Asymptotic Behavior And Well-Posedness Of Solutions To Two Types Of Equations Via Particle Methods

In recent years,particle methods have become one of the popular tools for solving partial differential equations,especially suitable for linear transmission equations,Vlasov type equations,dispersion equations,and so on.The main idea of the particle method is to establish the mean field limit theory between the partial differential equation and its corresponding particle system,i.e.to use the particle system solution to approximate the solution of the target partial differential equation.The characteristic flow method for solving transport partial differential equations is one important type of particle methods.The Cucker-Smale model is proposed to model the flocking behavior of self-driven particle systems,and their kinetic equations are Vlasov-type partial differential equations.This model can reflect the phenomenons in nature,like flocks of birds,swarms of fish,school of sheep,ect.asymptotic reaching a consistent speed.It can also be applied to drone control,spacecraft formation,etc.The generalized modified Camassa-Holm equation is used to describe the motion of shallow water waves and is a type of dispersion partial differential equation.This article will use particle methods to study the well-posedness and flocking behavior of solutions to these two types of equations.First,we study the critical exponent of the unconditional flocking of the CuckerSmale model under group-hierarchical multi-leadership topologies in multidimensional Euclidean space.These topologies have multiple leaders,with single leader hierarchical topology and undirected graph topology as its special cases.We prove that range of exponents for unconditional flocking of the continuous-time Cucker-Smale model and the discrete-time Cucker-Smale model under group-hierarchical multi-leadership topologies is[0,1/2].This result expands the exponents for unconditional flocking of the CuckerSmale model under single-leader hierarchical topologies and undirected graphs to 1/2.Next,the Cucker-Smale model with truncated environmental white noises is proposed,and its stochastic flocking behavior is studied.Its corresponding kinetic equation is a stochastic partial differential equation.We use the particle method to obtain the wellposedness of the measure-value solution of the kinetic equation.For the Cucker Smale communication weight function,it is proven that the system will reach stochastic flocking behavior without the positive lower bound assumption for communication weight function.We prove the existence and uniqueness of the measure value solutions to the kinetic equation corresponding to the initial values with compact support by the particle systems and the particle method.Finally,we consider the well-posedness of the solutions to the generalized modified Camassa-Holm equations.For initial values without compact support,the local existence and uniqueness of classical solutions are obtained via characteristics,and the existence of global weak solutions is constructed by double mollification characteristics.The results on the local existence and uniqueness of classical solutions and the global existence of weak solutions for the modified Camassa-Holm equation is extended to the case that the initial values do not have compact supports.Finally,the error estimates between the classical solutions and the numerical approximate solutions under the particle method to the generalized modified Camassa-Holm equations are provided.