Random pullback attractor of a non-autonomous local modified stochastic Swift–Hohenberg equation with multiplicative noise

Li, Yongjun; Wu, Hongqing; Zhao, Tinggang

doi:10.1063/5.0008895

Cited by 11 publications

(6 citation statements)

References 24 publications

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“…There have been lots of research on the subject of MSHEs. Roughly speaking, these works mainly include three aspects: attractors [22,36,55,57] and the regularity [41], bifurcations of solutions [5,6,56] and optimal control [12,42]. What is more, for the nonautonomous MSHE, Wang et al presented in [49] a lower number of recurrent solutions by topological methods (see more in [34,[46][47][48]); Wang et al studied the existence of invariant measures and statistical solutions in [50].…”

Section: Introductionmentioning

confidence: 99%

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

Wang

Zhang

2023

Nonlinearity

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We consider a 2D stochastic modified Swift–Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space H 2 m for each m ⩾ 1 . Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on H 2 m . At last, we improve the regularity of the obtained invariant measures to H 2 ( m + 1 ) . With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz et al in their situation (2014 J. Math. Phys. 55 277–304).

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Section: Introductionmentioning

confidence: 99%

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

Wang

Zhang

2023

Nonlinearity

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“…Moreover, what we study in this paper is the stochastic MSHE with a general multiplicative noise, which is allowed to be nonlinear. In this situation, we will not consider the mild solutions (studied in [36,45,53]), but discuss the weak and strong solutions in the stochastic sense, saying, the martingale and pathwise solutions, correspondingly, and establish their global existence. With the unique global pathwise solution in hand, we further explore the existence of ergodic invariant measures for the Markovian semigroup associated with the solution in spaces of high regularity.…”

Section: Introductionmentioning

confidence: 99%

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation

Wang¹,

Zhang²,

Li³

2022

Preprint

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We consider a 2D stochastic modified Swift-Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space H 2m for each m 1. Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on H 2m . At last, we improve the regularity of the obtained invariant measures to m+1) . With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by .

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“…Later, it has also played a valuable role extensively in the study of plasma confinement in toroidal devices, 5 viscous film flow, lasers, 6 and pattern formation. 7 In the previous work, most attention was paid to the existence of attractors (global attractor, 8,9 uniform attractor, 10 pullback attractor, 11,12 and random attractor [12][13][14] ), bifurcations (dynamical bifurcations 15,16 and nontrivial-solution bifurcations 17 ), and optimal control [18][19][20][21] of different types of modified Swift-Hohenberg equations. Wang et al presented in their work 22 a lower number of recurrent solutions for the nonautonomous case by topological methods (see more in other works [23][24][25][26] ).…”

Section: Introductionmentioning

confidence: 99%

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

Wang

Zhang

Zhao

2021

Math Methods in App Sciences

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We consider the nonautonomous modified Swift-Hohenberg equationon a bounded smooth domain Ω ⊂ R n with n ⩽ 3. We show that, if |b|<4 and the external force g satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space H 2 0 (Ω). Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

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Random pullback attractor of a non-autonomous local modified stochastic Swift–Hohenberg equation with multiplicative noise

Cited by 11 publications

References 24 publications

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift–Hohenberg equation

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

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