Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
Dirac Mass Dynamics in Multidimensional Nonlocal Parabolic Equations
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DOI：
10.1080/03605302.2010.538784
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发表时间：
2010-11
影响因子：
1.9
通讯作者：
A. Lorz;S. Mirrahimi;B. Perthame
中科院分区：
文献类型：
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作者：
A. Lorz;S. Mirrahimi;B. Perthame
非局部Lotka-Volterra模型具有在小扩散极限下解集中为狄拉克质量的性质。有没有可能描述极限集中点和狄拉克质量权重的动力学？狄拉克质量的长时间渐近性是什么？几个狄拉克质量可以共存吗？我们将解释这些问题如何与所谓的“约束哈密尔顿-雅可比方程”有关，以及如何建立正则方程的形式。这个方程是在假设平滑的情况下建立的。在这里，我们建立了一个框架，光滑的解决方案存在，从而可以严格地开发完整的理论。我们还表明，我们的形式的正则方程带有一种李雅普诺夫泛函。数值模拟表明，轨迹可以表现出意想不到的动力学很好地解释了这个方程。我们的动机来自于种群适应性进化，这是数学生态学的一个分支，它模拟了达尔文进化论.
Nonlocal Lotka–Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Is it possible to describe the dynamics of the limiting concentration points and of the weights of the Dirac masses? What is the long time asymptotics of these Dirac masses? Can several Dirac masses co-exist? We will explain how these questions relate to the so-called “constrained Hamilton–Jacobi equation” and how a form of canonical equation can be established. This equation has been established assuming smoothness. Here we build a framework where smooth solutions exist and thus the full theory can be developed rigorously. We also show that our form of canonical equation comes with a kind of Lyapunov functional. Numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation. Our motivation comes from population adaptive evolution a branch of mathematical ecology which models Darwinian evolution.