We consider a fractional chemotaxis-Navier-Stokes model in the whole space $\mathbb{R}^N , N \geq 2$, with a time-fractional variation in the Caputo sense, a fractional self-diffusion for the physical variables and a fractional dissipation mechanism for the chemoattraction process. We prove the existence and uniqueness of global mild solutions with small initial data in a larger class of critical spaces of Besov–Morrey type. Our result extend the well-posedness ones in the classical (no fractional regime) obtained by Postigo and Ferreira [16]. We also prove the long-time asymptotic stability of solutions.

中文翻译：

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分数趋化性-Navier-Stokes 系统解的全局存在性和渐近行为

我们考虑整个空间 $\mathbb{R}^N , N \geq 2$ 中的分数趋化性-Navier-Stokes 模型，在 Caputo 意义上具有时间分数变化，物理变量的分数自扩散和趋化过程的分数耗散机制。我们证明了在更大类的 Besov-Morrey 类型临界空间中具有小初始数据的全局温和解的存在性和唯一性。我们的结果扩展了 Postigo 和 Ferreira [16]获得的经典（无分数体系）中的适定性。我们还证明了解的长期渐近稳定性。