论文标题
$α$ conccavity的多孔介质方程式
Non-preservation of $α$-concavity for the porous medium equation
论文作者
论文摘要
我们表明，多孔介质方程通常不会保留$ 0 \ leleα<1/2 $或$ 1/2 <α\ le 1 $的压力的$α$ conccavity。特别是，这解决了Vázquez的开放问题，即是否由多孔培养基方程保留了压力的凹陷。我们的结果加强了Ishige-Salani的早期工作，后者考虑了小$α> 0 $的情况。由于Daskalopoulos-Hamilton-Lee表明保留了$ 1/2 $ - conceve富的票房，因此我们的结果很敏锐。 我们的明确示例表明，在初始数据支持的内部点可以立即断裂凹度。对于$ 0 \leα<1/2 $，我们给出另一组示例，以表明凹面可能会在边界点打破。
We show that the porous medium equation does not in general preserve $α$-concavity of the pressure for $0\leα<1/2$ or $1/2<α\le 1$. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small $α>0$. Since Daskalopoulos-Hamilton-Lee showed that $1/2$-concavity is preserved, our result is sharp. Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For $0\leα<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.
