Large Time Behavior for the 3D Navier-Stokes with Navier Boundary Conditions

IF 1.3 3区 数学 Q2 MATHEMATICS, APPLIED
James P. Kelliher, Christophe Lacave, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, Edriss S. Titi
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引用次数: 0

Abstract

We study the three-dimensional incompressible Navier-Stokes equations in a smooth bounded domain \(\Omega \) with initial velocity \(u_0\) square-integrable, divergence-free and tangent to \(\partial \Omega \). We supplement the equations with the Navier friction boundary conditions \(u \cdot {\varvec{n}}= 0\) and \([(2Su){\varvec{n}}+ \alpha u]_{{\scriptstyle {\textrm{tan}}}} = 0\), where \({\varvec{n}}\) is the unit exterior normal to \(\partial \Omega \), \(Su = (Du + (Du)^t)/2\), \(\alpha \in C^0(\partial \Omega )\) is the boundary friction coefficient and \([\cdot ]_{{\scriptstyle {\textrm{tan}}}}\) is the projection of its argument onto the tangent space of \(\partial \Omega \). 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 \(\alpha = 0\), we prove convergence of the solution to a steady rigid rotation, if the domain is a solid of revolution.

具有Navier边界条件的三维Navier- stokes大时间行为
我们研究了光滑有界区域\(\Omega \)上三维不可压缩的Navier-Stokes方程,其初始速度为\(u_0\)平方可积,无散度且与\(\partial \Omega \)相切。我们用纳维摩擦边界条件\(u \cdot {\varvec{n}}= 0\)和\([(2Su){\varvec{n}}+ \alpha u]_{{\scriptstyle {\textrm{tan}}}} = 0\)补充方程,其中\({\varvec{n}}\)是\(\partial \Omega \)的单位外法线,\(Su = (Du + (Du)^t)/2\), \(\alpha \in C^0(\partial \Omega )\)是边界摩擦系数,\([\cdot ]_{{\scriptstyle {\textrm{tan}}}}\)是其辐角在\(\partial \Omega \)的切空间上的投影。我们证明了所得到的初边值问题的一个弱leray型解的整体存在性以及当摩擦为正时这些解的能量模的指数衰减。我们还证明了摩擦非负且定义域不是旋转固体时的指数衰减。这两个结果在Dirichlet边界条件下是众所周知的,但是,即使它们已经隐式地用于Navier边界条件,在文献中也没有全面的分析。在仔细研究了这种边界条件下的Stokes半群之后,我们使用了Galerkin存在法、poincar型不等式(适当地适应了边界的微分几何)和一种新的积分gronwall型不等式。此外,在无摩擦情况下\(\alpha = 0\),我们证明了当定义域是旋转固体时,解的收敛性。
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来源期刊
CiteScore
2.00
自引率
15.40%
发文量
97
审稿时长
>12 weeks
期刊介绍: The Journal of Mathematical Fluid Mechanics (JMFM)is a forum for the publication of high-quality peer-reviewed papers on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. As an important part of that, the journal encourages papers dealing with mathematical aspects of computational theory, as well as with applications in science and engineering. The journal also publishes in related areas of mathematics that have a direct bearing on the mathematical theory of fluid mechanics. All papers will be characterized by originality and mathematical rigor. For a paper to be accepted, it is not enough that it contains original results. In fact, results should be highly relevant to the mathematical theory of fluid mechanics, and meet a wide readership.
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