We study the Cauchy problem and the mixed initial boundary value problem of a fluid-particle interaction system in \(\mathbb{R}^{3}\). A Serrin type criterion for the strong solution of the Cauchy problem is established in terms of \(\|\rho \|_{L^{\infty }_{t}L^{\infty }_{x}}\) and \(\|u\|_{L^{s}_{t}L^{r}_{x}}\), where \(2/s+3/r\le 1\) and \(3< r\le \infty \). In view of some useful integral inequalities, we prove the life span estimates of the regular solution.

中文翻译：

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R3 $ \ mathbb {R} ^ {3} $中流体-颗粒相互作用系统的爆破机理

我们研究了\（\ mathbb {R} ^ {3} \）中流体相互作用系统的柯西问题和混合初始边值问题。根据\（\ | \ rho \ | _ {L ^ {\ infty} _ {t} L ^ {\ infty} _ {x}} \）建立了柯西问题强解的Serrin类型准则。和\（\ | u \ | _ {L ^ {s} _ {t} L ^ {r} _ {x}} \），其中\（2 / s + 3 / r \ le 1 \）和\（ 3 <r \ le \ infty \）。鉴于一些有用的积分不等式，我们证明了正规解的寿命估计。