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

Arch Rational Mech Anal

Gołdys

Jegaraj

2017

Self Cite

We study a stochastic Landau-Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is compactness, or weak to strong continuity, of the solution map for a deterministic Landau-Lifschitz equation, when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications of ferromagnetic nanowires to the fabrication of magnetic memories.

Section: Preliminariesmentioning

confidence: 99%

Large Deviations and Transitions Between Equilibria for Stochastic Landau–Lifshitz–Gilbert Equation

Arch Rational Mech Anal

Gołdys

Jegaraj

2017

Self Cite

“…In this way the noise is rougher than in [7], [30], [24], [25]. Our original aim was to investigate the existence of invariant measures and stationary solutions for these stochastic Navier-Stokes equations, following on one side the work of [16] in bounded domains and on the other side the method by the first named authour with Motyl and Ondreját [9] for unbounded domains but with more regular noise, which is based on [23]. While working on this problem we realised that the existence or uniqueness of solutions was left open in some cases; indeed, [16] proved existence of martingale and stationary martingale solutions with rough multiplicative noise only for d = 2.…”

Section: Introductionmentioning

confidence: 99%

A note on stochastic Navier–Stokes equations with not regular multiplicative noise

Ferrario

2016

Stoch PDE: Anal Comp

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Dedicated to Professor Boles law Szafirski on his 80th birthday.Abstract. We consider the Navier-Stokes equations in R d (d = 2, 3) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so Itô calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for d = 2, 3 and pathwise uniqueness for d = 2.

“…Then there exists at least one invariant measure for (6.1). time homogeneous damped tamed NSEs, which can be proved analogously, see also [10,Theorem 4.11]. Theorem 6.6.…”

Section: Invariant Measuresmentioning

confidence: 83%

“…We also prove the existence of invariant measures, Theorem 6.1, for time homogeneous damped tamed Navier-Stokes equations 6.1 under the assumptions (H1) ′ − (H3) ′ (see Section 6). We use the technique (Theorem 6.4) of Maslowski and Seidler [23], see also [6,10], working with weak topologies to establish the existence of invariant measures. We show the two conditions of Theorem 6.4, boundedness in probability and sequentially weakly Feller property are satisfied for the semigroup (T t ) t≥0 , defined by (6.2).…”

Section: Introductionmentioning

confidence: 99%

Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Dhariwal

2020

J. Math. Fluid Mech.

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Röckner and Zhang in [31] proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary case using a result from [36]. In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the L 4 -norm of the solution from the torus to R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on R 3 for a time-homogeneous damped tamed 3D Navier-Stokes equation, given by (6.1). (2010). Primary 60H15; Secondary 35R60, 35Q30, 76D05. Mathematics Subject ClassificationKeywords. Stochastic tamed Navier-Stokes equations, stochastic damped Navier-Stokes equations, invariant measures. 1 i.e. the equation is satisfied in the weak sense and the solution u belongs to the space L ∞ (0, T ; H) ∩ L 2 (0, T ; V) ∩ L β+1 (0, T ; L β+1 ). 2 A pair of function (u, p) is a strong solution iff it is a weak solution and u ∈ L ∞ (0, T ; V) ∩ L 2 (0, T ; H 2 ) ∩ L ∞ (0, T ; L β+1 ).