and measures the number of degrees of freedom of the flow or resolution needed to describe a solution to the sqg equation

low dimensional attractors arise from forcing at small scales abstract standard estimates of the dimension of the attractor of the 2d navier stokes equations are given in terms of a dimensionless grashof number that measures the total amount of energy injected into the flow however this result takes no account of whether the forcing is concentrated at large or small scales by a simple modification of the usual argument this paper provides a bound that is linear in a modified grashof number that reflects the spatial structure of the forcing in particular as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases

necessary and sufficient conditions for the existence of exponential attractors for semigroups and applications abstract first we establish some necessary and sufficient conditions for the existence of exponential attractors by using limit compactness and a measure of non compactness then we provide a new method for proving the existence of exponential attractors we prove the existence of exponential attractors for reaction diffusion equations and 2d navier stokes equations as simple applications

periodic in time flow in thin structure equation on the graph abstract the dimension reduction for the non stationary time periodic navier stokes equations in thin structures leads to a non local in time reynolds type equation on the graph with kirchhoff type junction conditions in the vertices the existence and uniqueness of a solution to this problem is proved

asymptotic behavior of stochastic g navier stokes equations on a sequence of expanding domains the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain the limiting dynamics of stochastic 2d nonautonomous g navier stokes equations defined on a sequence of expanding domains are investigated where the limiting domain is unbounded by generalizing the energy equation method we show that the sequence of expanding cocycles is weakly equicontinuous and strongly equiasymptotically compact which lead to both existence and upper semicontinuity of the null expansion of the corresponding random attractor when the bounded domain approaches to the unbounded domain

entire nonradial solutions for non cooperative coupled elliptic system with critical exponents in r3 abstract we consider the following coupled elliptic system s n u 1 u n 2 n 2 u 2 n 2 v n n 2 in r n v 2 v n 2 n 2 v 2 n 2 u n n 2 in r n u v 0 u v d 1 2 r n where n 3 1 2 are two positive constants and 0 is the coupling constant when n 3 we prove the existence of infinitely many positive nonradial solutions

erratum to time periodic solutions of the navier stokes equations in unbounded cylindrical domains leray s problem for periodic flows poiseuille flows in infinite cylindrical pipes in spite of their enormous simplicity have a main role in many theoretical and applied problems as is well known the poiseuille flow is a stationary solution of the stokes and the navier stokes equations with a given constant flux time periodic flows in channels and pipes have a comparable importance however the problem of the existence of time periodic flows in correspondence to any given time periodic total flux is still an open problem a solution is known only in some very particular cases for instance the womersley flows our aim is to solve this problem in the general case the above existence result opens the way to further investigations as an example of this possibility we consider the extension of the classical leray s problem for poiseuille flows to arbitrary time periodic flows

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

asymptotic analysis of the non steady navier stokes equations in a tube structure ii general case abstract the non steady navier stokes equations with dirichlet boundary conditions are considered in thin tube structures these domains are connected finite unions of thin finite cylinders in the 2 d case respectively thin rectangles the complete asymptotic expansion of the solution is constructed it contains a regular part and three types of the boundary layer correctors in space in time and in space and in time the estimates for the difference of the exact solution and its j th asymptotic approximation are proved

a seven mode truncation of the plane incompressible navier stokes equations a model obtained by a seven mode truncation of the navier stokes equations for a two dimensional incompressible fluid on a torus is studied this model extending a previously studied five mode one exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis i e coexistence of attractors a stochastic behavior is found for high values of the reynolds number when no stable fixed points closed orbits or tori are present

determination of the solutions of the navier stokes equations by a set of nodal values we consider the navier stokes equations of a viscous incompresible fluid and we want to see to what extent these solutions can be determined by a discrete set of nodal values of these solutions the results presented here are exact results and not approximate ones we show in several cases that the solutions are entirely determined by their values on a discrete set provided this set contains enough points and these points are sufficiently densely distributed in a sense described in the article two typical results are the following ones two stationary solutions coincide if they coincide on a set sufficiently dense but finite similarly if the large time behavior of the solutions to the navier stokes equations is known on an appropriate discrete set then the large time behavior of the solution itself is totally determined

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

on the regularity for the boussinesq equations in a bounded domain we address the regularity for the 2d boussinesq equations with zero diffusivity and positive viscosity on an open bounded smooth domain in particular we prove that for the initial velocity in h2 and the initial density in h1 there exists a global in time solution for which the velocity and the density are locally in time bounded functions in h2 and h1 respectively

liouville type theorems monotonicity results and a priori bounds for positive solutions of elliptic systems this paper is a contribution to the study of boundary value problems for systems of elliptic partial differential equations

analytic and algebraic conditions for bifurcations of homoclinic orbits i saddle equilibria abstract we study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four dimensional systems which may be hamiltonian or not only one parameter is enough to treat these types of bifurcations in hamiltonian systems but two parameters are needed in general systems we apply a version of melnikov s method due to gruendler to obtain saddle node and pitchfork types of bifurcation results for homoclinic orbits furthermore we prove that if these bifurcations occur then the variational equations around the homoclinic orbits are integrable in the meaning of differential galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds we illustrate our theories with an example which arises as stationary states of coupled real ginzburg landau partial differential equations and demonstrate the theoretical results by numerical ones

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

asymptotic analysis of the non steady navier stokes equations in a tube structure i the case without boundary layer in time abstract the non steady navier stokes equations with dirichlet boundary conditions are considered in thin tube structures these domains are connected finite unions of thin finite cylinders in the 2 d case respectively thin rectangles the complete asymptotic expansion of the solution is constructed in the case without boundary layer in time the estimates for the difference of the exact solution and its j th asymptotic approximation is proved the method of asymptotic partial domain decomposition is formulated and justified for the non steady navier stokes equations in a tube structure it gives the asymptotically exact interface conditions of coupling of the 1d and 3d models of the flow note that the obtained results hold true for the important in applications case of time periodic flows

characteristic exponents for a viscous fluid subjected to time dependent forces we consider a viscous incompressible fluid enclosed in a bounded region of 2 or 3 and subjected to time dependent forces using bound state estimates for the schrodinger operator we obtain rigorous bounds for the characteristic exponents entropy kolmogorov sinai invariant and hausdorff dimension of attracting sets our methods are of potential use for more general time evolutions described by nonlinear partial differential equations

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

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

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

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