###
The Lubrication Approximation for Thin Viscous Films: Regularity
and Long Time Behavior of Weak Solutions

* with A. Bertozzi, CPAM 49(1996)2:85-123.
*

### Abstract

We consider the fourth order degenerate diffusion equation
in one space dimension.
This equation, derived from a `lubrication approximation',
models surface tension dominated motion
of thin viscous films and spreading droplets.
The equation with *f*(*h*)=|*h*| also models a
thin neck of fluid in the Hele-Shaw cell.
In such problems *h*(*x*,*t*) is the local thickness of the the film or neck.
This paper will consider the properties of weak solutions
which are more relevant to the droplet problem than to Hele-Shaw.
For simplicity we consider periodic boundary conditions
with the interpretation of modeling a periodic array of droplets.
We consider two problems:
The first has initial data
and *f*(*h*)=|*h*|^{n}, 0<*n*<3.
We show that there exists a weak *nonnegative* solution
for all time and that this solution
becomes a strong *positive* solution after some finite time
*T*^{*} and asymptotically approaches its mean as .The weak solution is in a classical sense of distributions
for and in a weaker sense
for the remaining .Furthermore, the solutions have sufficiently high
regularity to just include the unique `source type'
solutions with zero slope at
the edge of the support.
They do not include any of the less regular
solutions with positive slope at the edge of the support.
Secondly we consider strictly positive
initial data and *f*(*h*)=|*h*|^{n}, .For this problem we show that even if
a finite time singularity does occur
of the form , there
exists a weak *nonnegative* solution for all time *t*
and that this weak solution becomes strong
and *positive* again after some critical time *T*^{*}.
As in the first problem,
we show that the solution approaches its mean as .The main technical idea is to introduce new classes
of dissipative entropies to prove the existence
and higher regularity.
We show that these
entropies are related to norms of the difference between
the solution and its mean to prove the relaxation result.

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Both authors are supported by NSF postdoctoral fellowships.
AB is also partially supported by the DOE
and MP by NSF grant DMS-9305996.