AI 中文总结
研究粘性流体中刚体稳态自推进运动，在体固定系用耦合系统描述。通过规定物体表面边界速度产生自推进，在边界通量足够小假设下证明弱解存在，推广了加尔迪结果，关键是构造不依赖零通量等条件的无散度扩展。
AI 中文摘要
我们研究了沉浸在占据\(\mathbb{R}^3\)中外区域的不可压缩粘性流体中的刚体的稳态自推进运动。在体固定参考系中，该问题由在固定外区域中提出的耦合流体 - 刚体系统描述，其中纳维 - 斯托克斯方程与刚体未知的平移和角速度耦合。通过规定物体表面的边界速度来产生自推进。我们的主要结果断言，在规定的边界通量足够小的唯一假设下，存在弱解。关键步骤是构造边界数据的无散度扩展，该扩展不依赖于零通量条件或边界数据的小性。因此，我们推广了加尔迪的结果[《粘性不可压缩流体中物体的稳态自推进运动》中的定理5.1。《理性力学与分析档案》，148 (1999)，53 - 88]，其中需要这两个假设。
英文摘要
We study the steady self-propelled motion of a rigid body immersed in an incompressible viscous fluid occupying an exterior domain in $\mathbb{R}^3$. In a body-fixed reference frame, the problem is described by a coupled fluid-rigid body system posed in a fixed exterior domain, where the Navier--Stokes equations are coupled with the unknown translational and angular velocities of the rigid body. The self-propulsion is generated by prescribing a boundary velocity on the body surface. Our main result asserts the existence of weak solutions under the sole assumption that the prescribed boundary flux is sufficiently small. The crucial step is the construction of a divergence-free extension of the boundary data that does not rely on either the zero-flux condition or the smallness of the boundary data. As a result, we generalize the result of Galdi [Theorem 5.1 in On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rational Mech. Anal., 148 (1999), 53-88], where both assumptions were required.
