On a 1D nonlocal transport of the incompressible porous media equation

Abstract

Recently, Kiselev and Sarsam proposed the following nonlocal transport equation as a one-dimensional analogue of the 2D incompressible porous media (IPM) equation \begin{eqnarray*} \partial_t\rho+u\partial_x\rho= 0,~u=gH_a\rho, \end{eqnarray*} where the transform $H_a$ is defined by \begin{eqnarray*} H_af(x)=\frac{1}{\pi}P.V.\int\limits_{\mathbb{R}}\frac{a^2f(y)}{(x-y)((x-y)^2+a^2)}dy. \end{eqnarray*} In the work Kiselev-Sarsam (2025) [14], the authors proved the local well-posedness for this 1D periodic IPM model as well as finite time blow-up for a class of smooth initial data. In this paper, we present several new weighted inequalities for the transform $H_a$ in the setting of the real line. Based on these integral inequalities, we also prove the finite time blow-up for this 1D IPM model on the real line.