论文标题
关于多物种种群的临界大众Patlak-keller-segel系统：全球存在和无限时间聚集
On the critical mass Patlak-Keller-Segel system for multi-species populations: global existence and infinite time aggregation
论文作者
论文摘要
我们研究了整个欧几里得空间中多种种群的抛物线 - 纤维纤维纤维式patlak-keller-segel系统解决方案的全球全球生存和长时间的渐近学。此外，我们表明，随着时间$ t $接近无限，解决方案的所有组件都集中在单个点的狄拉克度量中。我们的方法利用了瓦斯恒星空间中De Giorgi的最小运动或JKO-Schemes的梯度流量结构。由于临界质量，JKO-SCHEMES中的最小化问题通常不承认解决方案。我们找到了一个必要的足够标准，任何最小化序列在适当的拓扑中保持均匀界定，以确保存在最小化器。
We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline case of critical mass there exists a global {\it free energy solution} subject to initial data with finite entropy and second moment. Moreover, we show that as time $t$ approaches to infinity, all the components of the solutions concentrate in the form of a Dirac measure at a single point. Our approach utilizes the gradient flow structure in Wasserstein space in the spirit of De Giorgi's minimizing movement or the JKO-schemes. Due to the critical mass, the minimization problem in JKO-schemes may not admit a solution in general. We find a necessary and sufficient criterion for which any minimizing sequence remains uniformly bounded in an appropriate topology to ensure the existence of a minimizer.
