We study the large time behavior of Fujita–Kato type solutions to the Vlasov–Navier–Stokes system set on $$\mathbb {T}^3 \times \mathbb {R}^3$$ T 3 × R 3 . Under the assumption that the initial so-called modulated energy is small enough, we prove that the distribution function converges to a Dirac mass in velocity, with exponential rate. The proof is based on the fine structure of the system and on a bootstrap analysis allowing us to get global bounds on moments.

中文翻译：

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Vlasov-Navier-Stokes 系统在环面上的大时间行为

我们研究了在 $$\mathbb {T}^3 \times \mathbb {R}^3$$ T 3 × R 3 上设置的 Vlasov-Navier-Stokes 系统的 Fujita-Kato 型解的大时间行为。在初始所谓的调制能量足够小的假设下，我们证明分布函数以指数速率收敛到狄拉克速度质量。证明基于系统的精细结构和引导分析，使我们能够获得矩的全局界限。