On a 1D nonlocal transport of the incompressible porous media equation
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| Format: | Preprint |
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2025
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| author | Liu, Caifeng Zhang, Wanwan |
| author_facet | Liu, Caifeng Zhang, Wanwan |
| contents | 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_tρ+u\partial_xρ= 0,~u=gH_aρ, \end{eqnarray*} where the transform $H_a$ is defined by \begin{eqnarray*} H_af(x)=\frac{1}πP.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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a 1D nonlocal transport of the incompressible porous media equation Liu, Caifeng Zhang, Wanwan Analysis of PDEs 35Q35, 76S05, 76B03 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_tρ+u\partial_xρ= 0,~u=gH_aρ, \end{eqnarray*} where the transform $H_a$ is defined by \begin{eqnarray*} H_af(x)=\frac{1}πP.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. |
| title | On a 1D nonlocal transport of the incompressible porous media equation |
| topic | Analysis of PDEs 35Q35, 76S05, 76B03 |
| url | https://arxiv.org/abs/2503.16238 |