This paper deals with the Keller–Segel–Navier–Stokes model with indirect signal production in a three-dimensional (3D) bounded domain with smooth boundary. When the logistic-type degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the associated no-flux/no-flux/no-flux/Dirichlet problem possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in L1(Ω) × Lp(Ω) × L2(Ω) × L2(Ω; ℝ3) with any p ≥ 1. Moreover, under an explicit condition on the chemotactic sensitivity, these solutions are shown to stabilize toward the corresponding spatially homogeneous state in the sense of some suitable norms. We underline that the same results were established for the corresponding system with direct signal production in a well-known result if the degradation is quadratic. Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the 3D Keller–Segel–Navier–Stokes system.

中文翻译：

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具有间接信号产生的三维 Keller-Segel-Navier-Stokes 系统的全局可解性和渐近稳定性

本文处理具有平滑边界的三维 (3D) 有界域中的间接信号产生的 Keller-Segel-Navier-Stokes 模型。当这里的逻辑类型退化弱于通常的二次情况时，证明对于任何足够规则的初始数据，相关的无通量/无通量/无通量/狄利克雷问题至少具有一个全局定义的解一个适当的广义意义，并​​且这个解决方案是一致有界的大号1(Ω) × 大号p(Ω) × 大号2(Ω) × 大号2(Ω; ℝ3)与任何p ≥ 1. 此外，在趋化敏感性的明确条件下，这些解决方案显示出在一些合适的规范意义上稳定向相应的空间均匀状态。我们强调，如果退化是二次的，那么对于具有直接信号产生的相应系统也建立了相同的结果，这是众所周知的结果。我们的结果严格证实了间接信号产生机制真正有助于 3D Keller-Segel-Navier-Stokes 系统的全局可解性。