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 dierential 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.
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