Nonlinear surface waves and the loss of uniform Lopatinski stability in IBVPs for hyperbolic conservation laws
Monday, July 27, 2009 - 11:00am - 11:50am
The Kriess-Sakamoto theory for the well-posedness of hyperbolic IBVPs and the Majda theory for shock-wave stability apply under the assumption that a suitable Lopatinski condition holds uniformly. The failure of uniformity is associated with the presence of surface waves on the boundary or discontinuity. We will derive asymptotic equations for genuine nonlinear surface waves that decay exponentially away from the surface, such as Rayleigh waves in elasticity or surface waves on a tangential discontinuity in MHD. We will give a short-time existence theorem for smooth solutions of the asymptotic equations under a tameness condition on the interaction coefficients between the Fourier components of the surface wave, which prevents the loss of derivatives. We will also show numerical solutions of the asymptotic equations that illustrate singularity formation on the boundary (a mechanism which differs from shock formation in the interior).