Relaxation Dynamics of Point Vortices

We study a model describing relaxation dynamics of point vortices, from quasi-stationary state to the stationary state. It takes the form of a mean field equation of Brownian point vortices derived from Chavanis, and is formulated by our previous work as a limit equation of the patch model studied by Robert-Someria. This model is subject to the micro-canonical statistic laws; conservation of energy, that of mass, and increasing of the entropy. We study the existence and nonexistence of the global-in-time solution. It is known that this profile is controlled by a bound of the negative inverse temperature. Here we prove a rigorous result for radially symmetric case. Hence E/M2 large and small imply the global-in-time and blowup in finite time of the solution, respectively. Where E and M denote the total energy and the total mass, respectively.

Boundary value problems for local and nonlocal Liouville type equations with several exponential type nonlinearities. Radial and nonradial solutions

This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc $B_{1} \subset {\mathbf{R}}^{2} $ B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at $\partial B_{1} $ ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for $r = 0 $ r = 0 , $r = \infty $ r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.

Research on twin-SIR rumor spreading model in online social network

In the study filed of rumor spreading, kill rumor or dispel rumor is very important in order to control rumor spreading and reduce the bad influence of the rumor. In the previous studies, rumor clarification is mostly finished by relying on external media or news reports instead of intervening and controlling from inside the network, which causes that the speed of rumor clarification is far lower than the speed of rumor spreading, and it is not ideal for the effect of rumor clarification. In this paper, a new Twin-SIR spreading model is proposed, in which, a rumor clarification node named as “rumor dispeller” with the spreading ability is introduced. The rumor dispeller is involved in the spreading process of the model together with the rumor spreader to control the spreading of rumor and thus to achieve the purpose of clarifying rumor. At the same time, during the process of building the model, we also apply the traditional media as a spreading parameter to the spreading process of the model. We built the mean-field equation of the model and then implemented further analysis of the model on homogeneous networks and heterogeneous networks. Through experimental simulations, the “rumor dispeller” was found to have the ability to reduce the spread of rumor spreading, and that the selection of the initial “rumor dispeller” node can affect the effect of rumor spreading, and at the same time, the external media have an important influence on rumor clarification. These conclusions have a new function for guiding us to study the mechanism of rumor spreading.