THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS - REGULARITY ANDLONG-TIME BEHAVIOR OF WEAK SOLUTIONS

We consider the fourth-order degenerate diffusion equation, h(f) = -de l .(f(h)del Delta h), in one space dimension. This equation, derived f rom a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = \h\ also models a thin neck of fluid in the Hele-Shaw cel l [10], [11], [23]. In such problems h(x, t) is the local thickness of the the film or neck. This paper considers the properties of weak sol utions that are more relevant to the droplet problem than to Hele-Shaw . For simplicity we consider periodic boundary conditions with the int erpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h(0) greater than or equal to 0 a nd f(h) = \h\(n), 0 < n < 3. We show that there exists a weak nonnegat ive solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T, and asymptotically approaches its mean as t --> infinity. The weak solution is in the cl assical sense of distributions for 3/8 n < 3 and in a weaker sense int roduced in [1] for the remaining 0 < n less than or equal to 3 Further more, the solutions have high enough regularity to just include the un ique source-type solutions [2] with zero slope at the edge of the supp ort. They do not include any of the less regular solutions with positi ve slope at the edge of the support. Second, we consider strictly posi tive initial data h(0) greater than or equal to m > 0 and f(h) = \h\(n ), 0 < n < infinity. For this problem we show that even if a finite-ti me singularity of the form h --> 0 does occur, there exists a weak non negative solution for all time t. This weak solution becomes strong an d positive again after some critical time T. As in the first problem we show that the solution approaches its mean as t --> infinity. The m ain technical idea is to introduce new classes of dissipative entropie s to prove existence and higher regularity. We show that these entropi es are related to norms of the difference between the solution and its mean to prove the relaxation result. (C) 1996 John Wiley & Sons, Inc.