Remarks on a multi-dimensional transport equation with nonlocal velocity

In this paper, we revisit a multi-dimensional nonlocal transport equation of the form \begin{eqnarray*} \partial_tρ+g$\mathcal{R}$aρ\cdot \nablaρ= 0,~ρ(\cdot,0)=ρ{0}, \end{eqnarray*} where the transform ${\mathcal{R}}_a$ is defined by \begin{eqnarray*} $\mathcal{R}$aρ(x,t)=\frac{Γ(\frac{n+1}{2})}{π^{\frac{n+1}{2}}}P.V.\int\limits{$\mathbb{R}$^n}\Big(\frac{x-y}{|x-y|^{n+1}}-\frac{x-y}{(|x-y|^2+a^2)^{\frac{n+1}{2}}}\Big)ρ(y,t)dy. \end{eqnarray*} This model was first introduced in [W. Zhang, J. Differential Equations 467 (2026) 114275] as a natural generalization of the well-known Kiselev-Sarsam equation proposed in [A. Kiselve and N.A. Sarsam, Nonlinearity 38 (2025) 055018]. By using an improved pointwise inequality for the transform ${\mathcal{R}}_a$, we obtain a better and more natural version of a nonlinear integral inequality proved in Zhang [31] for the transform ${\mathcal{R}}_a$. In addition, we also establish a new weighted nonlinear inequality for the transform ${\mathcal{R}}_a$ implying the finite-time blow-up for a class of smooth radial initial data.