Finite speed of propagation and continuity of the interface for thin viscous flows

Abstract

We consider the fourth-order nonlinear degenerate parabolic equationwhich arises in lubrication models for thin viscous films and spreading droplets as well as in the flow of a thin neck of fluid in a Hele-Shaw cell. We prove that if 0< n< 2 this equation has finite speed of propagation for nonnegative "strong"solutions and hence there exists an interface or free boundary separating the regions where u > 0 and u = 0. Then we provethat the interface is Hölder continuous if 1/2< n< 2 and right-continuous if 0 < n< 2; these ratesexactly match those of the source-type solutions. If 0< n< 1 the property of finite speed of propagation is also proved for changing sign solutions.