A rigorous but accessible introduction to the mathematical theory of the three-dimensional Navier–Stokes equations, this book provides self-contained proofs of some of the most significant results in the area, many of which can only be found in research papers. Highlights include the existence of global-in-time Leray–Hopf weak solutions and the local existence of strong solutions; the conditional local regularity results of Serrin and others; and the partial regularity results of Caffarelli, Kohn, and Nirenberg. Appendices provide background material and proofs of some 'standard results' that are hard to find in the literature. A substantial number of exercises are included, with full solutions given at the end of the book. As the only introductory text on the topic to treat all of the mainstream results in detail, this book is an ideal text for a graduate course of one or two semesters. It is also a useful resource for anyone working in mathematical fluid dynamics.

Suppose that u(t) is a solution of the three-dimensional Navier-Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev spaceḢ s must be bounded below by c s t

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We consider the three-dimensional Navier-Stokes equations on the whole space R 3 and on the three-dimensional torus T 3. We give a simple proof of the local existence of (finite energy) solutions in L 3 for initial data u 0 P L 2 L 3 , based on energy estimates and regularisation of the initial data with the heat semigroup. We also provide a lower bound on the existence time of a strong solution in terms of the solution v(t) of the heat equation with such initial data: there is an absolute constant e b 0 such that solutions remain regular on [0Y T] if ku 0 k 3 L 3 T 0 R 3 jrv(s)j 2 jv(s)j dx dt e. This implies the u P C 0 ([0Y T]Y L 3) regularity criterion due to von Wahl. We also derive simple a priori estimates in L p for p b 3 that recover the well known lower bound ku(T À t)k L p ! ct À(pÀ3)a2p on any solution that blows up in L p at time T. The key ingredients are a calculus inequality kuk p L 3p c juj pÀ2 jruj 2 (valid on R 3 and for functions on bounded domains with zero average) and the bound on the pressure kpk L r c r kuk 2 L 2r , valid both on the whole space and for periodic boundary conditions. MATHEMATICS SUBJECT CLASSIFICATION (2010). 35Q30, 46E35.

Abstract. We consider the behaviour of weak solutions of the unforced threedimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ( u 2 = |u| 2 ) but infinite enstrophy ( Du 2 = |curl u| 2 ). We show that this has to be reflected in the solution for small times, so that in particular Du(t) → +∞ as t → 0. We also give some limitations on this 'backwards blowup', and give an elementary proof that the upper box-counting dimension of the set of singular times can be no larger than one half. Although similar in flavour, this final result neither implies nor is implied by Scheffer's result that the 1/2-dimensional Hausdorff measure of the singular set is zero.