论文标题
1D随机多孔介质方程和传输噪声的有限时间灭绝
Finite time extinction for the 1D stochastic porous medium equation with transport noise
论文作者
论文摘要
我们建立有限的时间灭绝，概率为cauchy-dirichlet问题的弱解，用于1D随机多孔培养基方程，并具有Stratonovich传输噪声和紧凑的平滑初始基准。启发式上，这是有望坚持的，因为布朗运动具有平均利率$ o $ o（t^\ frac {1} {2}）$，而解决方案对确定性PME的支持仅以速率$ o（t^{\ frac {\ frac {1} {1} {m {+}}}）$。严格的证据依靠收缩原理来依赖于时间依赖的王子近似，将布朗途径的两个副本作为侧向边界的两个副本转换为确定性的PME，以及粘度解决方案理论的技术。
We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^\frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{\frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.
