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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.04660 |
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| _version_ | 1866914360819974144 |
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| author | Zhang, Wanwan |
| author_facet | Zhang, Wanwan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite time blow-up for a multi-dimensional model of the Kiselev-Sarsam equation Zhang, Wanwan Analysis of PDEs 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. |
| title | Finite time blow-up for a multi-dimensional model of the Kiselev-Sarsam equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.04660 |