一类退化微观动力学方程解的收敛性
Convergence of a degenerate microscopic dynamics to the porous medium equation
摘要：
我们从一个相互作用的粒子系统导出了多孔介质方程，该系统属于排斥过程族，具有最近邻交换。粒子遵循简并动力学，在这个意义上，跳跃率可以为零的某些配置，并存在阻塞的配置，不能演变。在[Gonçalves-Landim-Toninelli '09]中，证明了流体动力极限中的宏观密度分布受多孔介质方程（PME）控制，初始密度均匀地远离$0$和$1$。在本文中，我们考虑更一般的情况下，密度可以采取这些极值。在这种情况下，PME解决方案显示出更丰富的行为，如移动界面，有限的传播速度和规律性的破坏。因此，通常用于证明这种流体动力学极限的标准技术不能直接应用于我们的情况。我们在这里提出了一种推广\emph{relative entropy method}的方法，通过涉及流体动力学方程的近似解，而不是精确解。
英文摘要：
We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [Gonçalves-Landim-Toninelli '09] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from $0$ and $1$. In this paper we consider the more general case where the density can take those extreme values. In this context, the PME solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. As a consequence, the standard techniques that are commonly used to prove this hydrodynamic limits cannot be straightforwardly applied to our case. We present here a way to generalize the \emph{relative entropy method}, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.
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