We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e.

$$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$

As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

中文翻译：

---

多孔介质系统的 Hölder 正则性

我们为某些多孔介质类型的抛物线系统的局部有界弱解建立 Hölder 连续性，即

$$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m> 0. \end{对齐}$$

作为我们局部 Hölder 估计的结果，还建立了有界全局解的 Liouville 型结果。此外，还给出了充分条件来保证局部弱解的局部有界性。