By Sergei B. Kuksin

ISBN-10: 3037190213

ISBN-13: 9783037190210

This publication provides an account of contemporary achievements within the mathematical conception of two-dimensional turbulence, defined through the 2nd Navier-Stokes equation, perturbed through a random strength. the most effects provided right here have been acquired over the past 5 to 10 years and, during the past, were to be had in simple terms in papers within the fundamental literature. Their precis and synthesis right here, starting with a few preliminaries on partial differential equations and stochastics, make this publication a self-contained account that may attract readers with a common historical past in research. After laying the basis, the writer is going directly to fresh effects on ergodicity of random dynamical platforms, which the randomly compelled Navier-Stokes equation defines within the functionality house of divergence-free vector fields, together with a imperative restrict Theorem. The actual that means of those effects is mentioned in addition to their relatives with the idea of attractors. subsequent, the writer reports the behaviour of ideas while the viscosity is going to 0. within the ultimate part those dynamical equipment are used to derive the so-called stability relations--the infinitely many algebraical kin happy through the strategies. A e-book of the eu Mathematical Society (EMS). allotted in the Americas by means of the yankee Mathematical Society.

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**Randomly Forced Nonlinear Pdes and Statistical Hydrodynamics by Sergei B. Kuksin PDF**

This ebook supplies an account of contemporary achievements within the mathematical conception of two-dimensional turbulence, defined by means of the second Navier-Stokes equation, perturbed through a random strength. the most effects awarded right here have been acquired over the past 5 to 10 years and, during the past, were on hand merely in papers within the fundamental literature.

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Let us choose any number 2e C. R0 ≥ 1 ∨ e−1 Then 1 E R(k) ≥ R0 =⇒ E R( ) ≤ E R(k), ∀ ≥ k + 1 . s. ∀ ≥ k . 9) 46 6. Uniqueness of a stationary measure: kick-forces Let us ﬁx some 0 < d0 ≤ 1. 1), corresponding to the same kicks {η k }. Assume that R(0) ≤ R0 . Then, |u1 (0)|, |u2 (0)| ≤ R0 . So, if 0 = η1 = · · · = ηT , then |uj (T )| ≤ exp(−T )R0 , and |uj (T )| ≤ 14 d0 for j = 1, 2, if T = [ln (4 R0 d−1 0 )] + 1 . 10) where θ depends on R0 and d0 . 3 The theorem Let us denote by P1 (H) the class of measures µ with the ﬁnite ﬁrst moment M1 (µ) = |u| µ(du) < ∞.

10) u(t) is Ft -adapted . For any t0 ≥ 0, let us write ut0 (t) = u(t0 + t). 4) with u0 = u(t0 ) and ζ(t) replaced by ζ(t + t0 ) − ζ(t0 ). 9) to ut0 we see that u(t) , t ≥ t0 , is a function of ut0 and ζ(s) − ζ(t0 ), t0 ≤ s ≤ t . 4. The solution u(t) is an Ft -adapted Markov process with continuous trajectories. We omit details of the proof. , the Wiener measure, corresponding to the process ζ). 8), D u(·) = Φ ◦ (dW ) . Here Φ ◦ (dW ) denotes the push-forward of dW by the map Φ. 3. 5. 4) with the process ζ(t) replaced by some process ζ (t) such that D(ζ ) = D(ζ).

We claim that ν1 − ν2 var ≤ C∗ d . 4) Indeed, let vj = PN S(uj ), j = 1, 2. Then, due to (S2), |v1 − v2 | ≤ C(R) d . 5) Since bs = 0 for |s| ≤ N and D(ξsk ) = ps (r)dr, where the functions ps are Lipschitz, then D(PN η1 ) = q(x)dx, x ∈ RN∗ . Here N∗ = dim H(N ) and q is a Lipschitz function. 3). 4) and the Dobrushin lemma, the measures ν1 , ν2 admit a maximal coupling (Ξ1 , Ξ2 ), deﬁned on a probability space (Ω2 , F2 , P2 ), and depending on the parameter (u1 , u2 ). , Ξj = Ξj (ω2 ; u1 , u2 ) and P2 Ξ1 = Ξ2 = ν1 − ν2 var ≤ C∗ d .

### Randomly Forced Nonlinear Pdes and Statistical Hydrodynamics in 2 Space Dimensions by Sergei B. Kuksin

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