Abstract
While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space
H
0
1
(
Ω
)
{H_{0}^{1}(\Omega)}
, the Banach Sobolev space
W
0
1
,
q
(
Ω
)
{W^{1,q}_{0}(\Omega)}
,
1
<
q
<
∞
{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
0
1
,
q
(
Ω
)
{W^{1,q}_{0}(\Omega)}
-
W
0
1
,
q
′
(
Ω
)
{W_{0}^{1,q^{\prime}}(\Omega)}
functional setting,
1
q
+
1
q
′
=
1
{\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
′
{W^{1,q^{\prime}}}
-stability of the
H
0
1
{H_{0}^{1}}
-projector.