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  • 1
    Electronic Resource
    Electronic Resource
    Martingale and stationary solutions for stochastic Navier-Stokes equations (1995)
    Springer
    Probability theory and related fields 102 (1995), S. 367-391 
    Details
    ISSN: 1432-2064
    Keywords: 60H15 ; 76D05 ; 35Q10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192467
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  • 2
    Electronic Resource
    Electronic Resource
    Attractors for random dynamical systems (1994)
    Springer
    Probability theory and related fields 100 (1994), S. 365-393 
    Details
    ISSN: 1432-2064
    Keywords: 58F11 ; 58F12 ; 34D45 ; 35Q30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01193705
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  • 3
    Electronic Resource
    Electronic Resource
    Dissipativity and invariant measures for stochastic Navier-Stokes equations (1994)
    Springer
    Nonlinear differential equations and applications 1 (1994), S. 403-423 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A 2-dimensional stochastic Navier-Stokes equation with a general white noise is considered. The aim is to prove the existence of invariant measures, using a new dissipativity property of the stochastic dynamic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194988
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  • 4
    Electronic Resource
    Electronic Resource
    2-D Euler equation perturbed by noise (1999)
    Springer
    Nonlinear differential equations and applications 6 (1999), S. 35-54 
    Details
    ISSN: 1420-9004
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A 2-dimensional Euler equations subject to a stochastic perturbation (a noise) is investigated. An existence and uniqueness result is proved with some assumptions of spatial regularity on the noise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s000300050063
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  • 5
    Electronic Resource
    Electronic Resource
    Ergodicity of the 2-D Navier-Stokes equation under random perturbations (1995)
    Springer
    Communications in mathematical physics 172 (1995), S. 119-141 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02104513
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  • 6
    Electronic Resource
    Electronic Resource
    Direct solution of a Riccati equation arising in a stochastic control problem with control and observation on the boundary (1986)
    Springer
    Applied mathematics & optimization 14 (1986), S. 107-129 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control of a class of stochastic parabolic systems is studied. This class includes systems with noise depending on spatial derivatives of the state, Neumann boundary control, and Dirichlet boundary observation, and extends a class of stochastic systems with distributed control studied by Da Prato [3] and Da Prato and Ichikawa [4]. The work is based on the direct study of the Riccati equation arising in the optimal control problem over finite time horizon. The problem over infinite time horizon and the corresponding algebraic Riccati equation are also considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01442231
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  • 7
    Electronic Resource
    Electronic Resource
    Pathwise global attractors for stationary random dynamical systems (1993)
    Springer
    Probability theory and related fields 95 (1993), S. 87-102 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The asymptotic behaviour of random dynamical systems in Polish spaces is considered. Under the assumption of existence of a random compact absorbing set, assumption supposed to hold path by path, a candidate pathwise attractorA(ω) is defined. The goal of the paper is to show that, in the case of stationary dynamical systems,A(ω) attracts bounded sets, is measurable with respect to the σ-algebra of invariant sets, and is independent of ω when the system is ergodic. An application to a general class of Navier-Stokes type equations perturbed by a multiplicative ergodic real noise is discussed in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01197339
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  • 8
    Electronic Resource
    Electronic Resource
    Hausdorff Dimension of Invariant Sets for Random Dynamical Systems (1998)
    Springer
    Journal of dynamics and differential equations 10 (1998), S. 449-474 
    Details
    ISSN: 1572-9222
    Keywords: Hausdorff dimension ; random invariant set ; random attractor ; random dynamical system ; Navier–Stokes equation ; reaction diffusion equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Suppose ω↦X(ω) is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim H (X(ω)) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in ωɛΩ. The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022605313961
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  • 9
    Electronic Resource
    Electronic Resource
    Additive Noise Destroys a Pitchfork Bifurcation (1998)
    Springer
    Journal of dynamics and differential equations 10 (1998), S. 259-274 
    Details
    ISSN: 1572-9222
    Keywords: Random dynamical systems ; random attractors ; invariant measures ; Markov measures
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the deterministic pitchfork bifurcation the dynamical behavior of the system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed points split the state space into two macroscopically distinct regions. It is shown here that this bifurcation of the dynamical behavior disappears as soon as additive white noise of arbitrarily small intensity is incorporated the model. The dynamical behavior of the disturbed system remains the same for all parameter values. In particular, the system has a (random) global attractor, and this attractor is a one-point set for all parameter values. For any parameter value all solutions converge to each other almost surely (uniformly in bounded sets). No splitting of the state space into distinct regions occurs, not even into random ones. This holds regardless of the intensity of the disturbance.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022665916629
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  • 10
    Electronic Resource
    Electronic Resource
    Weak Solutions and Attractors for Three-Dimensional Navier–Stokes Equations with Nonregular Force (1999)
    Springer
    Journal of dynamics and differential equations 11 (1999), S. 355-398 
    Details
    ISSN: 1572-9222
    Keywords: Stochastic Navier–Stokes equation ; random attractor
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A three-dimensional Navier–Stokes equation is considered. The forcing term is the derivative of a continuous function; the case of white noise is also considered. The aim is to prove the existence of weak solutions and to construct an attractor for the corresponding shift dynamical system in path space, following an idea of Sell.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021937715194
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