Asymptotic Equipartition and Long Time Behavior of Solutions of a Thin-film Equation
by
Suleyman Ulusoy
Georgia Institute of Technology

We investigate the large{time behavior of strong solutions to the thin lm type equation ut = ��(uuxxx)x. It was shown in previous work of Carrillo and Toscani that for non negative initial data u0 that belongs to H1(R) and also has a nite mass and second moment, the solutions relax in the L1(R) norm at an explicit rate to the unique self{similar source type solution with the same mass. The equation itself is gradient ow for an energy functional that controls the H1(R) norm, and so it is natural to expect that one should also have convergence in this norm. Carrillo and Toscani raised this question, but their methods, using a different Lyapunov functions that arises in the theory of the porous medium equation, do not directly address this: Their Lyapunov functional does not involve derivatives of u. Here we show that the solutions do indeed converge in the H1(R) norm at an explicit, but slow, rate, though we require more than a second moment: we present the argument assuming a fourth moment. The key to establishing convergence this is an asymptotic equipartition of the excess energy part. Roughly speaking, the energy functional whose dissipation drives the evolution through gradient ow consists of two parts: one involving derivatives of u, and one that does not. We show that these must decay at related rates due to the asymptotic equipartition and then use the results of Carrillo and Toscani to control the rate for the part that does not depend on derivatives. From this one gets a rate on the dissipation for all of the excess energy. This is a joint work with Eric A. Carlen.

Date received: April 1, 2007