Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

Mohammed, Wael W.; Blömker, Dirk

doi:10.1137/140981952

Cited by 11 publications

(6 citation statements)

References 19 publications

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“…Lemma 14 Under Assumptions 2.6 and 2.7 and Definition 2.10, let X be a real‐valued stochastic process such that for some small r ≥ 0, we have

X false(0 false) = 풪 false(ε^{- r} false)

and

d X = G d T

with

G = 풪 false(ε^{- r} false)

. Then, for

l \in ℕ

with m ≥ l ≥ 1, it holds the following: 1.…”

Section: Boundedness and Error Analysismentioning

confidence: 99%

“…Assumption 2.6. 14 Let

W false(t false) = {false(W_{1} false(t false), W_{2} false(t false) false)}^{T}

be a Wiener process on X 0 , defined on a filtered probability space

false(normalΩ, F, ℙ false)

with covariance operator

풬 = {{false(풬}_{1} {, 풬}_{2} false)}^{T}

{false(풬}_{1} : H^{1} false(D false) \to H^{1} false(D false) {, 풬}_{2} : H^{1} false({normalΓ}_{1} false) \to H^{1} false({normalΓ}_{1} false) false)

defined by

풬_{i} f_{i, k} = α_{i, k}^{2} f_{i, k}

, where

{false{α_{i, k} false}}_{k = 0}^{\infty}

is a sequence of real numbers and

{false{f_{1, k} false}}_{k = 0}^{\infty} false({false{f_{2, k} false}}_{k = 0}^{\infty} false)

is any orthonormal basis on H 1 ( D )( H 1 (Γ 1 )) with f i , 0 ≡ Constant for

i = 1,2

. For t ≥ 0, we can write W i ( t ) as

W_{i} (t) = \sum_{k = 0}^{\infty} α_{i, k} β_{i, k} f_{i, k},

…”

Section: Preliminariesmentioning

confidence: 99%

“…For the singular boundary perturbation issue of stochastic partial differential equation, Mohammed and Blömker 14 stressed the impact of noise acting on the stochastic Neumann boundary in a reaction‐diffusion system. Da Prato and Zabczyk 9 discussed the difference between the problems with Dirichlet and Neumann boundary noises.…”

Section: Introductionmentioning

confidence: 99%

“…Originating from Da Prato and Zabczyk, 9 Mohammed and Blömker, 14 and Blömker and Mohammed, 16 it first formulates the original system as an abstract stochastic equation with non‐Lipschitz nonlinear term satisfying polynomial growth and fulfilling a suitable dissipative condition. And it has a unique mild solution, which is also the unique weak solution for the system 10 .…”

Section: Introductionmentioning

confidence: 99%

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Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

Chen

Lei

2021

Math Methods in App Sciences

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This work is concerned with a class of stochastic partial differential equations with a fast random dynamical boundary condition. In the limit of fast diffusion, it derives an effective stochastic partial differential equation to describe the evolution of the dominant pattern. Using the multiscale analysis and the averaging principle, it then establishes deviation estimates of the original stochastic system towards the effective approximating system. A concrete example further illustrates the result on a large time scale.

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“…Lemma 14 Under Assumptions 2.6 and 2.7 and Definition 2.10, let X be a real‐valued stochastic process such that for some small r ≥ 0, we have

X false(0 false) = 풪 false(ε^{- r} false)

and

d X = G d T

with

G = 풪 false(ε^{- r} false)

. Then, for

l \in ℕ

with m ≥ l ≥ 1, it holds the following: 1.…”

Section: Boundedness and Error Analysismentioning

confidence: 99%

“…Assumption 2.6. 14 Let

W false(t false) = {false(W_{1} false(t false), W_{2} false(t false) false)}^{T}

be a Wiener process on X 0 , defined on a filtered probability space

false(normalΩ, F, ℙ false)

with covariance operator

풬 = {{false(풬}_{1} {, 풬}_{2} false)}^{T}

{false(풬}_{1} : H^{1} false(D false) \to H^{1} false(D false) {, 풬}_{2} : H^{1} false({normalΓ}_{1} false) \to H^{1} false({normalΓ}_{1} false) false)

defined by

풬_{i} f_{i, k} = α_{i, k}^{2} f_{i, k}

, where

{false{α_{i, k} false}}_{k = 0}^{\infty}

is a sequence of real numbers and

{false{f_{1, k} false}}_{k = 0}^{\infty} false({false{f_{2, k} false}}_{k = 0}^{\infty} false)

is any orthonormal basis on H 1 ( D )( H 1 (Γ 1 )) with f i , 0 ≡ Constant for

i = 1,2

. For t ≥ 0, we can write W i ( t ) as

W_{i} (t) = \sum_{k = 0}^{\infty} α_{i, k} β_{i, k} f_{i, k},

…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

Chen

Lei

2021

Math Methods in App Sciences

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show abstract

“…The importance of including stochastic effects in complicated system modeling has been highlighted. For instance, there is a significant focus on using SPDEs to mathematically model complex phenomena in finance, materials sciences, electrical and mechanical engineering, information systems, condensed matter physics, biology, and climate systems [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

et al. 2022

Self Cite

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We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the exp(−Φ(η))-expansion method and sine–cosine method. Since this equation is used to explain the hydrodynamic model of shallow-water waves, the wave of leading fluid flow, and plasma physics, scientists will be able to characterize a wide variety of fascinating physical phenomena with these solutions. Furthermore, we evaluate the influence of noise on the behavior of the acquired solutions using 2D and 3D graphical representations.

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Approximate solutions for stochastic time‐fractional reaction–diffusion equations with multiplicative noise

Mohammed

2020

Math Methods in App Sciences

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In this paper, we consider the approximate solutions of time-fractional reactiondiffusion equations forced by multiplicative noise on a bounded domain. When the diffusion is large, one can approximate the solutions of the stochastic time-fractional reaction-diffusion equations with polynomial term by the solutions of a stochastic time-fractional ordinary equations. We illustrate our results by applying to time-fractional logistic and time-fractional Ginzburg-Landau equations.

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Fast Diffusion Limit for Reaction-Diffusion Systems with Stochastic Neumann Boundary Conditions

Cited by 11 publications

References 19 publications

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions

The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

Approximate solutions for stochastic time‐fractional reaction–diffusion equations with multiplicative noise

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