带非线性噪音的随机${g}$-Navier-Stokes方程的后向弱紧均值动力学

摘要/Abstract

摘要：

考虑由无限维柱形噪声驱动的随机二维$g$-Navier-Stokes方程的均值动力学, 且该方程具有非线性扩散项和依赖于时间的外力项. 当非线性扩散项是Lipschitz连续的并且外力项是局部可积时, 可得到一个均值随机动力系统(RDS). 若外力项是缓增的, 均值RDS在偶幂的Bochner空间中有唯一的弱拉回均值吸引子. 此外, 通过使用Bochner 空间相对于时间的单调性, 证明若外力项是后向缓增的, 则弱拉回均值吸引子的后向并集在渐进Bochner空间中是定义明确且弱紧的. 最后, 当外力项为零、周期或递增时分别给出后向弱紧弱吸引子的三个例子.

关键词: 弱拉回均值吸引子, $g$-Navier-Stokes方程, 非线性噪音, Bochner空间, 后向弱紧性

Abstract:

We study the mean dynamics for the stochastic 2D $g$-Navier-Stokes matrix driven by infinitely dimensional cylindrical noise with a nonlinear diffusion term and a time-dependent external force. We first obtain a mean random dynamical system if the nonlinear diffusion term is Lipschtz continuous and the force is locally integrable. We then show that the mean RDS possesses a unique mean pullback weak attractor in the Bochner space of even power if the force is also tempered. Moreover, by using the monotonicity of Bochner spaces with respect to the time, we show that the backward union of the mean pullback w-attractor is well-defined and weakly compact in progressive Bochner spaces if the force is backward tempered. We finally provide three examples of backward $w$-compact $w$-attractors when the force is null, periodic or increasing, respectively.

Key words: Mean pullback $w$-attractor, $g$-Navier-Stokes matrix, Nonlinear noise, Bochner spaces, Backward $w$-compactness

中图分类号:

O211.4

李扬荣, 王凤玲, 杨爽. 带非线性噪音的随机${g}$-Navier-Stokes方程的后向弱紧均值动力学[J]. 数学物理学报, 2023, 43(2): 531-548.

Li Yangrong, Wang Fengling, Yang Shuang. Backward $W$-compact Mean Dynamics for Stochastic ${g}$-Navier-Stokes Equations with Nonlinear Noise[J]. Acta mathematica scientia,Series A, 2023, 43(2): 531-548.

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