hyponormal operators with finite rank self commutators and quadrature domains this paper studies some classes of pure hyponormal operators with finite rank self commutators satisfying the condition that their spectra are unions of a finite collection of the closures of quadrature domains

existence of solutions of nonlinear stochastic integrodifferential inclusions in a hilbert space in this paper the existence of mild solutions of nonlinear stochastic integrodifferentialinclusions in a hilbert space is studied the results are obtained by using a fixed point theorem for condensing map due to martelli the result is illustrated with an example

the denjoy wolff theorem in cn abstract if d is a bounded and strictly convex domain in c n and f d d is holomorphic and fixed point free then there exists d such that the sequence f n of the iterates of f converges in the compact open topology to the constant mapping taking the value

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

the denjoy wolff theorem for condensing holomorphic mappings if b is the open unit ball of a strictly convex banach space x and f b b is holomorphic condensing with respect to and fixed point free then there exists b such that the sequence lcub fn of the iterates of f converges in the compact open topology to the constant map taking the value

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

a family of balance relations for the two dimensional navier stokes equations with random forcing for the 2d navier stokes equation perturbed by a random force of a suitable kind we show that if g f is an arbitrary real continuous function with at most polynomial growth then the stationary in time vorticity field t x satisfies mathbb e g omega t x nabla omega t x 2 frac 1 2 m 1 mathbb e g omega t x where m 1 is a number independent of g which measures the strength of the random forcing another way of stating this result is that in the unique stationary measure of this system the random variables g t x and t x 2 are uncorrelated for each t and each x

diffusion equation for multivalued stochastic differential equations abstract deterministic oscillations with bilinear hysteresis are governed by a multivalued differential equation of the type k b g where k is maximal monotonic and b is lipschitzian an existence and uniqueness result is proven for corresponding stochastic equation the diffusion equation satisfied by the laws of t is established in the particular case k 0 this equation is equivalent to the fokker planck equation

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

existence of solutions to differential inclusions and to time optimal control problems in the autonomous case we prove existence of solutions to upper semicontinuous differential inclusions and to time optimal control problems under conditions that are strictly weaker than the usual assumption of convexity

inequalities for proper contractions and strictly dissipative operators abstract some inequalities for proper contractions and strictly dissipative operators on a hilbert space are proved these inequalities for operators are related to the classical harnack s inequalities for analytic functions with a positive real part

remarks on the balance relations for the two dimensional navier stokes equation with random forcing we use the balance relations for the stationary in time solutions of the randomly forced 2d navier stokes equations found in 10 to study these solutions further we show that the vorticity t x of a stationary solution has a finite exponential moment and that for any a in mathbb r t geq 0 the expectation of the integral of nabla x xi over the level set x mid xi t x a up to a constant factor equals the expectation of the integral of nabla x xi 1 over the same set

probabilistic estimates for the two dimensional stochastic navier stokes equations we consider the navier stokes equation on a two dimensional torus with a random force white noise in time and analytic in space for arbitrary reynolds number r we prove probabilistic estimates for the long time behavior of the solutions that imply bounds for the dissipation scale and energy spectrum as r

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

iteration of holomorphic maps in hilbert spaces abstract previous work gong ning chen j math anal appl 98 1984 305 313 on iteration of holomorphic maps of cn is continued the purpose of this note is to extend results given in the above mentioned reference to the case of complex hilbert spaces other comments are appended

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

law of large numbers and central limit theorem for randomly forced pde s we consider a class of dissipative pde s perturbed by an external random force under the condition that the distribution of perturbation is sufficiently non degenerate a strong law of large numbers slln and a central limit theorem clt for solutions are established and the corresponding rates of convergence are estimated it is also shown that the estimates obtained are close to being optimal the proofs are based on the property of exponential mixing for the problem in question and some abstract slln and clt for mixing type markov processes

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

randomly forced cgl equation stationary measures and the inviscid limit we study a complex ginzburg landau cgl equation perturbed by a random force which is white in time and smooth in the space variable x assuming that dim x 4 we prove that this equation has a unique solution and discuss its asymptotic in time properties next we consider the case when the random force is proportional to the square root of the viscosity and study the behaviour of stationary solutions as the viscosity goes to zero we show that under this limit a subsequence of solutions in question converges to a nontrivial stationary process formed by global strong solutions of the nonlinear schrodinger equation

random iterations fixed points and invariant crf horospheres in complex banach spaces abstract for holomorphic noncontractive maps on not necessarily bounded domains in complex banach spaces we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations in particular we show that the problem concerning the existence of the horospheres determined by caratheodory reiffen finsler pseudometrics defined on unbounded domains has the solution and we prove new results of type of julia s lemma and wolff s theorem

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

multiplicative ergodic theorem for a non irreducible random dynamical system we study the asymptotic properties of the trajectories of a discrete time random dynamical system in an infinite dimensional hilbert space under some natural assumptions on the model we establish a multiplica tive ergodic theorem with an exponential rate of convergence the assumptions are satisfied for a large class of parabolic pdes including the 2d navier stokes and complex ginzburg landau equations perturbed by a non degenerate bounded random kick force as a consequence of this er godic theorem we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system in particular we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible the proof relies on a refined version of the uniform feller property combined with some contraction and bootstrap arguments

random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces conditions which guarantee the uniform convergence of random iterations of holomorphic contractions on unbounded domains in complex hausdorff locally convex and sequentially complete topological vector spaces are established also conditions concerning the convergence of random iterations of weaker contractions in hausdorff uniform spaces are given

absolute continuity of the invariant measures for some stochastic pdes we consider a free system and an interacting systems having invariant measures and respectively under suitable assumptions we prove an explicit formula relating with and implying the absolute continuity of with respect to we apply our result to a reaction diffusion equation and to the burgers equation

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

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

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

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 unique ergodicity in nonlinear stochastic partial differential equations we illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes examples exhibiting parabolic and hyperbolic structure are studied in detail in the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the krylov bogolyubov procedure and compactness fails