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The three-dimensional Navier-Stokes equations. Classical theory. (English) Zbl 1358.35002
Cambridge Studies in Advanced Mathematics 157. Cambridge: Cambridge University Press (ISBN 978-1-107-01966-9/hbk; 978-1-139-09514-3/ebook). xiv, 471 p. (2016).
This nice monograph on mathematical fluid dynamics describes rigorously many significant results in the area of three-dimensional Navier-Stokes system. The presentation is self-contained and yet accessible for graduate students, and the appendices contain all the necessary background. Numerous exercises, supplemented with solutions, illustrate main ideas. The content is presented in 17 chapters and 5 appendices: Chapters: Function spaces, The Helmholtz-Weyl decomposition, Weak formulation, Existence of weak solutions, The pressure, Existence of strong solutions, Regularity of strong solutions, Epochs of regularity and Serrin’s condition, Robustness of regularity and convergence of Galerkin approximations, Local existence and uniqueness in \(\dot H^{1/2}\), Local existence and uniqueness in \(L^3\), Vorticity, The Serrin condition for local regularity, The local energy inequality, Partial regularity I and II, Lagrangian trajectories; Appendices: Functional analysis: miscellaneous results, Calderón-Zygmund Theory, Elliptic equations, Estimates for the heat equation, A measurable-selection theorem.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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