Math @ Duke

Publications [#340899] of Thomas P. Witelski
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 Bowen, M; Witelski, TP, Pressuredipole solutions of the thinfilm equation,
European Journal of Applied Mathematics, vol. 30 no. 2
(April, 2019),
pp. 358399 [doi]
(last updated on 2020/11/25)
Abstract: Copyright © Cambridge University Press 2018. We investigate selfsimilar signchanging solutions to the thinfilm equation, h t = (h n h xxx ) x , on the semiinfinite domain x ≥ 0 with zeropressuretype boundary conditions h = h xx = 0 imposed at the origin. In particular, we identify classes of first and secondkind compactly supported selfsimilar solutions (with a freeboundary x = s(t) = Lt β ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct firstkind selfsimilar solutions on the entire halfline x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation pointlike behaviour of signchanges local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full timedependent partial differential equation (based on a nonlocal regularisation of the mobility coefficient) and discuss the computational results in relation to the selfsimilar solutions.


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