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.
Table of Contents:
1.Introduction.- 2.The linear heat equation.- 3.The Burgers equation.- 4.The heat equation in similarity variables.- 5.The nonlinear Cauchy problem.- 6.Weakly nonlinear asymptotic behavior.- 7.Self-similar asymptotic behavior.- 8.Strongly nonlinear asymptotic behavior.- 9.Further topics.- Index.
About the Author :
Enrique Zuazua (Eibar, Basque Country, Spain, \href{https: //dcn.nat.fau.eu/zuazua}{https: //dcn.nat.fau.eu/zuazua}) has held, since September 2019, the Chair for Dynamics, Control, Machine Learning and Numerics -- Alexander von Humboldt Professorship, at the Department of Mathematics of Friedrich-Alexander-Universit\"at Erlangen-N\"urnberg (FAU) in Germany, together with part-time appointments at Universidad Aut\'onoma de Madrid (UAM) and the University of Deusto, Bilbao, Spain. He is also Chief Algorithm Scientist at Sherpa.ai (\href{https: //sherpa.ai}{https: //sherpa.ai}), a member of the Basque Academy ``Jakiunde'', the Academia Europaea, and a Fellow of the Artificial Intelligence Industry Academy (AIIA), the Asia-Pacific Artificial Intelligence Association (AAIA), the Asia-National Academy of Artificial Intelligence (NAAI), and the CORE Academy.
He holds a degree in Mathematics (1984) from the University of the Basque Country, and a dual Ph.D.\ degree from the same university (1987) and Universit\'e Pierre et Marie Curie, Paris (1988).
He has been awarded the Euskadi (Basque Country) Prize for Science and Technology (2006), the National Julio Rey Pastor Prize (2007) in Mathematics and Information and Communication Technology, and three Advanced Grants of the European Research Council (ERC): NUMERIWAVES (2010), DyCon (2016), and CoDeFeL (2022). In 2022, he received the W. T.\ and Idalia Reid Prize of SIAM. He was an invited section speaker in ``Control and Optimization'' at the International Congress of Mathematicians (ICM 2006), Madrid, and will deliver a Special Section Lecture at the International Congress of Mathematicians (ICM 2026) in the fields of Control and Artificial Intelligence.
His current research agenda lies at the interface of partial differential equations (PDEs), numerical analysis, control theory, and machine learning (ML), to develop the mathematical foundations for the design of reliable, interpretable, and scalable algorithms for complex systems in science, engineering, and industry.
He ranks among the world's leading scientists. International citation and impact analyses consistently place him among the most influential applied mathematicians worldwide and within the top tier of researchers across all scientific disciplines. His work is widely cited both in core mathematics and in neighboring fields such as Partial Differential Equations (PDE), control theory, numerical analysis, and machine learning, reflecting its combination of depth, breadth, and long-lasting influence.
He holds leading editorial positions as editor-in-chief, co-editor-in-chief, and editorial board member of several major international journals, and serves on scientific committees of multiple research centers and funding agencies.
He was the first Mathematics Manager of the Spanish National Research Plan (1999--2002), the Founding Scientific Director of the Basque Center for Applied Mathematics (BCAM) in 2008--2012, and in 2016, he launched the Chair of Computational Mathematics at the University of Deustoin Bilbao, which he currently directs. Since 2021, he has been the Founding Speaker of the FAU Research Center for Mathematics of Data (FAU MoD).He also pursues an active outreach and dissemination agenda, which is gathered at: \href{https: //cmc.deusto.eus/enzuazua/}
