一类具有最大规模的质量聚集模型
On a Model for Mass Aggregation with Maximal Size
摘要：
我们研究了一个动力学平均场方程的不同尺寸的粒子系统，其中粒子被允许凝聚，只有当它们的大小总和到一个规定的时间依赖性值。我们主要通过数值模拟证明了该模型的适定性，研究了自相似解的存在性，并分析了其大时间行为。根据参数$\Dconst$，它控制的概率凝固，我们观察到两种不同的情况：对于$\Dconst>2$存在两个自相似的平均场方程的解决方案，其中一个是不稳定的。在数值模拟中，我们观察到，对于所有的初始数据，重新标度的解决方案收敛到稳定的自相似解决方案。然而，对于$\Dconst<2$，当解在原始变量中收敛到强烈依赖于初始数据的极限时，不会发生自相似行为。我们严格证明了$\Dconst\in (0,1/3)$的一个相应的陈述。对交叉情况$\Dconst=2$的模拟并不完全确定，但表明，根据初始数据，部分质量以自相似的方式演化，而另一部分质量保留在小粒子中。
英文摘要：
We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study the existence of self-similar solutions, and analyze the large-time behavior mostly by numerical simulations. Depending on the parameter $\Dconst$, which controls the probability of coagulation, we observe two different scenarios: For $\Dconst>2$ there exist two self-similar solutions to the mean field equation, of which one is unstable. In numerical simulations we observe that for all initial data the rescaled solutions converge to the stable self-similar solution. For $\Dconst<2$, however, no self-similar behavior occurs as the solutions converge in the original variables to a limit that depends strongly on the initial data. We prove rigorously a corresponding statement for $\Dconst\in (0,1/3)$. Simulations for the cross-over case $\Dconst=2$ are not completely conclusive, but indicate that, depending on the initial data, part of the mass evolves in a self-similar fashion whereas another part of the mass remains in the small particles.
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