Stochastic Navier-Stokes equation on a 2D rotating sphere with stable L'evy noise: existence and uniqueness of weak and strong solutions (original) (raw)

Random dynamical systems generated by stochastic Navier--Stokes equation on the rotating sphere

arXiv (Cornell University), 2014

In this paper we first prove the existence and uniqueness of solution to the stochastic Navier-Stokes equation on the rotating 2-dimensional sphere. Then we show the existence of an asymptotically compact random dynamical system associated with the equation. Contents 1. Introduction 2. The Navier-Stokes equations on a rotating unit sphere 2.1. Preliminaries 2.2. The weak formulation 3. Stochastic Navier-Stokes equation on a rotating unit sphere 10 4. Proofs of Theorems 3.2 and 3.3 11 4.1. Proof of Theorem 3.2 11 4.2. Proof of Theorem 3.3 18 5. Ornstein-Uhlenbeck processes 21 5.1. Preliminaries 21 5.2. Ornstein-Uhlenbeck process 24 6. Random dynamical systems generated by the stochastic NSEs on the sphere 27 7. Appendix 35 References 36

Stochastic Navier–Stokes Equations on a Thin Spherical Domain

Applied Mathematics & Optimization

Incompressible Navier–Stokes equations on a thin spherical domain Q_\varepsilon Qεalongwithfreeboundaryconditionsunderarandomforcingareconsidered.TheconvergenceofthemartingalesolutionoftheseequationstothemartingalesolutionofthestochasticNavier–StokesequationsonasphereQ ε along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier–Stokes equations on a sphereQεalongwithfreeboundaryconditionsunderarandomforcingareconsidered.TheconvergenceofthemartingalesolutionoftheseequationstothemartingalesolutionofthestochasticNavier–Stokesequationsonasphere\mathbb {S}^2$$ S 2 as the thickness converges to zero is established.

Probabilistic estimates for the Two Dimensional Stochastic Navier-Stokes Equations

Journal of Statistical Physics, 2000

We consider the Navier-Stokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number RRR. We prove probabilistic estimates for the long time behaviour of the solutions that imply bounds for the dissipation scale and energy spectrum as RtoinftyR\to\inftyRtoinfty.

Dynamics of stochastic 2D Navier–Stokes equations

Journal of Functional Analysis, 2010

In this paper, we study the dynamics of a two-dimensional stochastic Navier-Stokes equation on a smooth domain, driven by linear multiplicative white noise. We show that solutions of the 2D Navier-Stokes equation generate a perfect and locally compacting C 1,1 cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium/stationary solution. We give sufficient conditions on the parameters of the Navier-Stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.

Ergodicity for stochastic equation of Navier--Stokes type

arXiv: Probability, 2020

In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier-Stokes equation (N.S.E.) system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant measure, there exist another one. In...

Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains

Building upon a recent work by two of the authors and J. Seidler on bw-Feller property for stochastic nonlinear beam and wave equations, we prove the existence of an invariant measure to stochastic 2-D Navier-Stokes (with multiplicative noise) equations in unbounded domains. This answers an open question left after the first author and Y. Li proved a corresponding result in the case of an additive noise.