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The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

The Boltzmann equation which describes the time evolution of a large number of particles through the binary collision in statistics physics has close relation to the systems of fluid dynamics, that is, Euler equations and NavierStokes equations. As for a basic wave pattern to Euler equations, we consider the nonlinear stability of contact discontinuities to the Boltzmann equation. Even though the stability of the other two nonlinear waves, i.e., shocks and rarefaction waves has been extensively studied, there are few stability results on the contact discontinuity because unlike shock waves and rarefaction waves, its derivative has no definite sign, and decays slower than a rarefaction wave. Moreover, it behaves like a linear wave in a nonlinear setting so that its coupling with other nonlinear waves reveals a complicated interaction mechanism. Based on the new definition of contact waves to the Boltzmann equation

This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy , and . We also extend the result to higher dimensions.

Let G=(V,E) be a finite weighted graph, and Ω⊆V be a domain such that Ω^∘≠∅. In this paper, we study the following initial boundary problem for the non-homogenous wave equation ∂^2_t u(t,x) − Δ_Ω u(t,x) = f(t,x), (t,x)∈[0,∞)×Ω^∘ u(0,x)=g(x), x∈Ω^∘, ∂_t u(0,x)=h(x), x∈Ω^∘, u(t,x)=0, (t,x)∈[0,∞)×∂Ω, where Δ_Ω denotes the Dirichlet Laplacian on Ω^∘. Using Rothe's method, we prove that the above wave equation has a unique solution.