existence and regularity of invariant measures for the three dimensional stochastic primitive equations we establish the continuity of the markovian semigroup associated with strong solutions of the stochastic 3d primitive equations and prove the existence of an invariant measure the proof is based on new moment bounds for strong solutions the invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties

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

painleve expansions for nonintegrable evolution equations abstract the wtc method is applied to a variety of nonintegrable nonlinear evolution equations and shown to provide a systematic procedure for obtaining special solutions some of the special techniques developed for integrable systems are also shown to be applicable to nonintegrable equations

roth s theorem in the primes we show that any set containing a positive proportion of the primes contains a 3 term arithmetic progression an important ingredient is a proof that the primes enjoy the so called hardy littlewood majorant property we derive this by giving a new proof of a rather more general result of bourgain which because of a close analogy with a classical argument of tomas and stein from euclidean harmonic analysis might be called a restriction theorem for the primes

similarity reductions from extended painleve expansions for nonintegrable evolution equations abstract the use of singular manifold expansions to find exact solutions to nonintegrable evolution equations is extended to include arbitrary resonance coefficients in such a way as to make the resulting infinite series exactly resummable the technique involves the use of a rescaling ansatz analogous to that used to analyze the psi series of nonintegrable ordinary differential equations the result is a similarity reduction of the equation in which the constrained singular manifold plays the role of a similarity variable the method is capable of yielding new solutions corresponding to either classical or nonclassical conditional lie symmetries

sets of integers with no large sum free subset answering a question of p erd os from 1965 we show that for every 0 there is a set a of n integers with the following property every set a 0 a with at least 1 n elements contains three distinct elements x y z with x y z contents

non generic connections corresponding to front solutions a classification of special nonlinear front solutions for certain one time and one space reaction diffusion equations is presented using the method of weiss tabor and carnevale wtc these results are related to known stability criteria in particular the steepness criterion of van saarloos 1989 the wtc method is shown to be equivalent to a special first order reduction and both of these methods are shown to work for reaction diffusion equations with special nonlinearities of particular interest is the fact that the special first order reduction is shown to give separatrices in appropriate phase spaces an example of reaction diffusion equation is presented without these special nonlinearities while this equation is shown to have a special nonlinear connection and resulting stability properties it is intractable for either a singular manifold expansion of a first order reduction a lie symmetry analysis is carried out and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques however the rescaling ansatz of cariello and tabor 1991 suggests that some symmetries are present

pattern propagation in nonlinear dissipative systems abstract we discuss the problem of pattern selection in situations where a stable nonuniform state of a nonlinear dissipative system propagates into an initially unstable homogeneous region our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known and we point out that these known properties are consistent with a marginal stability hypothesis that has been suggested in the theory of dendritic crystal growth we then describe a more general interpretation of the marginal stability hypothesis and finally present numerical evidence for its validity from three different pattern forming models

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

gibbsian dynamics and invariant measures for stochastic dissipative pdes we present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative pdes it consists of two main steps the first step is the reduction to a finite dimensional gibbsian dynamics of the low modes the second step is to prove the equivalence between measures induced by different past histories using girsanov theorem as applications we prove ergodicity for ginzburg landau kuramoto sivashinsky and cahn hilliard equations with stochastic forcing

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

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

space time invariant measures entropy and dimension for stochastic ginzburg landau equations we consider a randomly forced ginzburg landau equation on an unbounded domain the forcing is smooth and homogeneous in space and white noise in time we prove existence and smoothness of solutions existence of an invariant measure for the corresponding markov process and we define the spatial densities of topological entropy of measure theoretic entropy and of upper box counting dimension we prove inequalities relating these different quantities the proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square integrable functions and a priori bounds on the semi flow in these spaces the bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution

existence and uniqueness of stationary solutions for 3d navier stokes system with small random forcing via stochastic cascades we consider the 3d navier stokes system in the fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition we explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with fourier transform majorized by a certain function h moreover we prove the following one force one solution principle the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t our explicit construction of the solution is based upon the stochastic cascade representation

random kick forced 3d navier stokes equations in a thin domain we consider the navier stokes equations in the thin 3d domain mathbb t 2 times 0 epsilon where mathbb t 2 is a two dimensional torus the equation is perturbed by a non degenerate random kick force we establish that firstly when e 1 the equation has a unique stationary measure and secondly after averaging in the thin direction this measure converges as e 0 to a unique stationary measure for the navier stokes equation on mathbb t 2 thus the 2d navier stokes equations on surfaces describe asymptotic in time and limiting in e statistical properties of 3d solutions in thin 3d domains

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

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

ergodicity for the dissipative boussinesq equations with random forcing we study the stationary measure for the two dimensional boussinesq equation with random forcing we prove the ergodicity for the two dimensional stochastically forced boussinesq equation we also study the galerkin truncations of the three dimensional boussinesq equations under degenerate stochastic forcing we follow closely the previous results on the stochastically forced navier stokes equations

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

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

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

stochastic 3d navier stokes equations in a thin domain and its α approximation abstract in the thin domain o e t 2 0 e where t 2 is a two dimensional torus we consider the 3d navier stokes equations perturbed by a white in time random force and the leray approximation for this system we study ergodic properties of these models and their connection with the corresponding 2d models in the limit e 0 in particular under natural conditions concerning the noise we show that in some rigorous sense the 2d stationary measure comprises asymptotical in time statistical properties of solutions for the 3d navier stokes equations in o e when e 1

uniqueness of the invariant measure for a stochastic pde driven by degenerate noise we consider the stochastic ginzburg landau equation in a bounded domain we assume the stochastic forcing acts only on high spatial frequencies the low lying frequencies are then only connected to this forcing through the non linear cubic term of the ginzburg landau equation under these assumptions we show that the stochastic pde has a unique invariant measure the techniques of proof combine a controllability argument for thelow lying frequencies with an infinite dimensional version of the malliavin calculus to show positivity and regularity of the invariant measure this then implies the uniqueness of that measure

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

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

discontinuous galerkin hp adaptive methods for multiscale chemical reactors quiescent reactors abstract we present a class of chemical reactor systems modeled numerically using a fractional multistep method between the reacting and diffusing modes of the system subsequently allowing one to utilize algebraic techniques for the resulting reactive subsystems a mixed form discontinuous galerkin method is presented with implicit and explicit imex timestepping strategies coupled to dioristic entropy schemes for h p adaptivity of the solution where the h and p are adapted based on an l 1 stability result finally we provide some numerical studies on the convergence behavior adaptation and asymptotics of the system applied to a pair of equilibrium problems as well as to general three dimensional nonlinear lotka volterra chemical systems

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

some limiting properties of randomly forced two dimensional navier stokes equations we consider random perturbations of two dimensional navier stokes equations under some natural conditions on random forces we study asymptotic properties of solutions and stationary measures

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

vinogradov s theorem with almost equal summands let 11 20 we prove that every sufficiently large odd integer n can be written as a sum of three primes n p1 p2 p3 with pi n 3 n for i 1 2 3

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

the small scales of the stochastic navier stokes equations under rough forcing we prove that the small scale structures of the stochastically forced navier stokes equations approach those of the naturally associated ornstein uhlenbeck process as the scales get smaller precisely we prove that the rescaled kth spatial fourier mode converges weakly on path space to an associated ornstein uhlenbeck process as k in addition we prove that the navier stokes equations and the naturally associated ornstein uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper viscosity this gives a simple proof of unique ergodicity for the hyperviscous navier stokes system we show how different strengthened hyperviscosity produce varying levels of equivalence

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

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

asymptotic coupling and a general form of harris theorem with applications to stochastic delay equations there are many markov chains on infinite dimensional spaces whose one step transition kernels are mutually singular when starting from different initial conditions we give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of harris theorem the first uses the existence of couplings which draw the solutions together as time goes to infinity such asymptotic couplings were central to mattingly and sinai in comm math phys 219 3 523 565 2001 mattingly in comm math phys 230 3 461 462 2002 hairer in prob theory relat field 124 345 380 2002 bakhtin and mattingly in commun contemp math 7 553 582 2005 on which this work builds as in bakhtin and mattingly 2005 the emphasis here is on stochastic differential delay equations harris celebrated theorem states that if a markov chain admits a lyapunov function whose level sets are small in the sense that transition probabilities are uniformly bounded from below then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed this convergence takes place in a total variation norm weighted by the lyapunov function a second aim of this article is to replace the notion of a small set by the much weaker notion of a d small set which takes the topology of the underlying space into account via a distance like function d with this notion at hand we prove an analogue to harris theorem where the convergence takes place in a wasserstein like distance weighted again by the lyapunov function this abstract result is then applied to the framework of stochastic delay equations in this framework the usual theory of harris chains does not apply since there are natural examples for which there exist no small sets except for sets consisting of only one point this gives a solution to the long standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it

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

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