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Viorel Barbu

  1. Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls.

    Thu, 02/11/2010 - 16:01 — SomNambul
    Authors: Viorel Barbu, Sergio S. Rodrigues, Armen Shirikyan
    Subjects: Optimization and Control
    Abstract

    We consider the Navier-Stokes system in a bounded domain with a smooth
    boundary. Given a sufficiently regular global solution, we construct a
    finite-dimensional feedback control that is supported by a given open set and
    stabilizes the linearized equation. The proof of this fact is based on a
    truncated observability inequality, the regularizing property for the
    linearized equation, and some standard techniques of the optimal control
    theory. We then show that the control constructed for the linear problem
    stabilizes locally also the full Navier-Stokes system.

  2. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space.

    Mon, 08/31/2009 - 17:16 — SomNambul
    Authors: Viorel Barbu, Giuseppe Da Prato, Luciano Tubaro
    Subjects: Probability
    Abstract

    We consider the stochastic reflection problem associated with a self-adjoint
    operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty
    interior and regular boundary $\Sigma$ in a Hilbert space $H$. We prove the
    existence and uniqueness of a smooth solution for the corresponding elliptic
    infinite-dimensional Kolmogorov equation with Neumann boundary condition on
    $\Sigma$.

  3. Strong solutions for stochastic porous media equations with jumps.

    Fri, 08/28/2009 - 23:56 — SomNambul
    Authors: Viorel Barbu, Carlo Marinelli
    Subjects: Analysis of PDEs
    Abstract

    We prove global well-posedness in the strong sense for stochastic generalized
    porous media equations driven by square integrable martingales with stationary
    independent increments.

  4. Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case.

    Thu, 08/20/2009 - 23:06 — SomNambul
    Authors: Viorel Barbu, Michael Roeckner, Francesco Russo
    Subjects: Probability
    Abstract

    We consider a possibly degenerate porous media type equation over all of
    $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear
    growth and prove a probabilistic representation of its solution in terms of an
    associated microscopic diffusion. This equation is motivated by some singular
    behaviour arising in complex self-organized critical systems. The main idea
    consists in approximating the equation by equations with monotone
    non-degenerate coefficients and deriving some new analytical properties of the
    solution.

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