Lecture notes on regularity theory for the Navier-Stokes equations / Gregory Seregin.

Print version record.

Includes bibliographical references and index.

Preface; Contents; 1. Preliminaries; 1.1 Notation; 1.2 Newtonian Potential; 1.3 Equation div u = b; 1.4 Necas Imbedding Theorem; 1.5 Spaces of Solenoidal Vector Fields; 1.6 Linear Functionals Vanishing on Divergence Free Vector Fields; 1.7 Helmholtz-Weyl Decomposition; 1.8 Comments; 2. Linear Stationary Problem; 2.1 Existence and Uniqueness of Weak Solutions; 2.2 Coercive Estimates; 2.3 Local Regularity; 2.4 Further Local Regularity Results, n = 2, 3; 2.5 Stokes Operator in Bounded Domains; 2.6 Comments; 3. Non-Linear Stationary Problem; 3.1 Existence of Weak Solutions.

3.2 Regularity of Weak Solutions3.3 Comments; 4. Linear Non-Stationary Problem; 4.1 Derivative in Time; 4.2 Explicit Solution; 4.3 Cauchy Problem; 4.4 Pressure Field. Regularity; 4.5 Uniqueness Results; 4.6 Local Interior Regularity; 4.7 Local Boundary Regularity; 4.8 Comments; 5. Non-linear Non-Stationary Problem; 5.1 Compactness Results for Non-Stationary Problems; 5.2 Auxiliary Problem; 5.3 Weak Leray-Hopf Solutions; 5.4 Multiplicative Inequalities and Related Questions; 5.5 Uniqueness of Weak Leray-Hopf Solutions. 2D Case; 5.6 Further Properties of Weak Leray-Hopf Solutions.

880-01 Appendix A Backward Uniqueness and Unique ContinuationA. 1 Carleman-Type Inequalities; A.2 Unique Continuation Across Spatial Boundaries; A.3 Backward Uniqueness for Heat Operator in Half Space; A.4 Comments; Appendix B Lemarie-Riesset Local Energy Solutions; B.1 Introduction; B.2 Proof of Theorem 1.6; B.3 Regularized Problem; B.4 Passing to Limit and Proof of Proposition 1.8; B.5 Proof of Theorem 1.7; B.6 Density; B.7 Comments; Bibliography; Index.

The lecture notes in this book are based on the TCC (Taught Course Centre for graduates) course given by the author in Trinity Terms of 2009-2011 at the Mathematical Institute of Oxford University. It contains more or less an elementary introduction to the mathematical theory of the Navier-Stokes equations as well as the modern regularity theory for them. The latter is developed by means of the classical PDE's theory in the style that is quite typical for St Petersburg's mathematical school of the Navier-Stokes equations. The global unique solvability (well-posedness) of initial boundary value.

eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - Worldwide