We study kinetic theories for isotropic, two-dimensional grain boundary networks which evolve by curvature flow. The number densities $$f_s(x,t)$$ f s ( x , t ) for s -sided grains, $$s =1,2,\ldots $$ s = 1 , 2 , … , of area x at time t , are modeled by kinetic equations of the form $$\partial _t f_s + v_s \partial _x f_s =j_s$$ ∂ t f s + v s ∂ x f s = j s . The velocity $$v_s$$ v s is given by the Mullins–von Neumann rule and the flux $$j_s$$ j s is determined by the topological transitions caused by the vanishing of grains and their edges. The foundations of such kinetic models are examined through simpler particle models for the evolution of grain size, as well as purely topological models for the evolution of trivalent maps. These models are used to characterize the parameter space for the flux $$j_s$$ j s . Several kinetic models in the literature, as well as a new kinetic model, are simulated and compared with direct numerical simulations of mean curvature flow on a network. The existence and uniqueness of mild solutions to the kinetic equations with continuous initial data is established.

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二维晶界网络：随机粒子模型和动力学极限

我们研究了由曲率流动演变的各向同性二维晶界网络的动力学理论。s 面晶粒的数密度 $$f_s(x,t)$fs ( x , t )，$$s =1,2,\ldots $$ s = 1 , 2 , ... , 在时间区域 x t ，由 $$\partial _t f_s + v_s \partial _x f_s =j_s$$ ∂ tfs + vs ∂ xfs = js 形式的动力学方程建模。速度 $$v_s$$ vs 由穆林斯-冯诺依曼规则给出，通量 $$j_s$$ js 由晶粒及其边缘消失引起的拓扑转变决定。这些动力学模型的基础通过更简单的颗粒尺寸演化粒子模型以及三价映射演化的纯拓扑模型进行了检查。这些模型用于表征通量 $$j_s$$ js 的参数空间。对文献中的几个动力学模型以及新的动力学模型进行了模拟，并与网络上平均曲率流的直接数值模拟进行了比较。建立了具有连续初始数据的动力学方程的温和解的存在唯一性。