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arXiv 2303.03489math.AP
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具有 Navier 边界条件的三维 Navier-Stokes 方程的大时间行为

Large time behavior for the 3D Navier-Stokes with Navier boundary conditions

James P. Kelliher, Christophe Lacave, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, Edriss S. Titi

备注：
Version accepted for publication in Journal of Mathematical Fluid Mechanics
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AI总结：

本文研究三维有界区域中带 Navier 摩擦边界条件的不可压缩 Navier-Stokes 方程，借助 Stokes 半群、Galerkin 方法及新型不等式证明弱解存在与能量指数衰减，并刻画无摩擦时向刚性旋转的收敛。

摘要：

我们研究光滑有界区域 $Ω$ 中的三维不可压缩 Navier-Stokes 方程，其初始速度 $u_0$ 平方可积、无散且与 $\partial Ω$ 相切。我们为方程补充 Navier 摩擦边界条件 $u \cdot n = 0$ 和 $[(2Su)n + αu]_{tang} = 0$，其中 $n$ 是 $\partial Ω$ 的单位外法向量，$Su = (Du + (Du)^t)/2$，$α\in C^0(\partialΩ)$ 是边界摩擦系数，而 $[\cdot]_{tang}$ 表示其参数到 $\partial Ω$ 切空间的投影。我们证明了由此产生的初边值问题存在全局 Leray 型弱解，并在摩擦为正时证明这些解在能量范数下指数衰减。若摩擦非负且区域不是旋转体，我们也证明了指数衰减。这两个结果在 Dirichlet 边界条件情形下广为人知，但是，即使它们已被隐式地用于 Navier 边界条件，文献中仍缺少全面分析。在仔细研究此类边界条件的 Stokes 半群之后，我们使用 Galerkin 方法证明存在性，使用针对边界微分几何作适当调整的 Poincaré 型不等式，以及一个新的 Gronwall 型积分不等式。此外，在无摩擦情形 $α=0$ 下，若区域是旋转体，我们证明解收敛到稳态刚性旋转。

英文摘要：

We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain $Ω$ with initial velocity $u_0$ square-integrable, divergence-free and tangent to $\partial Ω$. We supplement the equations with the Navier friction boundary conditions $u \cdot n = 0$ and $[(2Su)n + αu]_{tang} = 0$, where $n$ is the unit exterior normal to $\partial Ω$, $Su = (Du + (Du)^t)/2$, $α\in C^0(\partialΩ)$ is the boundary friction coefficient and $[\cdot]_{tang}$ is the projection of its argument onto the tangent space of $\partial Ω$. We prove global existence of a weak Leray-type solution to the resulting initial-boundary value problem and exponential decay in energy norm of these solutions when friction is positive. We also prove exponential decay if friction is non-negative and the domain is not a solid of revolution. These two results are well known in the case of Dirichlet boundary condition, but, even if they have been implicitly used for the Navier boundary conditions, the comprehensive analysis is not available in the literature. After carefully studying the Stokes semigroup for such a boundary condition, we use the Galerkin method for existence, Poincaré-type inequalities, with suitable adaptations to account for the differential geometry of the boundary, and a novel integral Gronwall-type inequality. In addition, in the frictionless case $α= 0$, we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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