Abstract

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H-0(1)(Omega), the Banach Sobolev space W-0(1,q) (Omega), 1 < q < infinity, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W-0(1,q) (Omega)-W-0(1,q') (Omega) functional setting, 1/q +1/q' = 1. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin's method in this setting, for a diffusion-dominated case and under the assumption of W-1,W-q'-stability of the H-0(1)-projector.