On an unstable thin-film equation in multi-dimensional domains

Taranets, Roman M.; King, John R.

doi:10.1007/s00030-013-0240-3

Cited by 12 publications

(12 citation statements)

References 29 publications

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“…The graphs can be interpreted as narrow grooves on a solid surface in which extends a viscous fluid. Our study allows to extend the previously obtained results (see [1,4,18,5]) to the case of surfaces with more complex geometry. To the best of our knowledge, this result is new and no other authors studied TFEs on graphs previously.…”

Section: Introductionsupporting

confidence: 58%

Thin film flow dynamics on fiber nets

Taranets

Chugunova

2018

Communications in Mathematical Sciences

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We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.

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Section: Introductionsupporting

confidence: 58%

Thin film flow dynamics on fiber nets

Taranets

Chugunova

2018

Communications in Mathematical Sciences

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“…This case is beyond our analytical results. Second, our new approach looks very promising for multidimensional case considered in [21] as all functional estimations that we used admit generalisations for higher dimensions. Finally, it would be interesting to compare numerical blow-up results obtained in lubrication limit that we studied with original Navier-Stokes simulations.…”

Section: Then For Everymentioning

confidence: 99%

Blow-up with mass concentration for the long-wave unstable thin-film equation

Chugunova

Taranets

2015

Applicable Analysis

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For a family of long-wave unstable thin-film equations, we prove existence of non-negative weak solutions blowing-up in a finite time. Specifically, building these solutions from initial data with negative energy, we show that their L ∞ -norms go to infinity as t → T * > 0, T * < ∞. In addition, using the Bourgain's type approach, we obtain qualitative information about the blow-up and prove mass concentration phenomenon.

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“…which says that the second moment is increasing in t. Now we prove that at least one of (34) and (35) holds. Suppose that…”

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confidence: 69%

“…To the best of our knowledge, there are no results on existence of weak and strong solutions to (1) with unstable diffusion in multi-dimension before year 2014. In 2014, Taranets and King [35] proved local existence of nonnegative weak and strong solutions in a bounded domain Ω with smooth boundary in R d under a more restrictive threshold…”

Section: Jian-guo Liu and Jinhuan Wangmentioning

confidence: 99%