Partially dissipative systems with applications to fluid-solid interaction problems

粘性不可压缩流体与(刚性或弹性)固体的相互作用在工程、地球物理和生物应用中普遍存在。我们研究计划的长期目标是为系统地研究流固相互作用系统的稳定性和长期行为提供新的分析技术。在这个提案中，我们将集中讨论几个多物理流体-刚体相互作用问题，这些问题可以在数学上描述为“部分耗散”的微分方程组。如果满足以下两个性质，则称一个非线性发展方程是部分耗散的：(P1)存在一个沿轨道不增加的能量泛函；(P2)存在一个始终守恒的矢量场(取决于解)。短期目标是描述三个相关的流体-刚体相互作用问题的一大类解决方案的长期行为。控制方程是部分耗散的，其特点是耦合了Navier-Stokes方程和存在磁场和/或热传递的固体的动量平衡。以下主题将建立一个激励性的研究计划，用于培训3名博士、2名硕士和2名NSERC USRA学生：(1)具有流体填充间隙的固体刚体运动的渐近稳定性。考虑一个由两个固体组成的系统，其中一个包含在另一个内部，它们之间的空隙完全由粘性不可压缩流体填充。我们将研究永久旋转(即系统作为一个整体以恒定角速度旋转所给出的稳态)的渐近稳定性。(2)充液刚体在磁场中的长期行为。我们考虑一个具有完全由导电流体填充的腔的刚体。我们将研究流固耦合系统的运动是否能够在系统内产生并维持磁场。(3)具有流体缝隙的刚体在温度梯度和重力场共同作用下的长时间动力学。目的是确定固体旋转是否在很长时间内抑制流体对流运动的形成。上面提出的每个问题的特点是，在保持保守成分的同时，具有不同的耗散来源。从数学角度看，这些问题是满足(P1)和(P2)的抽象非线性发展方程的例子。现有的关于解的稳定性和长期行为的理论主要适用于强函数环境中的解。其深远的影响是为部分耗散系统的弱解的长期行为提供了一个新的理论。问题(1)-(3)除了它们的数学意义外，还提供了在地球物理和工程中应用的唯象模型。

The interactions of viscous incompressible fluids with (rigid or elastic) solids are ubiquitous in engineering, geophysical and biological applications. The long-term goal of our research program is to provide new analytical techniques for a systematic investigation of stability and long-time behaviour of fluid-solids interacting systems. In this proposal, we will focus on several multiphysics fluid-rigid body interaction problems that can be mathematically cast as "partially dissipative" systems of differential equations. A nonlinear evolution equation is said to be partially dissipative if the following two properties are satisfied: (P1) there exists an energy functional that is non-increasing along the trajectories; and (P2) there exists a vector field (depending on the solution) that is conserved at all times. The short-term objective is to characterize the long-time behaviour of a large class of solutions to three related fluid-rigid body interaction problems. The governing equations are partially dissipative and feature the coupling of the Navier-Stokes equations with the balance of momentums for the solids in presence of magnetic fields and/or heat transfer. The following topics will establish a stimulating research programme for the training of 3 PhD, 2 Master's and 2 NSERC USRA students: (1) Asymptotic stability of rigid body motions for solids with a fluid-filled gap. Consider a system of two solids, one contained inside the other, and with the gap between them completely filled by a viscous incompressible fluid. We will study the asymptotic stability of permanent rotations (i.e., steady states given by the system rotating as a whole rigid body with constant angular velocity). (2) Long-time behaviour of fluid-filled rigid bodies in a magnetic field. We consider a rigid body with a cavity completely filled by an electrically conducting fluid. We will investigate whether motions of the coupled fluid-solid system are able to generate and sustain a magnetic field within the system. (3) Long-time dynamics of rigid bodies with a fluid-filled gap subject to both temperature gradients and a gravitational field. The objective is to determine whether solid rotations inhibit the formation of fluid convecting motions at large times. Every problem proposed above is characterized by having different sources of dissipation while keeping a conservative component. From a mathematical point of view, these problems are examples of abstract nonlinear evolution equations satisfying (P1) and (P2). Existing theories concerning stability and long-time behaviour of solutions are applicable mainly to solutions in a strong functional setting. The far-reaching impact is to provide a novel theory for the long-time behaviour of weak solutions to partially dissipative systems. Besides their mathematical significance, problems (1)-(3) provide phenomenological models for applications in geophysics and engineering.