The main objective of this book is to provide an introductory study to the variable density incompressible Navier-Stokes equations. We will deal with their motivation, the main related mathematical problems, the main techniques used to solve and/or control the systems and the main open questions arising in the subject. Additionally, the detailed description of some (more or less) elementary numerical techniques is given. The book is intended to be useful to Master and PhD students and young researchers interested by the analysis and control of nonlinear partial differential equations (PDEs), particularly those concerned with applications to mechanics, engineering, biology, economics, etc. We have tried to adapt the explanations, the proofs of the results and the proposed exercises to this level. Accordingly, we hope it will be readable, understandable, and useful for someone familiar with basic results on linear functional analysis, measure theory, and numerical analysis. We consider questions concerning fluid mechanics which are in part purely theoretical (existence, uniqueness, regularity, stability). In part, connected with applications (specific solutions, particular geometrical situations, the behavior with respect to some parameters, the limit as space dependent initial densities tend to a constant, the limit as viscosity tends to zero). Also, motivated by control theory (optimal control and controllability problems. At the end of each chapter, we have proposed a collection of exercises whose resolution can help to understand the arguments and techniques. Some of them are relatively easy, but others are more involved, may need nontrivial results and may even lead to new contributions to the field. Very frequently, help hints are provided. A collection of open problems can be found in the field. Some of them are recalled. Hopefully, this will generate interest among young researchers.
P. Braz e Silva is a full professor of Mathematics at the Federal University of Pernambuco, Brazil. He holds B.Sc. degrees in Mathematics and Economics and received his Ph.D. in Mathematics from the University of New Mexico, United States, in 2003. His research interests are PDEs of evolution type, specially those arising in fluid mechanics.
E. Fernandez-Cara is full professor of Mathematics in the University of Sevilla (Spain). He received Ph.D. in Mathematics from this University and also form the University Paris 6 (France) in 1981. He has been visiting professor in several French, Chilean and Brazilian universities. He has been the advisor of more than 30 PhD Theses and has participated in more than 20 Research Projects in Spain, France, Chile and Brazil. His main areas of interest are the theoretical and numerical analysis and control of PDEs, with special emphasis in nonlinear PDEs from physics, biology and other sciences and some additional questions (optimization, stability, inverse problems, etc.). Since 2023, he has been included in the Stanford List of world's top 2% scientists. M. Rojas-Medar is full professor at the University of Tarapacá (Chile). He received his Ph.D. in Mathematics from the State University of Campinas (Brazil) in 1991. He has worked in Campinas, Chillán (Chile) and Arica and has been a visiting professor at several Brazilian, Spanish and Chilean universities. He has been the advisor of more than 21 PhD Theses and has participated in more than 30 research projects in Chile, Spain, France and Brazil. His research interests are PDEs of evolution type, specially equations arising in fluid mechanics, optimization (including control problems, non-smooth problems, etc.) and fuzzy multi-valued analysis.
