The thin film equation with backwards second order diffusion

Details The thin film equation with backwards second order diffusion The thin film equation with backwards second order diffusion

2010-10-04

arxiv.org/abs/1010.0536v1

Mathematics

Analysis of PDEs

Scientific paper

In this paper, we focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equation $$u_t + \{u^n(u_{xxx} + \nu u^{m-n}u_x -A u^{M-n} u_x)\}_x=0,$$ where $\nu=\pm 1,$ $n>0,$ $M>m,$ and $A \ge 0.$ Global existence of weak nonnegative solutions is proven when $ m-n> -2$ and $A>0$ or $\nu=-1,$ and when $-2< m-n<2,$ $A=0,$ $\nu=1.$ From the weak solutions, we get strong entropy solutions under the additional constraint that $m-n> -{3}/{2}$ if $\nu=1.$ A local energy estimate is obtained when $2 \le n<3 $ under some additional restrictions. Finite speed of propagation is proven when $m>n/2,$ for the case of "strong slippage," $0

Affiliated with

Also associated with

No associations

Rating

The thin film equation with backwards second order diffusion does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The thin film equation with backwards second order diffusion, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The thin film equation with backwards second order diffusion will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-274268