Asymptotic Behavior of Scalar Convection-Diffusion Equations

Scalar convection–diffusion equations sit at the crossroads of two fundamental mechanisms: diffusion, which spreads mass, and convection, which transports it. A central question, both mathematically and in applications, is what remains of a solution after a long time, once the transient details of the initial state have vanished. This monograph develops a unified, rigorous theory of large-time asymptotics for scalar convection–diffusion equations in the whole space, with a particular focus on the roles of mass conservation, dimension, and the structure of the nonlinear convective flux. A guiding theme is the emergence of self-similar dynamics: special profiles that encode the scaling invariances of the model and describe the dominant long-time behavior for broad classes of initial data. Depending on the balance between convection and diffusion, three distinct regimes arise, namely, weakly nonlinear, critical self-similar, and strongly nonlinear, each characterized by a different effective asymptotic model and source-type solutions. Originating from lecture notes of a research course and substantially revised and updated, the book combines detailed proofs with a course-friendly presentation. It is designed for graduate students and researchers in analysis and PDEs, and is suitable for use in a Master’s or PhD course, assuming prior familiarity with basic functional analysis and partial differential equations.