具有齐次核的凝聚方程的不稳定性和振动性
Instabilities and oscillations in coagulation equations with kernels of homogeneity one
摘要：
我们讨论了Smoluchowski的凝聚方程的解决方案的长期行为的内核的同质性，结合形式的渐近性，启发式参数的基础上线性化，和数值模拟。我们称之为对角占优核的情况特别有趣。在这里，人们期望，在适当的变量变量变换之后，长时间行为与伯格斯方程相同。然而，对于接近对角线的核，我们得到了不稳定性，常数解和行波，一般不收敛到N-波的可积数据。另一方面，对于不接近对角线的核，这些结构是稳定的，但是行波具有强烈的振荡。这对可积数据的N波方法有影响，其特征在于激波阵面附近的强烈振荡。
英文摘要：
We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.
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