Blowup Mechanism for a Fluid-Particle Interaction System in $\mathbb{R}^{3}$

Abstract

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.