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_t\rho+g\mathcal{R}_a\rho\cdot \nabla\rho= 0,~\rho(\cdot,0)=\rho_{0}, \end{eqnarray*} where the transform $\mathcal{R}_a$ is defined by \begin{eqnarray*} \mathcal{R}_af(x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\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.