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Jonathan C Mattingly - One of the best experts on this subject based on the ideXlab platform.

  • malliavin calculus for the stochastic 2d navier Stokes Equation
    Communications on Pure and Applied Mathematics, 2006
    Co-Authors: Jonathan C Mattingly, Etienne Pardoux
    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, 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.

  • malliavin calculus for the stochastic 2d navier Stokes Equation
    arXiv: Probability, 2004
    Co-Authors: Jonathan C Mattingly, Etienne Pardoux
    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.

  • the dissipative scale of the stochastics navier Stokes Equation regularization and analyticity
    Journal of Statistical Physics, 2002
    Co-Authors: Jonathan C Mattingly
    Abstract:

    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.

  • ergodicity for the navier Stokes Equation with degenerate random forcing finite dimensional approximation
    Communications on Pure and Applied Mathematics, 2001
    Co-Authors: E Weinan, Jonathan C Mattingly
    Abstract:

    We study Galerkin truncations of the two-dimensional Navier-Stokes Equation under degenerate, large-scale, stochastic forcing. We identify the minimal set of modes that has to be forced in order for the system to be ergodic. Our results rely heavily on the structure of the nonlinearity. c 2001 John Wiley & Sons, Inc.

  • gibbsian dynamics and ergodicity for the stochastically forced navier Stokes Equation
    Communications in Mathematical Physics, 2001
    Co-Authors: E Weinan, Jonathan C Mattingly, Yakov G Sinai
    Abstract:

    We study stationary measures for the two-dimensional Navier–Stokes Equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes.

E Weinan - One of the best experts on this subject based on the ideXlab platform.

Etienne Pardoux - One of the best experts on this subject based on the ideXlab platform.

  • malliavin calculus for the stochastic 2d navier Stokes Equation
    Communications on Pure and Applied Mathematics, 2006
    Co-Authors: Jonathan C Mattingly, Etienne Pardoux
    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, 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.

  • malliavin calculus for the stochastic 2d navier Stokes Equation
    arXiv: Probability, 2004
    Co-Authors: Jonathan C Mattingly, Etienne Pardoux
    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.

Franco Flandoli - One of the best experts on this subject based on the ideXlab platform.

Xicheng Zhang - One of the best experts on this subject based on the ideXlab platform.

  • tamed 3d navier Stokes Equation existence uniqueness and regularity
    Infinite Dimensional Analysis Quantum Probability and Related Topics, 2009
    Co-Authors: Michael Röckner, Xicheng Zhang
    Abstract:

    In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier–Stokes Equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier–Stokes Equation, then this solution satisfies our tamed Equation. Moreover, using this tamed Equation we can give a new construction for a suitable weak solution of the classical 3D Navier–Stokes Equation introduced in Refs. 16 and 2.

  • A Tamed 3D Navier-Stokes Equation in Domains
    arXiv: Analysis of PDEs, 2008
    Co-Authors: Xicheng Zhang
    Abstract:

    In this paper, we analyze a tamed 3D Navier-Stokes Equation in uniform $C^2$-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed Equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes Equation, then this solution satisfies our tamed Equation. Moreover, the existence of a global attractor for the tamed Equation in bounded domains is also proved. As simple applications, some well known results for the classical Navier-Stokes Equations in unbounded domains are covered.

  • Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and Regularity
    arXiv: Probability, 2007
    Co-Authors: Michael Röckner, Xicheng Zhang
    Abstract:

    In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier-Stokes Equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier-Stokes Equation, then this solution satisfies our tamed Equation. Moreover, using this renormalized Equation we can give a new construction for a suitable weak solution of the classical 3D Navier-Stokes Equation introduced in [Scheffer: Hausdorff measure and the Navier-Stokes Equations. Comm. Math. Phys., 1977] and [Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes Equations. Comm. Pure Appl. Math., 1982].