Large time behaviour for the motion of a solid in a viscous incompressible fluid
Large time behaviour for the motion of a solid in a viscous incompressible fluid
复制标题
粘性不可压缩流体中固体运动的大时间行为
作者：
S. Ervedoza;D. Maity;M. Tucsnak
在本文中，我们研究了描述刚体和包含刚体的粘性不可压缩流体耦合运动的系统的长期行为。我们假设刚体和流体构成的系统充满了整个空间$${mathbb {R}}^3$$ R 3。在刚体为球的情况下，我们证明了温和解的局部存在性，当初始数据很小时，我们证明了该系统解的全局存在性，并精确描述了它们的大时间行为。我们的主要结果特别断言，如果初始基准在合适的范数下足够小，那么当时间趋于无穷时，刚性球的中心位置收敛于某个$$h_infty in {mathbb {R}}^3$$ h∞∈R 3。这一结果与我们已知的2维或1维空间系统的类似结果形成对比，在那里，如果我们等待足够长的时间，它已经证明了物体退出任何有界集合。为了达到这个结果，我们使用了一个“整体”类型的方法，这意味着我们考虑一个线性化的问题，其中固体和流体的方程仍然是耦合的。与此耦合线性化问题相关的称为流固半群的半群的性质起着重要作用。这个半群的产生器称为流固算符。我们的主要工具是新的$$L^p - L^q$$流固半群的L p - L q估计。请注意，这些估计是对任意形状的物体证明的。用于研究流固半群及其产生器的主要成分是分解估计，它提供了流固半群的可分析性（在Borchers和Sohr的经典著作的精神中）和$$L^p- L^q$$ L p - L q衰变估计（通过采用Iwashita的策略）。
In this article, we study the long-time behaviour of a system describing the coupled motion of a rigid body and of a viscous incompressible fluid in which the rigid body is contained. We assume that the system formed by the rigid body and the fluid fills the entire space $${mathbb {R}}^3$$ R 3 . In the case in which the rigid body is a ball, we prove the local existence of mild solutions and, when the initial data are small, the global existence of solutions for this system with a precise description of their large time behavior. Our main result asserts, in particular, that if the initial datum is small enough in suitable norms then the position of the center of the rigid ball converges to some $$h_infty in {mathbb {R}}^3$$ h ∞ ∈ R 3 as time goes to infinity. This result contrasts with those known for the analogues of our system in 2 or 1 space dimensions, where it has been proved that the body quits any bounded set, provided that we wait long enough. To achieve this result, we use a “monolithic” type approach, which means that we consider a linearized problem in which the equations of the solid and of the fluid are still coupled. An essential role is played by the properties of the semigroup, called fluid-structure semigroup , associated to this coupled linearized problem. The generator of this semigroup is called the fluid-structure operator . Our main tools are new $$L^p - L^q$$ L p - L q estimates for the fluid-structure semigroup. Note that these estimates are proved for bodies of arbitrary shape. The main ingredients used to study the fluid-structure semigroup and its generator are resolvent estimates which provide both the analyticity of the fluid-structure semigroup (in the spirit of a classical work of Borchers and Sohr) and $$L^p- L^q$$ L p - L q decay estimates (by adapting a strategy due to Iwashita).
DOI：
10.1090/s0002-9947-2012-05652-2
发表时间：
2012-08
影响因子：
1.3
作者：
M. Geissert;K. Götze;Matthias Hieber
通讯作者：
M. Geissert;K. Götze;Matthias Hieber