Shape derivative of sharp functionals governed by Navier-Stokes flow
Shape derivative of sharp functionals governed by Navier-Stokes flow
复制标题
纳维-斯托克斯流控制的锐泛函的形状导数
DOI：
10.1201/9780203744376-6
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发表时间：
2017
影响因子：
3.5
通讯作者：
Sébastien Boisgérault
中科院分区：
文献类型：
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作者：
Sébastien Boisgérault
纳维-斯托克斯方程的形状分析已在文献中考虑过。诸如隐函数定理之类的经典技术可用于表明某些泛函（例如阻力）是形状可微的。然而，该属性依赖于为压力场和速度场的基本规律建立的结果。许多其他物理兴趣标准不在此范围内：我们在这里考虑此类泛函的形状分析，例如，流体对物体施加的（总）力或这些力的力矩。假设速度场和压力场 u 和 p 是稳态不可压缩纳维-斯托克斯方程 −νΔu + [Du]u +∇p = f (Ω) 的解，边界条件为 u|Γ = 0, Γ = ∂Ω。这些新结果基于所谓的速度方法，该方法允许我们将矢量场从扰动域“恢复”到初始域，同时保留无散特性。为该对应关系建立正则性结果，并用于定义和显示形状导数 u 和边界形状导数 u Γ 的一些属性。 AMS 主题分类：35Q30、49J20。
The shape analysis of the Navier-Stokes equation has been already considered in the literature. Classical techniques, such as the Implicit Function Theorem, may be used to show that some functionals, the drag for example, are shape differentiable. However, this property relies on results established for the basic regularity of the pressure and the velocity fields. Many other criterions of physical interest are out of this scope: we consider here the shape analysis of such functionals, for example, the (total) force exerted by the fluid on a body or the moment of these forces. The velocity and pressure fields u and p are assumed to be solutions of the stationary incompressible Navier-Stokes equation −ν∆u + [Du]u +∇p = f in Ω with the boundary condition u|Γ = 0, Γ = ∂Ω. These new results are based on the so-called speed method which allows us to “bring back” vector fields from a perturbed domain to the initial one while preserving the divergence-free property. Regularity results are established for that correspondence and used to define and show some properties of the shape derivative u and of the boundary shape derivative u Γ . AMS subject classification: 35Q30, 49J20.