Finite time blow-up for a multi-dimensional model of the Kiselev-Sarsam equation

In this paper, we propose and study a multi-dimensional nonlocal active scalar 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*} $\mathcal{R}$af(x)=\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)f(y)dy. \end{eqnarray*} This model can be viewed as a natural generalization of the well-known Kiselev-Sasarm equation, which was introduced in [19] as a one-dimensional model for the two-dimensional incompressible porous media equation. We show the local well-posedness for this multi-dimensional model as well as the gradient blow-up in finite time for a class of radial initial data.