We consider a nonlinear 4th-order degenerate parabolic partial dierential equation that arises in modelling the dynamics of an incompressible thin liquid lm on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively dierent ows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a nite-time blow-up. Finally, we present numerical simulations of solutions.
MSC: QSuTSDQSuQSD QSQSD QSqPSD QSfRHD QSfWWD QShHSD UTePH keywords: fourthEorder degenerte proli equtionsD thin liquid (lmsD onvetionD rimming )owsD oting )ows 1 Introduction e onsider the dynmis of visous inompressile )uid on the outer surfe of horizontl irulr ylinder tht is rotting round I arXiv:0910.5526v1 [math.AP]
Abstract.We study the existence of weak and strong solutions to the initial-boundary value problem for a thin-film type equation with unstable diffusion in multi-dimensional domains. Depending on the initial data and the parameter values, we prove local and global in time existence of nonnegative weak and strong solutions.
Abstract. For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential (H 1 , L 2 ) asymptotic decay to the flat profile of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature (cf. Eggers and Fontelos in Singularities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics, Cambridge, 2015), that a viscous sheet cannot rupture in finite time in the absence of external forcing. Moreover, in the absence of surface tension we find a special class of initial data for which the Lagrangian solution exhibits L 2 -exponential decay to the flat profile.Mathematics Subject Classification. 35B40, 35G31, 76D45, 76D27, 35D30, 35D35.
Motivated by models for thin films coating cylinders in two physical cases proposed in [Ker94] and [KF94], we analyze the dynamics of corresponding thin film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long-time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models of Reed Ogrosky [Ogr13] in conjunction with the works [CFL + 12, COO14, COO17, CMOV16].
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