Study of an Advection–Diffusion Equation: A Variational Approach and the Zero-Diffusion Limit

The advection–diffusion equation with the homogeneous Dirichlet boundary conditions is studied in this paper. The analysis is carried out in a variational framework within a Sobolev space. We establish well-posedness of the equation by deriving explicit coercivity criteria that quantify how diffusion, reaction, and the divergence of the velocity field balance each other. Under a gradient vector field, we use a transformation that produces a second-order elliptic problem with an effective potential, thereby placing the problem in a coercive setting. We also establish continuity of the solution with respect to perturbations of the velocity and damping parameters, a property relevant to inverse modeling and numerical simulation. Finally, we derive a necessary and sufficient criterion for the zero-diffusion limit to exist in one-dimensional cases.