关于刚体与粘性流体相互作用的交互：弱解的存在性及结构定理的证明
On the interaction between a rigid-body and a viscous-fluid: existence of a weak solution and a suitable Théorème de Structure
AI总结：
本文证明了刚体与粘性流体相互作用系统弱解的存在性及部分正则性，采用新方法证明了Leray结构定理，但正则性仅适用于大时间。
摘要：
在本文中，我们证明了描述刚体与粘性不可压缩牛顿流体相互作用系统中弱解的存在性及部分正则性。系统体-流体的演化研究是在与体固连的参考系中进行的。由于流体动量平衡方程中存在$ω×x·∇u$项，这种特殊参考系的选择在分析上变得至关重要。因此，我们被迫寻找一种不同于通常用于Navier-Stokes问题弱解存在性及部分正则性证明的技术。因此，我们以原创的方式证明了弱解的存在性，并给出了Leray著名的结构定理的新证明。然而，我们所获得的弱解的正则性仅适用于大时间，因此我们的结果相比Leray所获得的结果要弱。
英文摘要：
In this paper, we prove the existence and a partial regularity of a weak solution to the system governing the interaction between a rigid body and a viscous incompressible Newtonian fluid. The evolution of the system body-fluid is studied in a frame attached to the body. The choice of this special frame becomes critical from an analytical point of view due to the presence of the term $ω\times x\cdot\nabla u$ in the balance of momentum equation for the fluid. As a consequence, we are forced to look for a technique that is different from the ones usually employed both for the existence and for the partial regularity of a weak solution to the Navier-Stokes problem. Hence, we prove the existence of a weak solution in an original way and give a new proof of the celebrated Théorème de Structure due to Leray. However, the regularity obtained for our weak solution is only for large times, hence our result is weaker compared to the one obtained by Leray.
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