In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in $\mathbf{R}^3$. We first obtain the higher order regularity estimates for the solutions to Prandtl's equation boundary layers. Furthermore, we prove that the strong solution to Navier-Stokes equations converges to the Eulerian one in $C([0,T];H^1(Ω))$ and $L^\infty((0,T)\timesø)$, where $T$ is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.
In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.
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 $\mathbb{R}^3_+$
The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface, despite the assumption of the no-slip condition is preferred to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the "static slip models", there are phenomena not accurately described by them, e.g. in the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. W…
We study the barotropic compressible Navier-Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with $C^{\infty}$ boundary $\partialΩ.$ By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Futhermore, we also prove that the oscillation of the density will grow unboundedly in the long run wit…
The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pip…
An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with N…
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
The aim of this work is to show the local null controllability of a fluid-solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier-Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid-rigid body system with Navier boundary conditions. This work i…
In this paper, we investigate the incompressible steady Navier-Stokes system with Navier slip boundary condition in a two-dimensional channel. As long as the width of cross-section of the channel grows more slowly than the linear growth, the existence of solutions with arbitrary flux is established. Furthermore, if the flux is suitably small, the solution is unique even when the width of the channel is unbounded, and approaches to the shear flows at far field where the channels tend to be straight at far fields. One of the major difficulties for the analysis on flows with Navier boundary condi…
We consider the initial-boundary-value problem of the isentropic compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation.
This paper concerns the global existence for classical solutions problem to the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in $Ω$ with slip boundary condition, where $Ω$ is a simply connected bounded $C^{\infty}$ domain in $\mathbb{R}^3$ and its boundary only has a finite number of 2-dimensional connected components. By a series of a priori estimates, we show that the classical solution to the system exists globally in time under the assumption that the initial energy is suitably small. The initial density of such a classical solution is allowed to have large oscillat…
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。
In some problems of fluid mechanics, it is possible to be confronted with data that are not regular, that is why we are interested here in the search for the so-called very weak solutions for the stationary Stokes problem with Navier-type boundary conditions in a three-dimensional exterior domain. The problem describes the flow of a viscous and incompressible fluid past an obstacle where we assume that the fluid may slip on the boundary of the obstacle. Because the flow domain is unbounded, we set the problem in weighted Sobolev spaces in order to control the behavior at infinity of the soluti…
We consider the two-phase flow model with slip boundary condition in a 3D exterior domains whose boundary is smooth. We establish the global existence of classical solutions of this system provided that the initial energy is suitably small. Moreover, the pressure has large oscillations and contains vacuum states when the initial pressure allows large oscillations and a vacuum. Finally, we also obtain the large-time behavior of the classical solutions.
