| 变区域上二维 Navier-Stokes 方程的拉回吸引子 |
| Alternative Title | Pullback attractors for 2D Navier-Stokes equations on time-varying domains
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| 宋小亚 |
| Thesis Advisor | 杨璐
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| 2016-06-07
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 2D Navier-Stokes方程
变区域
𝒟
𝜆
1 –拉回吸引子
𝒟
𝜆
1 –拉回渐近紧性
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| Abstract | 本文主要对二维Navier-Stokes方程
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휕푢/휕푡 −△푢 +Σ︀2푖=1푢푖휕푢/휕푥푖= 푓 − ∇푝, 푥 ∈ Ω푡, 푡 > 휏,
∇ · 푢 = 0, 푥 ∈ Ω푡, 푡 ≥ 휏,
푢 = 0, 푥 ∈ 휕Ω푡, 푡 > 휏,
푢(푥, 휏 ) = 푢휏 (푥), 푥 ∈ Ω휏 ,
在同胚情形的变区域上的动力学行为进行研究.
首先, 借助坐标变换以及Faedo-Galerkin 逼近方法, 我们证明上述变区域问题弱解的存在唯一性. 其次, 我们建立弱解在퐿2(휏,푇;푉푡) 和퐿∞(휏, 푇;퐻푡) 中的先验估计,以此证明系统存在풟휆1 –拉回吸收集族. 最后,为克服区域变化带来的困难,我们运用有限휀 –网的方法证明解过程是풟휆1 –拉回渐近紧的, 从而得到了变区域上2D Navier-Stokes方程풟휆1 –拉回吸引子的存在性. |
| Other Abstract | In this master paper, we investigate the dynamic behavior of two dimensional Navier-Stokes equations on time-varying domains in the case of diffeomorphism,
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⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
휕푢/휕푡 −△푢 +Σ︀2푖=1푢푖휕푢/휕푥푖 = 푓 − ∇푝, 푥 ∈ Ω푡, 푡 > 휏,
∇ · 푢 = 0, 푥 ∈ Ω푡, 푡 ≥ 휏,
푢 = 0, 푥 ∈ 휕Ω푡, 푡 > 휏,
푢(푥, 휏 ) = 푢휏 (푥), 푥 ∈ Ω휏 .
Firstly, we prove the existence and uniqueness of the weak solution of the variable regional problem by the coordinate transformation and Faedo-Galerkin approximation. Secondly, we establish priori estimates of the weak solution in 퐿2(휏, 푇; 푉푡) and 퐿∞(휏, 푇;퐻푡) to prove the existence of the family of 풟휆1 –pullback absorbing sets for the system. Finally, in order to overcome the difficulties from regional change, we prove that the solution process is 풟휆1 –pullback asymptotically compact by using the method
of finite 휀 -net, hence, we obtain the existence of 풟휆1 –pullback attractors of 2D Navier-Stokes equations on time-varying domains. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/225608
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
宋小亚. 变区域上二维 Navier-Stokes 方程的拉回吸引子[D]. 兰州. 兰州大学,2016.
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