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59
Stochastic dissipative PDE's and Gibbs measures
 Comm. Math. Phys
, 2000
"... We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in stat ..."
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Cited by 54 (16 self)
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We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in x and stationary, shortcorrelated in time t. In this paper, we confine ourselves to "kick forces" of the form # # (t, x) = +# X k=# #(t  kT )#k (x), 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...
Global Attractors in Partial Differential Equations
"... this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping. ..."
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Cited by 22 (0 self)
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this paper, we present the weakly damped Schrodinger equation, which is a system generated by a dispersive equation with weak damping.
Exponential mixing for 2D NavierStokes equations perturbed by an unbounded noise
 J. Math. Fluid Mech
"... The paper is devoted to the problem of mixing for twodimensional Navier–Stokes equations perturbed by an unbounded kick force. We develop the coupling approach suggested in [16] to show that any solution exponentially converges to the stationary measure in the dual Lipschitz norm. This property co ..."
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Cited by 17 (5 self)
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The paper is devoted to the problem of mixing for twodimensional Navier–Stokes equations perturbed by an unbounded kick force. We develop the coupling approach suggested in [16] to show that any solution exponentially converges to the stationary measure in the dual Lipschitz norm. This property complements some earlier results established in [15] for the same model.
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show 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 Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 17 (8 self)
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We develop a general method that allows to show 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 Ł ptype 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 infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour 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 twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems
 Math. Comput
, 1997
"... Abstract. We show that the longtime behavior of the projection of the exact solutions to the NavierStokes equations and other dissipative evolution equations on the finitedimensional space of interpolant polynomials determines the longtime behavior of the solution itself provided that the spatia ..."
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Cited by 17 (6 self)
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Abstract. We show that the longtime behavior of the projection of the exact solutions to the NavierStokes equations and other dissipative evolution equations on the finitedimensional space of interpolant polynomials determines the longtime 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 finitedimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation. 1.
On recent progress for the stochastic Navier Stokes equations
 In Journées Équations aux dérivées partielles, ForgeslesEaux, XI:1–52, 2003. see http://www.math.sciences.univnantes.fr/edpa/2003/html/. [MY02] [Pro90] [Sin94] Nader Masmoudi and LaiSang
"... We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific examp ..."
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Cited by 14 (7 self)
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We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed. This article attempts to collect a number of ideas which have proven useful in the study of stochastically forced dissipative partial differential equations. The discussion will center around those of ergodicity but will also touch on the regularity of both solutions and transition densities. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. Though we have not tried to give any great generality, we also present a number of abstract results to help isolate what assumptions are used in which arguments. Though a few results are presented in new ways and a number of proofs are streamlined, the core ideas remain more or less the same as in the originally cited papers. We do improve sightly the exponential mixing results given in [Mat02c]; however, the techniques used are the same. Lastly, we do not claim to be exhaustive. This is not meant to be an all encompassing review article. The view point given here is a personal one; nonetheless, citations are given to good starting points for related works both by the author and others. Consider the twodimensional NavierStokes equation with stochastic forcing:
Analytic study of shell models of turbulence
 PHYSICA D
"... In this paper we study analytically the viscous ‘sabra’ shell model of energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant ine ..."
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Cited by 10 (3 self)
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In this paper we study analytically the viscous ‘sabra’ shell model of energy turbulent cascade. We prove the global regularity of solutions and show that the shell model has finitely many asymptotic degrees of freedom, specifically: a finite dimensional global attractor and globally invariant inertial manifolds. Moreover, we establish the existence of exponentially decaying
STOCHASTIC DYNAMICS OF A COUPLED ATMOSPHERE–OCEAN MODEL
"... Abstract. The investigation of the coupled atmosphereocean system is not only scientifically challenging but also practically important. We consider a coupled atmosphereocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the ai ..."
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Cited by 10 (9 self)
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Abstract. The investigation of the coupled atmosphereocean system is not only scientifically challenging but also practically important. We consider a coupled atmosphereocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the airsea interface. We reformulate this model as a random dynamical system. First, we have shown that the asymptotic dynamics of the coupled atmosphereocean model is described by a random climatic attractor. Second, we have estimated the atmospheric temperature evolution under oceanic feedback, in terms of the freshwater flux, heat flux and the external fluctuation at the airsea interface, as well as the earth’s longwave radiation coefficient and the shortwave solar radiation profile. Third, we have demonstrated that this system has finite degree of freedom by presenting a finite set of determining functionals in probability. Finally, we have proved that the coupled atmosphereocean model is ergodic under suitable conditions for physical parameters and randomness, and thus for any observable of the coupled
MALLIAVIN CALCULUS FOR THE STOCHASTIC 2D NAVIER STOKES EQUATION
, 2004
"... Abstract. 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 s ..."
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Cited by 9 (4 self)
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Abstract. 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 density with respect to Lebesgue measure. We also show that under natural assumptions the density of such a projection is everywhere strictly positive. In particular, our conditions are viscosity independent. We are mainly interested in forcing which excites a very small number of modes. All of the results rely on the nondegeneracy of the infinite dimensional Malliavin matrix. 1.