a new kernel based approach for linear system identification this paper describes a new kernel based approach for linear system identification of stable systems we model the impulse response as the realization of a gaussian process whose statistics differently from previously adopted priors include information not only on smoothness but also on bibo stability the associated autocovariance defines what we call a stable spline kernel the corresponding minimum variance estimate belongs to a reproducing kernel hilbert space which is spectrally characterized compared to parametric identification techniques the impulse response of the system is searched for within an infinite dimensional space dense in the space of continuous functions overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization the proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two step procedure that projects the nonparametric estimate onto the space of nominal models the continuous time derivation immediately extends to the discrete time case on several continuous and discrete time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques

brief optimal trajectory planning and smoothing splines in this paper some of the relationships between optimal control and trajectory planning are examined when planning trajectories for linear control systems a demand that arises naturally in air traffic control or noise contaminated data interpolation is that the curve passes close to given points or through intervals at given times in this paper we produce these curves by solving an optimal control problem for linear control systems while driving the output of the system close to the waypoints we furthermore show how this optimal control problem reduces to a finite quadratic programming problem and we thus provide a constructive yet theoretically sound framework for producing a rich set of curves called smoothing splines

identification of time varying systems in reproducing kernel hilbert spaces we consider a class of linear systems in state space form whose parameters evolve in time according to a continuous time gauss markov process our problem is to identify the system from a finite set of noisy output measurements we derive a connection between this problem and tikhonov regularization in reproducing kernel hilbert spaces relying upon this result conditions that ensure the existence of the maximum a posteriori estimate of the parameter trajectory are provided and an identification algorithm is worked out simulated data are used to test the goodness of the proposed approach

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

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

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

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

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

prediction error identification of linear systems a nonparametric gaussian regression approach a novel bayesian paradigm for the identification of output error models has recently been proposed in which in place of postulating finite dimensional models of the system transfer function the system impulse response is searched for within an infinite dimensional space in this paper such a nonparametric approach is applied to the design of optimal predictors and discrete time models based on prediction error minimization by interpreting the predictor impulse responses as realizations of gaussian processes the proposed scheme describes the predictor impulse responses as the convolution of an infinite dimensional response with a low dimensional parametric response that captures possible high frequency dynamics overparameterization is avoided because the model involves only a few hyperparameters that are tuned via marginal likelihood maximization numerical experiments with data generated by armax and infinite dimensional models show the definite advantages of the new approach over standard parametric prediction error techniques and subspace methods both in terms of predictive capability on new data and accuracy in reconstruction of system impulse responses

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

nonparametric identification of population models via gaussian processes population models are used to describe the dynamics of different subjects belonging to a population and play an important role in drug pharmacokinetics a nonparametric identification scheme is proposed in which both the average impulse response of the population and the individual ones are modelled as gaussian stochastic processes assuming that the average curve is an integrated wiener process it is shown that its estimate is a cubic spline an empirical bayes algorithm for estimating both the average and the individual curves is worked out the model is tested on simulated data sets as well as on xenobiotics pharmacokinetic data

a new kernel based approach for nonlinearsystem identification we present a novel nonparametric approach for identification of nonlinear systems exploiting the framework of gaussian regression the unknown nonlinear system is seen as a realization from a gaussian random field its covariance encodes the idea of fading memory in the predictor and consists of a mixture of gaussian kernels parametrized by few hyperparameters describing the interactions among past inputs and outputs the kernel structure and the unknown hyperparameters are estimated maximizing their marginal likelihood so that the user is not required to define any part of the algorithmic architecture e g the regressors and the model order once the kernel is estimated the nonlinear model is obtained solving a tikhonov type variational problem the hilbert space the estimator belongs to is characterized benchmarks problems taken from the literature show the effectiveness of the new approach also comparing its performance with a recently proposed algorithm based on direct weight optimization and with parametric approaches with model order estimated by aic or bic

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

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 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

on infinite volume mixing in the context of the long standing issue of mixing in infinite ergodic theory we introduce the idea of mixing for observables possessing an infinite volume average the idea is borrowed from statistical mechanics and appears to be relevant at least for extended systems with a direct physical interpretation we discuss the pros and cons of a few mathematical definitions that can be devised testing them on a prototypical class of infinite measure preserving dynamical systems namely the random walks

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

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

efficient spatio temporal gaussian regression via kalman filtering abstract we study the non parametric reconstruction of spatio temporal dynamical processes via gaussian processes gps regression from sparse and noisy data gps have been mainly applied to spatial regression where they represent one of the most powerful estimation approaches also thanks to their universal representing properties their extension to dynamical processes has been instead elusive so far since classical implementations lead to unscalable algorithms or require some sort of approximation we propose a novel procedure to address this problem by coupling gps regression and kalman filtering in particular assuming space time separability of the covariance kernel of the process and rational time spectrum we build a finite dimensional discrete time state space process representation amenable to kalman filtering with sampling over a finite set of fixed spatial locations our major finding is that the current kalman filter state represents a sufficient statistic to compute the minimum variance estimate of the process at any future time over the entire spatial domain in machine learning a representer theorem states that an important class of infinite dimensional variational problems admits a computable and finite dimensional exact solution in view of this our result can be interpreted as a novel dynamic representer theorem for gps we then extend the study to situations where the spatial input locations set varies over time the proposed algorithms are tested on both synthetic and real field data providing comparisons with standard gp and truncated gp regression techniques

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

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

regularization networks fast weight calculation via kalman filtering regularization networks are nonparametric estimators obtained from the application of tychonov regularization or bayes estimation to the hypersurface reconstruction problem their main drawback back is that the computation of the weights scales as o n sup 3 where n is the number of data in this paper we show that for a class of monodimensional problems the complexity can be reduced to o n by a suitable algorithm based on spectral factorization and kalman filtering moreover the procedure applies also to smoothing splines

generalized lyapunov exponent as a unified characterization of dynamical instabilities the lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits a positive lyapunov exponent exponential dynamical instability is a manifestation of chaos here we propose the lyapunov pair which is based on the generalized lyapunov exponent as a unified characterization of nonexponential and exponential dynamical instabilities in one dimensional maps chaos is classified into three different types i e superexponential exponential and subexponential chaos using one dimensional maps we demonstrate superexponential and subexponential chaos and quantify the dynamical instabilities by the lyapunov pair in subexponential chaos we show superweak chaos which means that the growth of the difference of nearby orbits is slower than a stretched exponential growth the scaling of the growth is analytically studied by a recently developed theory of a continuous accumulation process which is related to infinite ergodic theory

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

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

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

wavelet estimation by bayesian thresholding and model selection many wavelet based algorithms have been proposed in recent years to solve the problem of function estimation from noisy samples in particular it has been shown that threshold approaches lead to asymptotically optimal estimation and are extremely effective when dealing with real data working under a bayesian perspective in this paper we first study optimality of the hard and soft thresholding rules when the function is modelled as a stochastic process with known covariance function next we consider the case where the covariance function is unknown and propose a novel approach that models the covariance as a certain wavelet combination estimated from data by bayesian model selection simulated data are used to show that the new method outperforms traditional threshold approaches as well as other wavelet based bayesian techniques proposed in the literature

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

control theoretic smoothing splines some of the relationships between optimal control and statistics are examined we produce generalized smoothing splines by solving an optimal control problem for linear control systems minimizing the l sup 2 norm of the control signal while driving the scalar output of the control system close to given prespecified interpolation points we then prove a convergence result for the smoothing splines using results from the theory of numerical quadrature finally we show in simulations that our approach works in practice as well as in theory

bayesian online multitask learning of gaussian processes standard single task kernel methods have recently been extended to the case of multitask learning in the context of regularization theory there are experimental results especially in biomedicine showing the benefit of the multitask approach compared to the single task one however a possible drawback is computational complexity for instance when regularization networks are used complexity scales as the cube of the overall number of training data which may be large when several tasks are involved the aim of this paper is to derive an efficient computational scheme for an important class of multitask kernels more precisely a quadratic loss is assumed and each task consists of the sum of a common term and a task specific one within a bayesian setting a recursive online algorithm is obtained which updates both estimates and confidence intervals as new data become available the algorithm is tested on two simulated problems and a real data set relative to xenobiotics administration in human patients

a dirichlet process gaussian state machine model for change detection in transient processes abstractthe ability to detect incipient and critical changes in real world process esessential for system integrity assurance is currently impeded by the mismatch between the key assumption of stationarity underlying most change detection methods and the nonlinear and nonstationary transient dynamics of most real world processes the current approaches are slow or outright unable to detect qualitative changes in the behaviors that lead to anomalies we present a dirichlet process gaussian state machine dpgsm model to represent dynamic intermittency which is one of the most ubiquitous real world transient behaviors the dpgsm model treats a signal as a random walk among a dirichlet process mixture of gaussian clusters hypothesis tests and a numerical scheme based on this nonparametric representation were developed to detect subtle changes in the transient intermittent dynamics experimental investigations suggest that the dpgsm approach can consistently detect incipient critical changes in intermi

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

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

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

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

intermittency and regularized fredholm determinants we consider real analytic maps of the interval i 0 1 which are expanding everywhere except for a neutral fixed point at 0 we show that on a certain function space the associated perron frobenius operator has a continuous and residual spectrum contained in the line segment c 0 1 and a point spectrum p which has no points of accumulation outside 0 and 1 furthermore points in p 0 1 are eigenvalues of finite multiplicity we construct a regularized fredholm determinant d which has a holomorphic extension to c and can be analytically continued from each side of c to an open neighborhood of c 0 1 on different riemann sheets in c the zero set of d is in one to one correspondence with the point spectrum of through the conformal transformation the function d z extends to a holomorphic function in a domain which contains the unit disc

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

invariance principles for interval maps with an indifferent fixed point in this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points including the manneville pomeau map this implies a number of well known corollaries including the weak invariance principle and the law of the iterated logarithm

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

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

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

cluster approximation for the farey fraction spin chain we consider the farey fraction spin chain in an external field h utilising ideas from dynamical systems the free energy of the model is derived by means of an effective cluster energy approximation this approximation is valid for divergent cluster sizes and hence appropriate for the discussion of the magnetizing transition we calculate the phase boundaries and the scaling of the free energy at h 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy for h 0 our results are largely consistent with those found previously using mean field theory and renormalization group arguments