Undercompressive Shocks in Thin Film Flows

                                   by

                             Andrea Bertozzi
          Departments of Mathematics and Physics, Duke University

                JTB 110, 3:05pm  Friday, March 12, 1999
 
                               Abstract
                        
Nonlinear hyperbolic conservation laws have solutions with
propagating `shocks' or discontinuities. Classically admissible shocks in
scalar hyperbolic conservation laws satisfy a well known `entropy
condition', in which characteristics enter the shock from both sides. 
Undercompressive shocks, in which characteristics pass through the shock, 
can occur for special combined dispersive/diffusive limits of scalar 
laws with non-convex flux functions.  We show that fourth order diffusion
alone also produces undercompressive fronts, yielding such unusual behavior
as double shock structures from simple jump (Riemann) initial data. 
Thermal/gravity driven thin film flow is described by such equations and the
signature of undercompressive fronts has been observed in recent 
experiments [1]. The undercompressive front is an accumulation point 
for a countably infinite family of compressive waves having the same speed. 
The family appears through a cascade of bifurcations parameterized by 
the shock speed [2].

[1] A. L. Bertozzi, A. Muench, X. Fanton, and A. M. Cazabat, 
Phys. Rev. Lett. 81(23), December 7, 1998. 

[2]  A. L. Bertozzi, A. Muench, M. Shearer, submitted.


Requests for preprints and reprints send to 
                        Andrea Bertozzi <bertozzi@math.duke.edu>
This source can be found at http://www.math.utah.edu/research/