Oscillatory travelling wave solutions for coagulation equations

Barbara Niethammer; Juan J. J. L. Velazquez

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arXiv · 1702.02437

Oscillatory travelling wave solutions for coagulation equations

Barbara Niethammer·Juan J. J. L. Velazquez

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Abstract

We consider Smoluchowski's coagulation equation with kernels of homogeneity one of the form $K_{\varepsilon }(\xi,\eta) =\big( \xi^{1-\varepsilon }+\eta^{1-\varepsilon }\big)\big ( \xi\eta\big) ^{\frac{\varepsilon }{2}}$. Heuristically, in suitable exponential variables, one can argue that in this case the long-time behaviour of solutions is similar to the inviscid Burgers equation and that for Riemann data solutions converge to a traveling wave for large times. Numerical simulations in \cite{HNV16} indeed support this conjecture, but also reveal that the traveling waves are oscillatory and the oscillations become stronger with smaller $\varepsilon$. The goal of this paper is to construct such oscillatory traveling wave solutions and provide details of their shape via formal matched asymptotic expansions.

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Barbara Niethammer, Juan J. J. L. Velazquez. 2017-02-08. Oscillatory travelling wave solutions for coagulation equations. https://arxiv.org/abs/1702.02437

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