关于泊松介质方程拉格朗日粒子离散化的Aronson-Bénilan估计
The Aronson-Bénilan estimate for a Lagrangian particle discretization of the Porous Medium Equation
AI总结：
本文研究了一维泊松介质方程的拉格朗日粒子离散化方法，证明了其满足Aronson-Bénilan估计并证明了收敛性。
摘要：
我们考虑了一维泊松介质方程的最近邻拉格朗日粒子离散化方法。我们证明该粒子模型满足一个著名的Aronson-Bénilan估计的离散形式，利用该估计证明了支撑集演化的增长估计和一个已知在连续情况下成立的$ L^\infty $衰减估计，且这些估计对粒子数是均匀的。我们还证明了该方案在$ L^1 $初始数据的全一般性下向泊松介质方程解的收敛性。
英文摘要：
We consider a nearest neighbor, Lagrangian particle discretization of the one dimensional porous medium equation. We prove that the particle model satisfies a discrete analog of the celebrated Aronson-Bénilan estimate, which we use to prove a growth estimate for the evolution of the support and an $L^\infty$ decay estimate which are both known to hold in the continuum. These estimates are uniform with respect to the number of particles. We also prove convergence of the scheme towards the solution to the porous medium equation in the full generality of $L^1$ initial data.
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