The Navier-Stokes Problem

This book provides a clear and rigorous introduction to the mathematical analysis of the Navier–Stokes equations, guiding the reader from fundamental concepts to key results in modern research. Aimed at Master's and PhD students, as well as researchers in partial differential equations and applied analysis, it functions both as a graduate-level textbook and a reference work.The book first establishes the functional framework for incompressible fluid flows, presenting the main notions of solutions and their equivalence. The two-dimensional case is studied in detail, leading to a complete theory of global existence, uniqueness, and regularity.The central chapters address the challenges of the three-dimensional problem. They introduce modern analytical methods based on critical functional spaces and small-data theory, and explain how geometric properties of the flow can influence regularity. By combining classical theory with contemporary techniques, this book equips readers with essential tools for studying one of the most important problems in mathematical fluid mechanics.