کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4609890 1338532 2016 35 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
The Filippov characteristic flow for the aggregation equation with mildly singular potentials
چکیده انگلیسی

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. (2011) [17], 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 λ-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 λ  -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 LpLp-norms.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Differential Equations - Volume 260, Issue 1, 5 January 2016, Pages 304–338
نویسندگان
, , , ,