The internal stabilization by noise of the linearized Navier-Stokes equation

Viorel Barbu

Displaying similar documents to “The internal stabilization by noise of the linearized Navier-Stokes equation”

On the existence of steady-state solutions to the Navier-Stokes system for large fluxes

Antonio Russo, Giulio Starita (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper we deal with the stationary Navier-Stokes problem in a domain $Ω$ with compact Lipschitz boundary $\partial Ω$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial Ω$ , with possible countable exceptional set, provided $a$ is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for $Ω$ bounded.

A short note on $L^{q}$ theory for Stokes problem with a pressure-dependent viscosity

Václav Mácha (2016)

Czechoslovak Mathematical Journal

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We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$ , namely, it is represented by a stress tensor $T (D u, p) : = ν (p, | D |^{2}) D$ which satisfies $r$ -growth condition with $r \in (1, 2]$ . In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

Profile decomposition for solutions of the Navier-Stokes equations

Isabelle Gallagher (2001)

Bulletin de la Société Mathématique de France

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We consider sequences of solutions of the Navier-Stokes equations in $ℝ^{3}$ , associated with sequences of initial data bounded in ${\dot{H}}^{1 / 2}$ . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in ${\dot{H}}^{1 / 2}$ , up to a remainder term small in $L^{3}$ ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If $𝒜$ is an “admissible” space (in...

On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces

Wisam Alame (2005)

Banach Center Publications

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We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to $W_{p}^{2 s + 2, s + 1} (Ω^{T})$ or to $B_{p, q}^{2 s + 2, s + 1} (Ω^{T})$ with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in $W_{p}^{2 k + 2, k + 1}$ for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

An $L^{q} (L ²)$ -theory of the generalized Stokes resolvent system in infinite cylinders

Reinhard Farwig, Myong-Hwan Ri (2007)

Studia Mathematica

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Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $Σ \subset ℝ^{n - 1}$ , a bounded domain of class $C^{1, 1}$ , are obtained in the space $L^{q} (ℝ; L ² (Σ))$ , q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

Existence of solutions to the nonstationary Stokes system in $H_{- μ}^{2, 1}$ , μ ∈ (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski (2007)

Applicationes Mathematicae

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We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f \in L_{2, - μ}$ , $v ₀ \in H_{- μ} ¹$ , μ ∈ (0,1). We prove an estimate of the velocity in the $H_{- μ}^{2, 1}$ norm and of the gradient of the pressure in the norm of $L_{2, - μ}$ . We apply the Fourier transform with respect to the variable along...

On reliability analysis of consecutive $k$ -out-of- $n$ systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

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In this paper, we consider the linear and circular consecutive $k$ -out-of- $n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n - r + 1$ components ( $r \leq n$ ) of the system are working at time $t$ , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$ -out-of- $n$ systems. In the following, we investigate the inactivity time of the component with lifetime...

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form $\frac{d X^{ε} (t)}{d t} = ε B (X^{ε} (t)$ , $𝛯 (q_{1} (t)), 𝛯 (q_{2} (t)), ..., 𝛯 (q_{ℓ} (t)))$ where $𝛯 (t)$ , $t \geq 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j} (t) = α_{j} t$ , $α_{1} l t; α_{2} l t; \dots l t; α_{k}$ and $q_{j}$ , $j = k + 1, ..., ℓ$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Remarks on regularity criteria for the Navier-Stokes equations with axisymmetric data

Zujin Zhang (2016)

Annales Polonici Mathematici

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We consider the axisymmetric Navier-Stokes equations with non-zero swirl component. By invoking the Hardy-Sobolev interpolation inequality, Hardy inequality and the theory of $* A_{β}$ (1 < β < ∞) weights, we establish regularity criteria involving $u^{r}$ , $ω^{z}$ or $ω^{θ}$ in some weighted Lebesgue spaces. This improves many previous results.

Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

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We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$ -Laplacian and the Navier $p$ -biharmonic operator on a ball of radius $R$ in $ℝ^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty$ . Let $p$ tend to $\infty$ . There is a critical radius $R_{C}$ of the ball such that the principal eigenvalue goes to $\infty$ for $0 < R \leq R_{C}$ and to $0$ for $R > R_{C}$ . The critical radius is $R_{C} = 1$ for any $N \in ℕ$ for the $p$ -Laplacian and $R_{C} = \sqrt{2 N}$ in the case of the $p$ -biharmonic operator. When $p$ approaches $1$ , the principal eigenvalue...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process $ℐ^{u} \subset ℤ^{d}$ at level $u$ , $d \geq 3$ . Roughly speaking, we show that observations of $ℐ^{u}$ restricted to two disjoint subsets $A_{1}$ and $A_{2}$ of $ℤ^{d}$ are approximately independent, once we add a sprinkling to the process $ℐ^{u}$ by slightly increasing the parameter $u$ . Our results differ from previous ones in that we allow the mutual distance between the sets $A_{1}$ and $A_{2}$ to be much smaller than their diameters. We then provide an important application...

Extensions of the Cugiani-Mahler theorem

Yann Bugeaud (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In 1955, Roth established that if $ξ$ is an irrational number such that there are a positive real number $ε$ and infinitely many rational numbers $p / q$ with $q \geq 1$ and $| ξ - p / q | < q^{- 2 - ε}$ , then $ξ$ is transcendental. A few years later, Cugiani obtained the same conclusion with $ε$ replaced by a function $q \mapsto ε (q)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $| ξ - p / q | < q^{- 2 - ε (q)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it...