具有轻度奇异势的聚集方程的 Filippov 特征流
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
AI总结：
本文针对轻度奇异势下的多维聚集方程，利用 Filippov 不连续动力系统理论构造特征流，证明测度解的存在唯一性及其与梯度流解的等价性，并给出数值格式的收敛性。
摘要：
本文分析了多维聚集方程在时间上全局的测度解的存在性与唯一性。该系统可写为一个连续性方程，其速度场通过自洽相互作用势计算。在 Carrillo 等人（Duke Math J (2011)）的工作中，基于测度度量空间中梯度流的几何方法，已在势具有 λ-凸性这一基本假设下，为原点处轻度奇异的势建立了适定性理论。我们在此提出一种使用偏微分方程经典工具的替代方法。我们证明了基于 Filippov 不连续动力系统理论的特征流的存在性，使得弱测度解正是该流的推前测度。借助势的 λ-凸性，通过在传输距离中的压缩论证得到唯一性。此外，我们证明了该解与梯度流解的等价性。最后，我们在此框架下证明了一个数值格式对一般测度解的收敛性，从而能够模拟初始光滑密度在其 Lp 范数首次爆破之后的解。
英文摘要：
Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
