Mild solutions of stochastic Navier‐Stokes equation with jump noise in ‐spaces

Fernando, B. P. W.; Rüdiger, Barbara; Sritharan, S. S.

doi:10.1002/mana.201300248

Cited by 14 publications

(18 citation statements)

References 24 publications

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“…Remark 4.11. Stochastic Navier-Stokes equations with Poisson noise in L p -spaces are investigated in [18].…”

Section: Application To Stochastic 2d Quasi-geostrophic Equationsmentioning

confidence: 99%

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Zhu¹,

Brzeźniak²,

Liu³

2019

SIAM J. Math. Anal.

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We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.

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“…Remark 4.11. Stochastic Navier-Stokes equations with Poisson noise in L p -spaces are investigated in [18].…”

Section: Application To Stochastic 2d Quasi-geostrophic Equationsmentioning

confidence: 99%

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Zhu¹,

Brzeźniak²,

Liu³

2019

SIAM J. Math. Anal.

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“…for a certain stopping time with values in [0, ] independent of ∈ ℕ. The idea of the proof may be found in [3,14].…”

Section: The Stochastic Navier-stokes Equationsmentioning

confidence: 99%

“…For the local mild solution with additive noise given by Wiener processes, we refer to [3]. In [14,34], the stochastic Navier-Stokes equations with additive Lévy noise are considered. In [25], an existence and uniqueness result for strong pathwise solutions is given.…”

Section: Introductionmentioning

confidence: 99%

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Benner

Trautwein

2019

Mathematische Nachrichten

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We consider the controlled stochastic Navier-Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier-Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. K E Y W O R D SLévy process, local mild solution, stochastic Navier-Stokes equations, stochastic optimal control M S C ( 2 0 1 0 ) 93E20, 49J20, 35Q30 1444

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“…For the local mild solution with additive noise given by Wiener processes, we refer to [13]. In [14,15], the stochastic Navier-Stokes equations with additive Lévy noise are considered. A generalization to multiplicative Lévy noise can be found in [16].…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

Benner

Trautwein

2021

Appl Math Optim

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We analyze the control problem of the stochastic Navier–Stokes equations in multi-dimensional domains considered in Benner and Trautwein (Math Nachr 292(7):1444–1461, 2019) restricted to noise terms defined by a Q-Wiener process. The cost functional related to this control problem is nonconvex. Using a stochastic maximum principle, we derive a necessary optimality condition to obtain explicit formulas the optimal controls have to satisfy. Moreover, we show that the optimal controls satisfy a sufficient optimality condition. As a consequence, we are able to solve uniquely control problems constrained by the stochastic Navier–Stokes equations especially for two-dimensional as well as for three-dimensional domains.

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Mild solutions of stochastic Navier‐Stokes equation with jump noise in ‐spaces

Abstract: In this paper, we study solvability of the local mild solution of stochastic Navier‐Stokes equation with jump noise in Lp‐spaces. This research work has mainly carried out by exploiting some interesting works of Tosio Kato.

Cited by 14 publications

References 24 publications

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

A Stochastic Maximum Principle for Control Problems Constrained by the Stochastic Navier–Stokes Equations

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