Structure Analysis Of Approximate Solutions For A Class Of Pressureless Flow With Dirac Measures As Initial Conditions

In this paper we consider the systems of pressureless flow of the formwith initial conditionwhere Q_T = (?)×[0,T],δ(x) is a Dirac measure whose center is on the origin, andδ(x—λ) is a Dirac measure whose center is on the point "x =λ",λis a constant.The results of the systems of pressureless flow studied in the past is following :The existence of the global weak solution of (1.1) is first obtained independently by Brenier ,Grenier [3] and E,Rykov and Sinai [4].An explicit formula of weak solution was obtained in [4] by Generalized Variational Principles. Wang, Huang and Ding [8] extended their results to the general case when u₀ is not continuous. Boudin [10] showed that the weak solution could also be obtained as limits of solutions to a viscous pressureless model.The authors of [3,4,6] found that the Lax entropy condition was insufficient to guarantee the uniqueness for (1.1). In [4], E, Rykov and Sinai pointed out further that the Oleinik entropy condition might be necessary. Along this direction, Bouchut and James [7] proved the uniqueness of the weak solution for the case when the initial data ρ₀ is of bounded measurable function. Similar result were also obtained by Wang and Ding [9]. they introduced the method of generalized potential, whereρ(x,t),u(x,t) were both bounded measurable functions. F. Huang and Z. Wang[2] established the existence and uniqueness of the weak solution to (1.1), where the initial dataρ₀ is Radon measure and u₀ is bounded measurable function. They also considered the generalized potential and used the method of generalized characteristics, and indicated that the energy condition is necessary and sufficient for the uniqueness.In this paper, we consider the initial dataρ₀, u₀, which are both Dirac measure. And now the generalized potential is no longer appropriate, so we consider the problem with the method of the characteristics. However, the singularity and discontinuity of u, we consider the approximate problem of (1.1), then established the estimate of the solution.Firstly, we obtain the simplified model of (1.1) :By the paper of T P Liu, M Pierre, initial problemhave the following solution :So we consider the following initial problem By (1.4) we obtain the following approximate equationwhere j_ε, u_εis defined by (2.12), (2.13), respectively.By the characteristics, we get the approximate solutionρ_εof (1.5). At last, we study the asymptotic behavior ofρ_ε, asε→0.In the paper, we have the following results:Theorem 1.1. Supposeρ_ε(x,t) is the solution of the initial problem of (1.5), asλ= 1, thenρ_ε(x,t) is convergence to p(x,t) in the sense of generalized function, whereTheorem 1.2. Supposeρ_ε(x,t) is the solution of the initial problem of (1.5), asλ= - 1, thenρ_ε(x,t) is convergence toρ(x,t) in the sense of generalized function, whereTheorem 1.3. Supposeρ_ε(x,t) is the solution of the initial problem of (1.5), asλ= 0, thenρ_ε(x,t) is convergence toρ(x,t) in the sense of generalized function, where...