Title: Whitham averaging and nonlinear stability of periodic traveling
waves of viscous balance laws.  

Abstract: For gas or elastic flows
where thermodynamic stability is violated- for a van der Waal material
for example, or for a viscous relaxation system for which the
subcharacteristic condition fails - there may appear solitary-wave
and periodic traveling wave solutions along with the more familiar
traveling front, or shock wave, solutions. Behavior of a perturbed
periodic wave train is described formally by the Whitham modulation
equations.  It has been shown by Serre and Oh-Zumbrun using Evans
function techniques that this formal approximation may be connected
rigorously to spectral stability. However, the connection between
spectral and nonlinear stability until recently remained unclear. 

 In this talk, we show how to resolve this issue by finding a deeper
 connection to the Whitham system at the level of eigenmodes 
 rather than just eigenvalues. With the resulting sharpened estimates
 we are able to show that spectral stability implies nonlinear bounded
 stability from L1 ∩ H^s to L∞ , and asymptotic convergence in H^s to
 an appropriately modulated wave. The resulting stability theory is
 strikingly parallel to that of traveling fronts, even though the
 associated behavior is quite different in appearance. We conclude by
 describing applications to periodic solutions both stable and 
 unstable of the viscous Saint Venant equations for shallow water flow
 down an incline, a viscous balance law with relaxation.