Steady states and dynamics of a thin-film-type equation with non-conserved mass

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2020-12

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Abstract

<jats:p>We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.</jats:p>

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10.1017/s0956792519000330

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Ji, Hangjie, and Thomas Witelski (2020). Steady states and dynamics of a thin-film-type equation with non-conserved mass. European Journal of Applied Mathematics, 31(6). pp. 968–1001. 10.1017/s0956792519000330 Retrieved from https://hdl.handle.net/10161/23301.

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Witelski

Thomas P. Witelski

Professor in the Department of Mathematics

My primary area of expertise is the solution of nonlinear ordinary and partial differential equations for models of physical systems. Using asymptotics along with a mixture of other applied mathematical techniques in analysis and scientific computing I study a broad range of applications in engineering and applied science. Focuses of my work include problems in viscous fluid flow, dynamical systems, and industrial applications. Approaches for mathematical modelling to formulate reduced systems of mathematical equations corresponding to the physical problems is another significant component of my work.


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