Advances in Differential Equations

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

Francisco Bernis

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the fourth-order nonlinear degenerate parabolic equation $$ u_t+(|u|^n u_{xxx})_x=0 $$ which 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 prove that the interface is Hölder continuous if $1/2<n<2$ and right-continuous if $0<n\leq 1/2$. Finally we study the Cauchy problem and obtain optimal asymptotic rates as $t\to\infty$ for the solution and for the interface when $0<n<2$; these rates exactly 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.

Article information

Source
Adv. Differential Equations Volume 1, Number 3 (1996), 337-368.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896043

Mathematical Reviews number (MathSciNet)
MR1401398

Zentralblatt MATH identifier
0846.35058

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35K55: Nonlinear parabolic equations 35R35: Free boundary problems

Citation

Bernis, Francisco. Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differential Equations 1 (1996), no. 3, 337--368. https://projecteuclid.org/euclid.ade/1366896043.


Export citation