论文标题
径向对称溶液与弱相互作用力的2D聚集扩散方程的紧密度
Tightness of radially-symmetric solutions to 2D aggregation-diffusion equations with weak interaction forces
论文作者
论文摘要
我们证明了径向对称溶液与2D聚集 - 扩散方程的紧密度，其中成对吸引力可能在较大距离下退化。首先，我们通过引入对相互作用电位的新假设（称为基本径向合同（ERC）属性）的新假设，将问题降低为一个在界区域上的时间积分的有限性。然后，我们通过使用2-Wasserstein梯度流结构来证明这种有限性，并与连续的施泰纳对称曲线和局部聚类曲线结合使用。这是对一般弱限制电位的2D聚合扩散方程的第一个紧密度结果，即，这些$ w（r）$带有$ \ lim_ {r \ rightArrow \ rightarrow \ infty} w（r）<\ infty $，并且是迈向平衡研究的重要步骤。
We prove the tightness of radially-symmetric solutions to 2D aggregation-diffusion equations, where the pairwise attraction force is possibly degenerate at large distance. We first reduce the problem into the finiteness of a time integral in the density on a bounded region, by introducing a new assumption on the interaction potential called the essentially radially contractive (ERC) property. Then we prove this finiteness by using the 2-Wasserstein gradient flow structure, combining with the continuous Steiner symmetrization curves and the local clustering curve. This is the first tightness result on the 2D aggregation-diffusion equations for a general class of weakly confining potentials, i.e., those $W(r)$ with $\lim_{r\rightarrow\infty}W(r)<\infty$, and serves as an important step towards the study of equilibration.
