非负 Ricci 曲率流形上的分数阶多孔介质方程：通过位势方法研究解的存在性与光滑化效应

中文摘要 AI 辅助

我们研究了在具有非负 Ricci 曲率的完备非紧黎曼流形上，对于 $s\in(0,1]$ 且 $n>2s$ 的分数阶多孔介质方程。假设流形是 $s$-非抛物型的，使得分数阶拉普拉斯算子具有一个合适的正极小 Green 函数，我们利用该 Green 函数引入一个自然的加权初值空间，该空间严格大于 $L^1$。这导致方程的弱对偶（或位势）形式，我们证明了在该加权空间中非负初值解的存在性。接着，我们建立了对于 $L^1$ 或 Green 加权空间中初值的定量局部光滑化估计。在额外的非塌缩和一致体积增长假设下，我们获得了全局光滑化估计。我们还表明，即使在 $\mathbb{R}^n$ 中，Green 加权空间中的数据也不一定产生有界解，除非施加适当的均匀加权可积性条件。我们的估计所预测的短时和长时行为在适当意义上被证明是最优的。当 $s\in(0,1/2)$ 时，结果仅需 $\operatorname{Ric}\geq0$，并在需要处加上非塌缩假设。我们的估计也涵盖了 $s=1$ 的情形，其中若干结果在该情形下是新的。最后，我们将该方法推广到更一般的渗流方程。

英文摘要

We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for $s\in(0,1]$ and $n>2s$. Assuming that the manifold is $s$-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than $L^1$. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either $L^1$ or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in $\mathbb{R}^n$, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When $s\in(0,1/2)$, the results require only $\operatorname{Ric}\geq0$, together with a noncollapsing assumption where needed. Our estimates also cover the case $s=1$, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.

发表机构

• Politecnico di Milano（米兰理工大学）

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