Well-posedness For Two Types Of Partial Differential Equations By Particle Methods

The past several decades have seen significant development in the design and numerical analysis of particle methods for approximating solutions of PDEs.Even though the most “natural”application of the particle methods is linear transport equations,over the years,the range of these methods has been extended for approximating solutions of convection-diffusion and dispersive equations and general nonlinear problems.In this thesis,we mainly introduce some particle methods to study two different types of partial differential equations.One is a nonlinear dispersive equation and the other one is a Vlasov type equation.Firstly,we study the modified Camassa-Holm(m CH)equation with cubic nonlinearity in one dimension.Similarly to the famous Kd V equation,the m CH equation is also used to describe shallow water waves.It is a nonlinear dispersive equation and a completely integrable system,which has a bi-Hamiltonian structure and a Lax-pair.As it is known,the strong solutions to this equation exit locally and they are unique.However,strong solutions may blow up in finite time for some initial data.One of our main purpose is to study how to extend the solutions after the blow up time,which means to study the global existence of weak solutions.We start with the special solutions,N-peakon solutions,to the m CH equation.Due to the collision between peakons,the characteristic equations for the trajectories of peakons have a non-Lipschitz vector field.Hence,the nature question is how to extend the trajectories globally.Inspired by the sticky particle model in astrophysics,we give a sticky particle model for the m CH equation.This method gives the global sticky N-peakon solutions.For general initial data in Radon measure space,we use the mean field limit of the sticky particle model to obtain global weak solutions.Moreover,we prove that the weak solutions are stable in some solution classes.We also provide some examples to show the non-uniqueness of N-peakon weak solutions.Secondly,we present a dispersive regularization for the uniqueness of the weak solutions to the m CH equation.This method is similar to the vortex blob method for incompressible Euler equation.As mentioned above,the peakons may collide in finite time,which makes the vector field of characteristic equations non-Lipschitz.To obtain unique global N-peakon solutions,we present a regularized system by a double mollification for the characteristic equations.From this regularized system of ODEs,we obtain approximated N-peakon solutions with no collision between peakons.Then,a global N-peakon solution for the m CH equation is obtained,whose trajectories are globally Lipschitz functions and do not cross each other.Then,by a mean field limit process,we obtain global weak solutions for general initial data in Radon measure space.At last,we use a particle method to study a Vlasov type equation with local alignment which is the kinetic partial differential equation of Motsch-Tadmor(MT)model when the number of particles goes to infinity.MT model is a N particle model with self-organized behavior,which is used to describe the flocking phenomenon of animals.The MT model provides a natural particle method.This particle method is different from the above two particle methods,which satisfies a second order ODE system.For N particle system,we study the unconditional flocking behavior for a weighted MT model and a model with a “tail”.When N goes to infinity,global existence and stability(hence uniqueness)of measure valued solutions to the kinetic equation of this model are obtained.We also prove that measure valued solutions converge to a flock.