Localization, Smoothness, and Convergence to Equilibrium for a Thin Film Equation

PDE Seminar
Tuesday, January 25, 2011 - 14:00
1 hour (actually 50 minutes)
Skiles 006
University of Maryland

Note the unusual time and room

We investigate the long-time behavior of weak solutions to the  thin-film type equation $$v_t =(xv - vv_{xxx})_x\ ,$$ which arises in the Hele-Shaw problem. We estimate the rate of  convergence of solutions  to the Smyth-Hill equilibrium solution, which has the form  $\frac{1}{24}(C^2-x^2)^2_+$,  in the norm $$|| f ||_{m,1}^2 = \int_{\R}(1+ |x|^{2m})|f(x)|^2\dd x +    \int_{\R}|f_x(x)|^2\dd x\ .$$ We obtain exponential convergence in the $|\!|\!| \cdot  |\!|\!|_{m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining  rates of convergence in norms measuring both smoothness and  localization. The localization is the main novelty, and in fact, we  show that there is a close connection between the localization bounds and the smoothness  bounds: Convergence of second moments implies convergence in the $H^1$ Sobolev norm.   We then use methods of optimal mass  transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak  solutions for which all of the estimates on which our convergence  analysis depends may be rigorously derived. Though  our main results  on convergence can be stated without reference to optimal mass  transportation, essential use of this theory is made throughout our analysis.This is a joint work with Eric A. Carlen.