Stochastic 2D hydrodynamical type systems: Well posedeness and large deviations (original) (raw)
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Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations
Applied Mathematics and Optimization, 2010
We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.
Stochastic 2D hydrodynamical systems: Support theorem, preprint arXiv arXiv:0907.2100v1
2013
Abstract. We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equation, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general result of convergence in probability for non linear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process. 1.
Stochastic 2D hydrodynamical systems: Wong-Zakai approximation and Support theorem
2009
We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong-Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.
Stochastic Two-Dimensional Hydrodynamical Systems: Wong-Zakai Approximation and Support Theorem
2011
We deal with a class of abstract nonlinear stochastic models with multiplicative noise, which covers many 2D hydrodynamical models including the 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problems as well as some shell models of turbulence. Our main result describes the support of the distribution of solutions. Both inclusions are proved by means of a general Wong-Zakai type result of convergence in probability for nonlinear stochastic PDEs driven by a Hilbert-valued Brownian motion and some adapted finite dimensional approximation of this process.
SIAM Journal on Mathematical Analysis, 2012
Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces.
2D stochastic Navier–Stokes equations driven by jump noise
Nonlinear Analysis: Theory, Methods & Applications, 2013
In the paper, we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of Lévy type. Our result is formulated in an abstract setting. This type of equation covers the stochastic 2D Navier-Stokes Equations, the 2D stochastic Magneto-Hydrodynamic Equations, the 2D stochastic Boussinesq Model for the Bénard Convection, the 2D stochastic Magnetic Bérnard Problem, the 3D stochastic Leray α-Model for the Navier-Stokes Equations and several stochastic Shell Models of turbulence.
Stochastic 2D Navier-Stokes Equation and Applications to 2D Turbulence
2016
Author(s): Karimi, Shahab | Advisor(s): Birnir, Bjorn | Abstract: We will consider the 2-dimensional Navier-Stokes equation for an incompressible fluid with periodic boundary condition, and with a random perturbation that is in the form of white noise in time and a deterministic perturbation due to the large deviation principle. Our ultimate goal is to find appropriate conditions on the initial data and the forcing terms so that global existence and uniqueness of a mild solution is guaranteed. We will use the Picard's iteration method to prove existence of local mild solution and then prove the existence of a maximal solution which then leads to global existence. The result is applied to the backward Kolmogorov-Obukhov energy cascade and the forward Kraichnan enstrophy cascade in 2D turbulence.
Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition
2012
Using a weak convergence approach, we prove a LPD for the solution of 2D stochastic Navier Stokes equations when the viscosity converges to 0 and the noise intensity is multiplied by the square root of the viscosity. Unlike previous results on LDP for hydrodynamical models, the weak convergence is proven by tightness properties of the distribution of the solution in appropriate functional spaces.
Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type
Nonlinear Differential Equations and Applications NoDEA, 2015
In this paper we prove the existence and uniqueness of maximal strong (in PDE sense) solution to several stochastic hydrodynamical systems on unbounded and bounded domains of R n , n = 2, 3. This maximal solution turns out to be a global one in the case of 2D stochastic hydrodynamical systems. Our framework is general in the sense that it allows us to solve the Navier-Stokes equations, MHD equations, Magnetic Bénard problems, Boussinesq model of the Bénard convection, Shell models of turbulence and the Leray-α model with jump type perturbation. Our goal is achieved by proving general results about the existence of maximal and global solution to an abstract stochastic partial differential equations with locally Lipschitz continuous coefficients. The method of the proofs are based on some truncation and fixed point methods.