Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise
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| Main Authors: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866913324271140864 |
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| author | Wang, Jintao Li, Chunqiu Yang, Lu Jia, Mo |
| author_facet | Wang, Jintao Li, Chunqiu Yang, Lu Jia, Mo |
| contents | In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space $H_0^2(U)$, as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_00271 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise Wang, Jintao Li, Chunqiu Yang, Lu Jia, Mo Probability Analysis of PDEs Dynamical Systems 60H15, 86A05, 37L55, 37A25 In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper semi-continuity of random attractors in the Sobolev space $H_0^2(U)$, as the coefficient of the multiplicative noise approaches zero. Then, we extend the classical "stochastic Gronwall's lemma", making it more convenient in applications. Based on this improvement, we are allowed to use the analytic semigroup theory to establish the existence of ergodic invariant measures. |
| title | Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise |
| topic | Probability Analysis of PDEs Dynamical Systems 60H15, 86A05, 37L55, 37A25 |
| url | https://arxiv.org/abs/2012.00271 |