Includes bibliographical references (pages 199-216) and index.
Contents:
Preface -- Ideal incompressible fluids: the Euler equations -- Eulerian vs Lagrangian representations -- Incompressibility and Transport -- The incompressible homogeneous Euler equations -- Vorticity -- Symmetries and conservation laws -- Special explicit solutions -- Exercises -- Existence of solutions and continuation criteria for Euler -- Local existence of Hs solutions -- The Lipschitz continuation criterion -- The Beale-Kato-Majda theorem -- The global existence of strong solutions in 2D -- The Constantin-Fefferman-Majda criterion -- Exercises -- Incompressible viscous fluids: the Navier-Stokes equations -- Viscosity -- Non-dimensionalization -- Vorticity, symmetries, and balance laws -- Special explicit solutions -- Local existence of Hs solutions -- Strong solutions with initial datum in H1 : local and global -- Exercises -- Leray-Hopf weak solutions of Navier-Stokes -- Weak solutions -- Existence of weak solutions on the whole space via mollification -- The uniqueness of weak solutions in 2D -- Weak-strong uniqueness and the Prodi-Serrin class -- Partial regularity in time for Leray-Hopf weak solutions -- Existence of weak solutions on the periodic box via Galerkin -- Exercises -- Mild solutions of Navier-Stokes -- Mild formulation -- Scaling criticality -- Local-in-time well-posedness in H 1/2 -- Local-in-time well-posedness in L3 -- Local regularization -- Continuation of smooth solutions -- Exercises -- A survey of some advanced topics -- Local regularity and the Prodi-Serrin conditions -- Partial regularity of suitable weak solutions in 3D -- Bounded domains -- Stationary solutions of the Navier-Stokes equations -- Ruling out backward self-similar finite-time singularities -- Critical and supercritical well-posedness for Navier-Stokes -- Yudovich theory and 2D Euler with Lp vorticity -- Gradient growth in the 2D Euler equations -- The search for finite-time singularties in 3D Euler -- Hydrodynamic stability: Euler -- Hydrodynamic stability: Navier-Stokes -- The energy balance and Onsager's conjecture -- Appendix -- The contraction mapping principle -- Existence and uniqueness for ODEs -- Fourier transform -- Integral operators -- Sobolev Spaces -- Basic properties of the Poisson and heat equations -- Mollifiers -- Sobolev and Gagliardo-Nirenberg inequalities -- Compactness -- Bibliography -- Index.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.