Local tracking and stability for degenerate viscous shock waves

Abstract

We study the pointwise behavior of perturbed degenerate (sonic) shock waves for scalar conservation laws with nonconstant diffusion. Building on the pointwise Green's function approach of Zumbrun and Howard (Indiana U. Math. J. 47(3) (1998) 741) we extend the linear analysis to an equation with nonintegrable coefficients. In lieu of working with the integrated equation, we employ a tracking mechanism that we expect will allow degenerate waves to be incorporated into the general framework for nondegenerate systems (Indiana U. Math. J. 47(3) (1998) 741).