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Dynamics of solutions for the three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains

Ho Thi Hang|Bui Kim My|Pham Tri Nguyen
대한수학회 (The Korean Mathematical Society)|2024.09|대한수학회보 (Bulletin of the KMS)|제61권 제5호|pp. 1369-1393|DOI 10.4134/BKMS.b230635

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키워드

pullback random attractorsmultiplicative noiseunbounded domains
주제분류수학

초록

This paper is devoted to studying the long-time behavior for the three-dimensional stochastic globally modified Navier-Stokes equations driven by a linear multiplicative white noise on some unbounded domains $\mathcal{O}.$ By using the Ornstein-Uhlenbeck process, we first transfer the original equation to a random dynamical system, and then prove the existence of pullback attractors as well as the upper semicontinuity of the attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, the asymptotic compactness of the solutions is proved by Ball's idea of energy equations. The periodicity of the attractors is also obtained when the deterministic non-autonomous external terms are periodic in time. Our results extend and generalize some existing results.

Abstract

This paper is devoted to studying the long-time behavior for the three-dimensional stochastic globally modified Navier-Stokes equations driven by a linear multiplicative white noise on some unbounded domains $\mathcal{O}.$ By using the Ornstein-Uhlenbeck process, we first transfer the original equation to a random dynamical system, and then prove the existence of pullback attractors as well as the upper semicontinuity of the attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, the asymptotic compactness of the solutions is proved by Ball's idea of energy equations. The periodicity of the attractors is also obtained when the deterministic non-autonomous external terms are periodic in time. Our results extend and generalize some existing results.

참고문헌 (30)

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