Instabilities and oscillations in coagulation equations with kernels of homogeneity one

Herrmann, Michael; Niethammer, Barbara; Velázquez, Juan

doi:10.1090/qam/1454

Authors: Michael Herrmann, Barbara Niethammer and Juan J. L. Velázquez
Journal: Quart. Appl. Math. 75 (2017), 105-130
MSC (2010): Primary 70F99, 82C22, 45M10
DOI: https://doi.org/10.1090/qam/1454
Published electronically: September 6, 2016
MathSciNet review: 3580097
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Abstract: 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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