Stationary states of quadratic diffusion equations with long-range attraction
Stationary states of quadratic diffusion equations with long-range attraction
复制标题
具有长程吸引力的二次扩散方程的稳态
DOI：
10.4310/cms.2013.v11.n3.a3
复制
发表时间：
2011
影响因子：
1
通讯作者：
Marzena Franek
中科院分区：
文献类型：
--
作者：
Martin Burger;M. D. Francesco;Marzena Franek
本文研究了一类具有二次扩散的非局部聚集方程在种群动力学中非平凡平稳解的存在唯一性。该方程是能量E产生的Wasserstein梯度ow，能量E是二次自由能和相互作用能的总和。相互作用核采取径向和吸引力，非负和可积的，进一步的技术光滑性假设。这种解的存在与不存在由阈值现象决定，即非平凡稳态存在当且仅当扩散系数常数严格小于相互作用核的总质量。在一维情况下，我们证明了稳定状态是唯一的平移和质量约束。该策略基于强版本的Krein-Rutman定理。稳态是对称的关于他们的质量中心x 0，compactly支持在集的形式（x 0 L; x 0 + L），C2在他们的支持，严格减少（x 0; x 0 +L）。而且，它们是能量泛函E的全局极小元。数值模拟的结果是补充。
We study the existence and uniqueness of nontrivial stationary solutions to a nonlocal aggregation equation with quadratic diusion arising in many contexts in population dynamics. The equation is the Wasserstein gradient ow generated by the energy E, which is the sum of a quadratic free energy and the interaction energy. The interaction kernel is taken radial and attractive, nonnegative and integrable, with further technical smoothness assumptions. The existence vs. nonexistence of such solutions is ruled by a threshold phenomenon, namely nontrivial steady states exist if and only if the diusivity constant is strictly smaller than the total mass of the interaction kernel. In the one dimensional case we prove that steady states are unique up to translations and mass constraint. The strategy is based on a strong version of the Krein-Rutman theorem. The steady states are symmetric with respect to their center of mass x0, compactly supported on sets of the form (x0 L;x0 + L), C 2 on their support, strictly decreasing on (x0;x0+L). Moreover, they are global minimizers of the energy functional E. The results are complemented by numerical simulations.