1,720,987 research outputs found
Convergence of transport noise to Ornstein-Uhlenbeck for 2D Euler equations under the enstrophy measure
Full text linkWe consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition.
It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the spacetime white noise
Point vortex approximation for 2D Navier-Stokes equations driven by space-time white noise
Full text linkWe show that the system of point vortices, perturbed by a certain transport type noise,
converges weakly to the vorticity form of 2D Navier–Stokes equations driven by the spacetime
white noise
On the Boussinesq hypothesis for a stochastic Proudman-Taylor model
Full text linkWe introduce a stochastic version of Proudman-Taylor model, a 2D-3C fluid
approximation of the 3D Navier-Stokes equations, with the small-scale
turbulence modeled by a transport-stretching noise. For this model we may
rigorously take a scaling limit leading to a deterministic model with
additional viscosity on large scales. In certain choice of noises without
mirror symmetry, we identify an AKA effect. This is the first example with a 3D
structure and a stretching noise term.Comment: 38 page
Mean Field Limit of Point Vortices with Environmental Noises to Deterministic 2D Navier–Stokes Equations
No full textWe consider point vortex systems on the two-dimensional torus perturbed by environmental noise. It is shown that, under a suitable scaling of the noises, weak limit points of the empirical measures are solutions to the vorticity formulation of deterministic 2D Navier-Stokes equations
Quantitative convergence rates for scaling limit of SPDEs with transport noise
Full text linkWe consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamical equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic arguments, especially heat kernel properties and maximal estimates for stochastic convolutions. Similar ideas are applied to the stochastic 2D Keller-Segel model, yielding explicit choice of noise to ensure that the blow-up probability is less than any given threshold. Our
approach also gives rise to some mixing property for stochastic linear transport equations and dissipation enhancement in the viscous case
A numerical approach to Kolmogorov equation in high dimension based on Gaussian analysis
Full text linkFor Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples
Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation
No full textStochastic Differential Equations (SDEs) in high dimension, having the structure of finite dimensional approximation of Stochastic Partial Differential Equations (SPDEs), are considered. The aim is to numerically compute the expected values and probabilities associated to their solutions, by solving the corresponding Kolmogorov equations, with a partial use of Monte Carlo strategy - precisely, using Monte Carlo only for the linear part of the SDE. The basic idea was presented in [16], but here we strongly improve the numerical results by means of a shift of the auxiliary Gaussian process. For relatively simple nonlinearities, we have good results in dimension of the order of 100
On the relation between the Girsanov transform and the Kolmogorov equations for SPDEs
Full text linkThe Girsanov transform and Kolmogorov equations are two useful methods for
studying SPDEs. It is shown that, under suitable conditions, the series
expansion obtained from the Girsanov transform coincides with the one generated
by an iteration scheme for Kolmogorov equations. We also apply the iteration
approach to extend the well posedness theory for Kolmogorov equations beyond
the boundedness condition on the nonlinear term.Comment: 26 page
2D Smagorinsky type large eddy models as limits of stochastic PDEs
Full text linkWe prove that a version of Smagorinsky large eddy model for a 2D fluid in vorticity form is the scaling limit of suitable stochastic models for large scales, where the influence of small turbulent eddies is modeled by a transport-type noise
Global well-posedness of the 3D Navier–Stokes equations perturbed by a deterministic vector field
No full textFlandoli F, Hofmanová M, Luo D, Nilssen T. Global well-posedness of the 3D Navier–Stokes equations perturbed by a deterministic vector field . Annals of Applied Probability. 2022;32(4):2568-2586.We are concerned with the problem of global well-posedness of the 3D Navier-Stokes equations on the torus with unitary viscosity. While a full answer to this question seems to be out of reach of the current techniques, we establish a regularization by a deterministic vector field. More precisely, we consider the vorticity form of the system perturbed by an additional transport type term. Such a perturbation conserves the enstrophy and therefore a priori it does not imply any smoothing. Our main result is a construction of a deterministic vector field v = v(t, x) which provides the desired regularization of the system and yields global well-posedness for large initial data outside arbitrary small sets. The proof relies on probabilistic arguments developed by Flandoli and Luo, tools from rough path theory by Hofmanova, Leahy and Nilssen and a new Wong-Zakai approximation result, which itself combines probabilistic and rough path techniques
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