d维球面上Navier-Stokes流的周期性与惯性流形
Periodicity and Inertial Manifolds for Navier-Stokes Flows on $d$-Spheres
- Hanoi University of Science and Technology（河内科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在d维球面的向量场L^p空间框架下，证明了Navier-Stokes方程周期解的存在唯一性，并利用Ebin-Marsden拉普拉斯算子特征值分布，得到该周期解附近解的惯性流形存在性。
AI中文摘要：
考虑d维球面$\boldsymbol{\text{S}}^d$，记$\boldsymbol{\text{T}\text{S}}^d$上所有向量场的集合为$\boldsymbol{\text{Γ}}(\boldsymbol{\text{T}\text{S}}^d)$。我们研究Navier-Stokes方程（NSE）：$\boldsymbol{\text{∂}}_t \boldsymbol{u} + \boldsymbol{\nabla}_{\boldsymbol{u}} \boldsymbol{u} + \boldsymbol{\text{grad}} \boldsymbol{\text{π}} = \boldsymbol{\text{Δ}} \boldsymbol{u} + \boldsymbol{\text{div}} \boldsymbol{f}(\boldsymbol{\text{·}}, t)$，且满足$\boldsymbol{\text{div}} \boldsymbol{u}=0$，其中向量场$\boldsymbol{u}(\boldsymbol{\text{·}}, t) \boldsymbol{\text{∈}} \boldsymbol{\text{Γ}}(\boldsymbol{\text{T}\text{S}}^d)$，$\boldsymbol{\text{Δ}}$表示由$\boldsymbol{\text{Δ}} \boldsymbol{u}= \boldsymbol{\text{div}} (\boldsymbol{\nabla} \boldsymbol{u} + \boldsymbol{\nabla} \boldsymbol{u}^t)^\boldsymbol{\text{♯}}$定义的Ebin-Marsden拉普拉斯算子，$\boldsymbol{\text{div}} \boldsymbol{f}(\boldsymbol{\text{·}}, t)$为周期外力。我们在$\boldsymbol{\text{S}}^d$上向量场的$L^p$空间框架下研究该方程，证明其周期解的存在性与唯一性；此外，利用Ebin-Marsden拉普拉斯算子的特征值分布，证明该解附近解的惯性流形存在性。
英文摘要：
Consider a $d$-sphere $\mathbb{S}^d$ and denote by $Γ(T\mathbb{S}^d)$ the set of all vector fields on $\mathbb{S}^d$. We study the Navier-Stokes equations (NSE): $$ \partial_t u + \nabla_u u + grad π= \mathbfΔ u + \operatorname{div} f(\cdot, t);\, \operatorname{div} u=0,$$ for the vector field $u(\cdot, t)\in Γ(T\mathbb{S}^d)$, where $\mathbfΔ$ denotes the Ebin-Marsden Laplace operator defined by $\mathbfΔ u= \operatorname{div} (\nabla u + \nabla u^t)^{\sharp}$, and $\operatorname{div} f(\cdot, t)$ is the periodic external force. We investigate the Navier-Stokes equations in the framework of $L^p$-spaces over vector fields on $\mathbb{S}^d$ and prove the existence and uniqueness of a periodic solution to such equations. Moreover, exploiting the distribution of eingenvalues of Ebin-Marsden Laplace operator we show the existence of an inertial manifold for solutions around that solution.
