AI 中文总结
本文研究带非负源项的Hele-Shaw流动自由边界的时空几何，证明其扩张速率、自由边界类型及正则性等性质，相关结果适用于肿瘤生长模型，奇点分析为该领域新进展。
AI 中文摘要
我们研究带有非负源项的Hele-Shaw流动的自由边界的动态行为，其中压力在{p>0}内满足-Δp=c，且自由边界以法向速度V=|∇p|移动。对于满足内部球条件的初始区域，我们证明了区域{p>0}以局部正速率扩张，因此其撞击时间是局部Lipschitz的；这是最优的，因为该区域的不同部分可能会发生碰撞。我们还表明，自由边界在局部要么是正则的，要么是碰撞型的，要么经历不同维度的抛物尺度孔洞闭合：这种奇点的时空表征在Hele-Shaw流动中是全新的。我们进一步证明，在任何未发生碰撞的时刻附近，时空自由边界是C^1超曲面。这意味着在这些时刻，每个自由边界点都有时空法向：时空单位法向与时间方向平行的位置，恰好是空间自由边界具有奇异几何的位置。这些结果尤其适用于Perthame、Quirós和Vázquez提出的带有营养物的肿瘤生长模型，即使对于经典注入问题，奇点分析也是全新的。
英文摘要
We study the dynamic behavior of the free boundary of the Hele-Shaw flow with a nonnegative source, in which the pressure satisfies $-Δp=c$ in $\{p>0\}$ and the free boundary moves with normal velocity $V=|\nabla p|$. For initial domains satisfying an interior ball condition, we prove that the patch $\{p>0\}$ expands at a locally positive rate, so that its hitting time is locally Lipschitz; this is optimal, since distinct portions of the patch may collide. We also show that the free boundary is locally either regular, or of collision type, or undergoing parabolic-scale hole closings of various dimensions: such a space-time characterization of singularities is new in Hele-Shaw flow. We further show that, near any time at which no collision occurs, the space-time free boundary is a $C^1$ hypersurface. This implies that at such times every free boundary point has a space-time normal: the space-time unit normal is parallel to the time direction precisely where the spatial free boundary has singular geometry. The results apply in particular to the tumor growth model with nutrients introduced by Perthame, Quirós, and Vázquez, but the analysis of singularities is new even for the classical injection problem.
