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.

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < ␣ < 1. The limiting case ␣ 3 0 corresponds to 2D Euler equations, and ␣ ‫؍‬ 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.alpha-patches ͉ quasi-geostrophic equation ͉ blow-up ͉ Euler equations ͉ self-similar behavior O ne of the most important open problems in mathematical fluid dynamics is whether the solutions to the Euler and Navier-Stokes equations modeling the evolution of incompressible inviscid and viscous fluids, respectively, may develop singularities in finite time. Several possible scenarios for a singularity have been proposed in the past (see ref. 1 for an account of some of them), although none of them led to a rigorous proof of the formation of singularities. One of these scenarios corresponds to the so-called vortex patch problem that we briefly describe below.A vortex patch consist of a 2D region D(t) (simply connected and bounded) that evolves with a velocity given at each instant of time bywhere the stream function is such that ϭ Ϫ⌬ , and the vorticity has a constant value 0 over D(t). Vortex patches are, therefore, weak solutions of Euler equations. The appearance of finite time singularities at the contour of D(t) was the subject of strong debate based on numerical results showing that the curvature of the boundary might grow superexponentially in time (see refs. 2-4). Nevertheless, work of Chemin (5) and Bertozzi and Constantin (6) rigorously proved global existence of regular solutions for the dynamics of the vortex patch, and, at least in this case, singularities cannot appear.A very natural singularity scenario would correspond to a vortex patch of the so-called surface quasi-geostrophic equation (see ref. 7) for which the relation between the stream function and potential temperature (that play the role that vorticity plays in 2D Euler equations) is ϭ (Ϫ⌬) 1/2 . The interest of this equation lies in its strong analogies to the 3D Euler equation as it has been argued in ref. 8 and its physical relevance as a model for the formation of temperature fronts in some geophysical contexts (see refs. 9 and 10).Hence, a natural question to ask is whether models for which ϭ (Ϫ⌬) 1Ϫ(␣/2) representing an interpolation between 2D Euler and quasi-geostrophic patches develop singularities for 0 Ͻ ␣ Յ 1. In this work, we provide numerical evidence showing that this scenario is indeed the case and describe the self-similar structure of such singularities. This result represents a previously undescribed singularity scenario for incompressible flows. The Model Following Zabusky et al. (4), we can invert the relation betweenand to obtain the following formula for the velocity with 0where x ជ(␥, t) is the position of points over C(t), the boundary of D(t), parameterized with ␥, and 0 ϭ ⅐c ␣ . Here is the value of in the patch and the factor c ␣ ϭ ⌫(␣/2)͞2 1Ϫ␣ ⌫(2...

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This paper establishes the local-in-time existence and uniqueness of strong solutions in H s for s > n/2 to the viscous, non-resistive magnetohydrodynamics (MHD) equations in R n , n = 2, 3, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato

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Abstract. In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of R n , n = 2, 3, for divergence-free initial data in certain Besov spaces, namely u 0 ∈ B n/2−1 2,1 and B 0 ∈ B n/2 2,1 . The a priori estimates include the termH n/2 ds on the right-hand side, which thus requires an auxiliary bound in H n/2−1 . In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in H 1/2 is required, which we prove using the splitting method of Calderón (Trans. Amer. Math. Soc. 318(1), 1990). By contrast, we prove that such solutions are unique in 3D, but the proof of uniqueness in 2D is more difficult and remains open.

We consider the problem of the evolution of sharp fronts for the surface quasigeostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two-dimensional Euler equation.The special interest of the quasi-geostrophic equation lies in its strong similarities with the three-dimensional Euler equation, while being a two-dimensional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the threedimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot-Savart law still needs to be understood.We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve.Finally, using a Nash-Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C ∞ case.