Nonnegative Solutions for a Long-Wave Unstable Thin Film Equation with Convection

Abstract

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]

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