Weak Solution of a Doubly Degenerate Parabolic Equation

Abstract

This paper studies a fourth-order, nonlinear, doubly-degenerate parabolic equation derived from the thin film equation in spherical geometry. A regularization method is used to study the equation and several useful estimates are obtained. The main result of this paper is a proof of the existence of a weak solution of the equation in a weighted Sobolev space.

Strong Solutions of the Thin Film Equation in Spherical Geometry

“…In the present article, we obtain the existence of weak solutions in a wider weighted classes of functions than it was done in [11]. Moreover, we show the existence of non-negative strong solutions and we also prove that this solution decays asymptotically to the flat profile.…”

“…which describes the behavior of a thin viscous film on a flat surface under the effect of surface tension, the equation (1.1) is not yet well analysed. To the best of our knowledge, there is only one analytical result [11] where the authors proved existence of weak solutions in a weighted Sobolev space. In 1990, Bernis and Friedman [2] constructed non-negative weak solutions of the equation (1.3) when n 1, and it was also shown that for n 4, with a positive initial condition, there exists a unique positive classical solution.…”

Modeling coating flow and surfactant dynamics inside the alveolar compartment

“…After the change of variable x = − cos θ, the coupled system of evolution equations for h(θ, t) (14) and Γ (θ, t) (20) and their conservation properties (17) and ( 22) can be written more compactly as…”

“…The lubrication approximation is the classical approach for studying the dynamics of thin viscous films. In spherical geometry, which is the one we adopt in our work, well-posedness of the thin film model was analyzed in [30,20]. The motion of a Newtonian viscous fluid layer on a solid horizontal plane, with a monolayer of insoluble surfactant on its surface was modeled by Jensen and Grotberg [17] resulting in:…”

Finite Speed of Propagation for the Thin-Film Equation in Spherical Geometry

“…Equation (1.2) is a particular case of (1.1) with a = b = 0 with an absence of the second-order diffusion term. Existence of weak solutions for (1.2) in a weighted Sobolev space was shown in [13] and existence of more regular non-negative strong solutions of (1.2) was recently proved in [16]. Unlike the classical thin film equation…”