Measure solutions for non-local interaction PDEs with two species
Measure solutions for non-local interaction PDEs with two species
复制标题
测量两个物种的非局部相互作用偏微分方程的解
DOI：
10.1088/0951-7715/26/10/2777
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发表时间：
2013
期刊：
影响因子：
1.7
通讯作者：
S. Fagioli
中科院分区：
文献类型：
--
作者：
M. D. Francesco;S. Fagioli
本文给出了两种群非定域相互作用偏微分方程组弱测度解的系统存在唯一性理论，该方程组是两种群确定性相互作用粒子系统的偏微分方程对应.这些模型背后的主要动机来自细胞生物学、行人运动和意见形成。在可对称化系统的情况下（即交叉相互作用势的一个倍数），我们提供了一个完整的存在性和唯一性理论内Ambrosio等人的Wasserstein梯度流理论（的适当推广）2008 Gradient Flows in Metric Spaces and in the Space of Probability Measures（数学讲座）（巴塞尔：Birkhäuser））和Carrillo等人（2011杜克数学杂志156 229-71），其允许考虑在原点处具有不连续梯度的相互作用势。在非对称系统的一般情况下，我们提供了测量解的存在性结果，该结果使用了Jordan-Kinderlehrer-Otto（JKO）方案的半隐式版本（Jordan et al 1998 SIAM J. Math. Anal. 29 1-17），这在相互作用势的合理非光滑设置中成立。在非对称化的情况下，证明了C2电位使用的一种变体的方法的特征。
This paper presents a systematic existence and uniqueness theory of weak measure solutions for systems of non-local interaction PDEs with two species, which are the PDE counterpart of systems of deterministic interacting particles with two species. The main motivations behind those models arise in cell biology, pedestrian movements, and opinion formation. In case of symmetrizable systems (i.e. with cross-interaction potentials one multiple of the other), we provide a complete existence and uniqueness theory within (a suitable generalization of) the Wasserstein gradient flow theory in Ambrosio et al (2008 Gradient Flows in Metric Spaces and in the Space of Probability Measures (Lectures in Mathematics ETH Zürich) 2nd edn (Basel: Birkhäuser)) and Carrillo et al (2011 Duke Math. J. 156 229–71), which allows the consideration of interaction potentials with a discontinuous gradient at the origin. In the general case of non-symmetrizable systems, we provide an existence result for measure solutions which uses a semi-implicit version of the Jordan–Kinderlehrer–Otto (JKO) scheme (Jordan et al 1998 SIAM J. Math. Anal. 29 1–17), which holds in a reasonable non-smooth setting for the interaction potentials. Uniqueness in the non-symmetrizable case is proven for C2 potentials using a variant of the method of characteristics.
DOI：
10.1098/rspa.2009.0239
发表时间：
2009-12-08
影响因子：
3.5
作者：
Duering, Bertram;Markowich, Peter;Wolfram, Marie-Therese
通讯作者：
Wolfram, Marie-Therese