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

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


We consider the fourth order degenerate diffusion equation

h_t=-\nabla\cdot(f(h) \nabla\Delta h)\end{displaymath}

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 $h_0 \geq 0$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 $t\to\infty$.The weak solution is in a classical sense of distributions for ${3\over 8}<n<3$ and in a weaker sense for the remaining $0<n\leq {3\over 8}$.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 $h_0\geq m \gt$ and f(h)=|h|n, $0<n<\infty$.For this problem we show that even if a finite time singularity does occur of the form $h\to 0 $, 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 $t\to\infty$.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.