Deterministic and Stochastic Control of Navier--Stokes Equation with Linear, Monotone, and Hyperviscosities

Sritharan, S. S.

doi:10.1007/s0024599110140

Cited by 47 publications

(37 citation statements)

References 36 publications

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“…We can therefore follow the same way of reasoning as before (see also [46]) to get (6.16). The proof is completed.…”

Section: Nmentioning

confidence: 85%

“…With all the properties of the operator A ε (among which the strict monotonicity, the maximality and the hemicontinuity) the existence and uniqueness of a martingale solution (and hence from the uniqueness, the strong) solution to (6.14) follows exactly the way of proceeding as in [46], and we can formulate the following result without proof.…”

Section: Homogenization Of a Stochastic Ladyzhenskaya Model For Incommentioning

confidence: 98%

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Homogenization in algebras with mean value

Woukeng¹

2015

Banach J. Math. Anal.

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Abstract. In several works, the theory of strongly continuous groups is used to build a framework for solving stochastic homogenization problems. Following this idea, we construct a detailed and comprehensive theory of homogenization. This enables to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not, thereby responding affirmatively to the question raised by Zhikov and Krivenko [V.V. Zhikov, E.V. Krivenko, Homogenization of singularly perturbed elliptic operators. Matem. Zametki, 33 (1983) 571-582 (english transl.: Math. Notes, 33 (1983) 294-300)] to know whether it is possible to homogenize problems in nonergodic algebras. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.

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“…We can therefore follow the same way of reasoning as before (see also [46]) to get (6.16). The proof is completed.…”

Section: Nmentioning

confidence: 85%

Section: Homogenization Of a Stochastic Ladyzhenskaya Model For Incommentioning

confidence: 98%

Homogenization in algebras with mean value

Woukeng¹

2015

Banach J. Math. Anal.

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“…The Aldous criterion for tightness is a sufficient condition for proving the tightness, refer to [9,10]. In [13,14], by using this criterion, one can get the tightness in Skorokhod space once the energy inequality is proved, thus martingale problem is formulated on vector valued Skorokhod space and with Gaussian as the limit measure. We use the same Aldous criterion to get the tightness in vector valued Skorokhod space, see Lemmas 2.4 and 2.5.…”

Section: Introductionmentioning

confidence: 99%

Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equations with jump

Dong

Zhai

2011

Journal of Differential Equations

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“…There are many papers which deal with the optimal control theory of a viscous incompressible fluid ( [5], [11], [15] for example). In most of them the distributed deterministic or distributed stochastic control problems are considered.…”

Section: Introductionmentioning

confidence: 99%

Optimal control of a coefficient in modification Navier-Stokes equations

Bilić

2009

Mathematica Slovaca

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ABSTRACT. This paper deals with the optimal control of a coefficient in the modification of Navier-Stokes equations. Namely, the motion of the viscous incompressible fluid for a small gradient of velocity is described by Navier-Stokes equations where the coefficient of the kinematic viscosity ν is the positive constant (ν 0 ). For a greater gradient of velocity the coefficient of kinematic viscosity is a positive function of the gradient of velocity, that is ν (|∇u|). In our case ν (|∇u|) = ν 0 + ν 1 a (|∇u|) where ν 0 , ν 1 ∈ R + . The function a is positive and monotone and it is taken as a control variable. The existence of a solution of the optimal control problem is proved. Further, the approximation of the control problem by the finite-dimensional control problem is performed. The proof of the existence of a solution of that aproximate problem has been brought into light. Finally, the connection between the solution of the control problem and the solution of the approximate control problem is established.

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Deterministic and Stochastic Control of Navier--Stokes Equation with Linear, Monotone, and Hyperviscosities

Cited by 47 publications

References 36 publications

Homogenization in algebras with mean value

Homogenization in algebras with mean value

Martingale solutions and Markov selection of stochastic 3D Navier–Stokes equations with jump

Optimal control of a coefficient in modification Navier-Stokes equations

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