Particle dynamics subject to impenetrable boundaries: existence and uniqueness of mild solutions
Particle dynamics subject to impenetrable boundaries: existence and uniqueness of mild solutions
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受不可逾越边界影响的粒子动力学：温和解的存在性和唯一性
DOI：
10.1137/18m1235922
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Z.X. Yang
中科院分区：
文献类型：
--
作者：
M. Kimura;P. van Meurs;Z.X. Yang
我们考虑相互作用粒子系统的动力学，其中粒子被限制在一个有界的，可能的非凸域。当粒子到达边界时，我们考虑速度的瞬间变化，这将描述粒子动力学的系统变成一个右侧不连续的ODE。作为对具有多值右手边的ode使用弱解来分析这类系统的典型方法（即应用Filippov在1988年引入的理论）的一种替代方法，我们定义并构造了温和解。温和解的优点是，这种解的唯一性很容易从Gronwall引理推导出来，解的某些性质很容易建立，它们为证明多粒子极限提供了一个方便的框架。我们补充了温和解理论，并应用于具有奇异相互作用势的相互作用粒子能量的梯度流，并通过对（非凸）域的各种选择的数值模拟来说明其特征。
We consider the dynamics of interacting particle systems where the particles are confined to a bounded, possibly nonconvex domain. When particles hit the boundary, we consider an instant change in velocity, which turns the system describing the particle dynamics into an ODE with a discontinuous right-hand side. As an alternative to the typical approach of analyzing such a system by using weak solutions to ODEs with multivalued right-hand sides (i.e., applying the theory introduced by Filippov in 1988), we define and construct mild solutions. The merit of mild solutions is that uniqueness of such solutions follows easily from Gronwall's lemma, that certain properties of the solutions are easy to establish, and that they provide a convenient framework for proving many-particle limits. We supplement our theory of mild solutions with an application to gradient flows of interacting particle energies with a singular interaction potential and illustrate its features by means of numerical simulations on various choices for the (nonconvex) domain.