Diffusion, attraction and collapse
Diffusion, attraction and collapse
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DOI：
10.1088/0951-7715/12/4/320
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发表时间：
1999-07-01
期刊：
影响因子：
1.7
通讯作者：
Venkataramani, SC
中科院分区：
文献类型：
--
作者：
Brenner, MP;Constantin, P;Venkataramani, SC
我们研究了一个抛物椭圆系统的偏微分方程，产生在建模的过阻尼重力相互作用的粒子云或细菌的趋化性。该系统具有丰富的动力学和可能的行为的解决方案，包括收敛到时间无关的解决方案和形成有限时间奇点。我们的目标是描述导致这些结果的不同类型的解决方案。我们限制我们的注意力径向解决方案，并发现该系统的行为强烈依赖于空间维数d。对于2 < d < 10，存在两种稳定的爆破模式（自相似和Burgers样）和一种稳定的稳态。在无界区域上，存在一个单参数的不稳定解族和可数个不稳定爆破行为。我们记录一个不稳定的爆破行为和稳定的稳态和稳定的爆破之间的连接，以及一个不稳定的爆破和两个不同的稳定爆破之间的连接。有一个拓扑和稳定性之间的对应关系的各种渐近行为，这表明有可能构建一个全球相图的系统，对待全球的时间解决方案和爆破解决方案在平等的基础上。
We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2 < d < 10 there are two stable blowup modalities (self-similar and Burgers-like) and one stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.