We consider the flow of an { ideal} fluid in a 2D-bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with \textit{non-homogeneous } Navier slip boundary conditions. These conditions can be written in the form $\mathbf{v}\cdot \mathsf{n}=a,$ $2D(\mathbf{v})\mathsf{n}\cdot \mathsf{s}+α\mathbf{v}% \cdot {\mathsf{s}}=b,$ where the tensor $D(\mathbf{v})$ is the rate-of-strain of the fluid's velocity $\mathbf{v}$ and\ $\mathsf{(n},% \mathsf{s)}$ is the pair formed by the normal and tangent vectors to the boundary. We establish the solvability of this problem in the class of solutions with $L_{p}-$bounded \ vorticity, $p\in (2,\infty ].$ To prove the solvability we realize the passage to the limit in Navier-Stokes equations with vanishing viscosity.
The inviscid limit of Navier-Stokes with critical Navier-slip boundary conditions for analytic data
本文建立了半空间上解析数据的不可压Navier-斯托克斯方程的短时无粘边界条件，该边界条件是由Boltzmann方程的流体力学极限和Maxwell边界条件物理推导出来的。分析是建立在最近的框架发展的T。T. Nguyen和T. T. Nguyen（Arch. Ration.机械分析：230（3）：1103-1129，2018.）在经典的无滑移边界条件的情况下。本文的新颖之处在于推导出了具有非局部边界条件的Stokes问题的格林核的精确逐点界，并传播了涡量的边界层行为。
Existence, regularity and uniqueness of weak solutions for a class of incompressible generalized Navier-Stokes system with slip boundary conditions in $\mathbb{R}^3_+$
Global Solutions to an initial boundary problem for the compressible 3-D MHD equations with Navier-slip and perfectly conducting boundary conditions in exterior domains
Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains
On a Navier-Stokes-Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions
Global Classical Solutions to the 3D Density-Dependent Viscosity Compressible Navier-Stokes Equations with Navier-Slip Boundary Condition in a Simply Connected Bounded Domain
Regularity Criteria for Navier-Stokes Equations with Slip Boundary Conditions on Non-flat Boundaries via Two Velocity Components
H.- O. Bae和H. J. Choe在1997年的论文中建立了基于两个速度分量的全空间$\R^3$中不可压Navier-斯托克斯方程的正则性准则。最近，本文作者之一将这一结果推广到半空间情况$\R^3_+\,.$。此外，本文作者与J. Bemelmans和J. Brand合作，将结果推广到物理滑移边界条件下的圆柱形区域。本文在光滑任意边界的情况下得到了类似的结果，但在一个不同的，显然非常相似的，滑移边界条件。它们只在边界的平坦部分重合。否则，两个结果之间的倒数减少看起来不明显，如下面最后一节所示。
Global stability for compressible isentropic Navier-Stokes equations in 3D bounded domains with Navier-slip boundary conditions
School of Mathematics and Statistics, Xiamen University of Technology（厦门理工学院数学与统计学院）；School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Department of Mathematics, CY Cergy Paris University, CNRS UMR 8088（CY塞吉-巴黎大学数学系，法国国家科学研究中心联合研究单位8088）
On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case
本文讨论了发展Navier-斯托克斯方程解的正则性的充分条件，文献中称为Prodi-Serrin条件。H.- O. Bae和H J. Choe在1999年的一篇论文中证明，在整个空间R^3中，速度的两个分量满足上述条件就足够了。下面，我们将结果推广到滑移边界条件下的半空间情形R^n_+。我们证明了平行于边界的速度分量满足上述条件是足够的。平边界几何形状似乎不是必要的，在圆柱形域的一些初步计算建议。
Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；BCAM, Basque Center for Applied Mathematics（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊卡巴斯科，巴斯克科学基金会）
Small-time global approximate controllability for incompressible MHD with coupled Navier slip boundary conditions
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Mathematical Sciences, Center for Applied Mathematics, MOE-LSC, CMA-Shanghai and SHL-MAC, Shanghai Jiao Tong University（上海交通大学数学科学学院，应用数学中心，教育部拉曼ujan数学科学中心，上海高等应用数学实验室）
Characterization of smooth solutions to the Navier-Stokes equations in a pipe with two types of slip boundary conditions
本文研究了具有Navier-Slip或Navier-Hodge-Lions边界条件的无限长管道中定常Navier-斯托克斯方程的光滑解。三个主要结果。首先，当配备Navier滑移边界条件，它表明，$W^{1,\infty}$轴对称的解决方案，零通量在一个横截面上，必须是涡旋的解决方案：$u=(- C x_2, C x_1,0)$，和$x_3-$周期的解决方案必须是螺旋的解决方案：$u=(-C_1x_2,C_1x_1,C_2)$。第二，同样采用Navier滑移边界条件，如果轴对称解的旋涡分量或垂直分量与垂直变量$x_3$无关，则解也是螺旋解。在垂直分量独立于$x_3$的情况下，不需要$W^{1,\infty}$假设。在涡流分量与$x_3$无关的情况下，$W^{1,\infty}$假设可以大大放宽，使得速度的水平径向分量$u_r$可以相对于到原点的距离呈指数增长。通过构造一个反例，证明了关于$u_r$的增长假设是最优的.第三，当满足Navier-Hodge-Lions边界条件时，我们可以证明，如果速度梯度次线性增长，则满足刘维定理的解是平凡剪切流：$(0,0,C)$。
