The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin’s Method

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^{1,q}_{0}(\Omega)}, {1<q<{\infty}}, 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^{1,q}_{0}(\Omega)}-{W_{0}^{1,q^{\prime}}(\Omega)} functional setting, {\frac{1}{q}+\frac{1}{q^{\prime}}=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,q^{\prime}}}-stability of the {H_{0}^{1}}-projector.