Pointwise Estimates on the Greenʹs Function for a Scalar Linear Convection–Diffusion Equation

Pointwise estimates are found on Greenʹs functions for the scalar linear convection–diffusion equations that arise when a scalar conservation law with non-constant diffusion is linearized about a viscous shock profile of arbitrary strength. The estimates take the form of Gaussian kernels centered around paths determined by the (typically different) asymptotic states of the convection function. The analysis extends the spectral transform method to the non-constant coefficient case.