This contribution covers the topic of my talk at the 2016-17 Warwick-EPSRC Symposium: 'PDEs and their applications'. As such it contains some already classical material and some new observations. The main purpose is to compare several avatars of the Kato criterion for the convergence of a Navier-Stokes solution, to a regular solution of the Euler equations, with numerical or physical issues like the presence (or absence) of anomalous energy dissipation, the Kolmogorov 1/3 law or the Onsager C^{0,1/3} conjecture. Comparison with results obtained after September 2016 and an extended list of references have also been added.

We investigate existence, uniqueness and regularity of time-periodic solutions to the Navier-Stokes equations governing the flow of a viscous liquid past a three-dimensional body moving with a time-periodic translational velocity. The net motion of the body over a full time-period is assumed to be non-zero. In this case, the appropriate linearization is the time-periodic Oseen system in a three-dimensional exterior domain. A priori L^q estimates are established for this linearization. Based on these "maximal regularity" estimates, existence and uniqueness of smooth solutions to the fully nonlinear Navier-Stokes problem is obtained by the contraction mapping principle.

We give a survey of recent results on weak-strong uniqueness for compressible and incompressible Euler and Navier-Stokes equations, and also make some new observations. The importance of the weak-strong uniqueness principle stems, on the one hand, from the instances of nonuniqueness for the Euler equations exhibited in the past years; and on the other hand from the question of convergence of singular limits, for which weak-strong uniqueness represents an elegant tool.

By their use of mild solutions, Fujita-Kato and later on Giga-Miyakawa opened the way to solving the initial-boundary value problem for the Navier-Stokes equations with the help of the contracting mapping principle in suitable Banach spaces, on any smoothly bounded domain $$\Omega \subset \R^n, n \ge 2$$ , globally in time in case of sufficiently small data. We will consider a variant of these classical approximation schemes: by iterative solution of linear singular Volterra integral equations, on any compact time interval J, again we find the existence of a unique mild Navier-Stokes solution under smallness conditions, but moreover we get the stability of each (possibly large) mild solution, inside a scale of Banach spaces which are imbedded in some $$C^0 (J, L^r (\Omega))$$ , $$1 < r < \infty$$ .

We address the decay and the quantitative uniqueness properties for solutions of the elliptic equation with a gradient term, $$\Delta u=W\cdot \nabla u$$ . We prove that there exists a solution in a complement of the unit ball which satisfies $$|u(x)|\le C\exp (-C^{-1}|x|^2)$$ where $$W$$ is a certain function bounded by a constant. Next, we revisit the quantitative uniqueness for the equation $$-\Delta u= W \cdot \nabla u$$ and provide an example of a solution vanishing at a point with the rate $${\rm const}\Vert W\Vert_{L^\infty}^2$$ . We also review decay and vanishing results for the equation $$\Delta u= V u$$ .

This paper reviews and summarizes two recent pieces of work on the Rayleigh-Taylor instability. The first concerns the 3D Cahn-Hilliard-Navier-Stokes (CHNS) equations and the BKM-type theorem proved by Gibbon, Pal, Gupta, & Pandit (2016). The second and more substantial topic concerns the variable density model, which is a buoyancy-driven turbulent flow considered by Cook & Dimotakis (2001) and Livescu & Ristorcelli (2007, 2008). In this model $\rho^* (x, t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $$\rho^*_2 > \rho^*_1$$ and $$\rho^*_0$$ is a reference normalisation density. Following the work of a previous paper (Rao, Caulfield, & Gibbon, 2017), which used the variable $$\theta = \ln \rho^*/\rho^*_0$$ , data from the publicly available Johns Hopkins Turbulence Database suggests that the L2-spatial average of the density gradient $$\nabla \theta$$ can reach extremely large values at intermediate times, even in flows with low Atwood number At = $$(\rho^*_2 - \rho^*_1)/(\rho^*_2 + \rho^*_1) = 0.05$$ . This implies that very strong mixing of the density field at small scales can potentially arise in buoyancy-driven turbulence thus raising the possibility that the density gradient $$\nabla \theta$$ might blow up in a finite time.

In this contribution we focus on a few results regarding the study of the three-dimensional Navier-Stokes equations with the use of vector potentials. These dependent variables are critical in the sense that they are scale invariant. By surveying recent results utilising criticality of various norms, we emphasise the advantages of working with scale-invariant variables. The Navier-Stokes equations, which are invariant under static scaling transforms, are not invariant under dynamic scaling transforms. Using the vector potential, we introduce scale invariance in a weaker form, that is, invariance under dynamic scaling modulo a martingale (Maruyama-Girsanov density) when the equations are cast into Wiener path-integrals. We discuss the implications of this quasi-invariance for the basic issues of the Navier-Stokes equations.

Regularity criteria for solutions of the three-dimensional Navier-Stokes equations are derived in this paper. Let $$\Omega(t, q) := \left\{x:|u(x,t)| > C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\}, \tilde\Omega(t,q) := \left\{x:|u(x,t)| \le C(t,q)\normVT{u}_{L^{3q-6}(\mathbb{R}^3)}\right\} \cap\left\{x:\widehat{u}\cdot\nabla|u|\neq0\right\},$$ where $$q\ge3$$ and $$C(t,q) := \left(\frac{\normVT{u}_{L^4(\mathbb{R}^3)}^2\normVT{|u|^{(q-2)/2}\,\nabla|u|}_{L^2(\mathbb{R}^3)}}{cq\normVT{u_0}_{L^2(\mathbb{R}^3)} \normVT{p+\mathcal{P}}_{L^2(\tilde\Omega)}\normVT{|u|^{(q-2)/2}\, \widehat{u}\cdot\nabla|u|}_{L^2(\tilde\Omega)}}\right)^{2/(q-2)}.$$ Here $$u_0=u(x,0)$$ , $$\mathcal{P}(x,|u|,t)$$ is a pressure moderator of relatively broad form, $$\widehat{u}\cdot\nabla|u|$$ is the gradient of $$|u|$$ along streamlines, and $$c=(2/\pi)^{2/3}/\sqrt{3}$$ is the constant in the inequality $$\normVT{f}_{L^6(\mathbb{R}^3)}\le c\normVT{\nabla f}_{L^2(\mathbb{R}^3)}$$ .

The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain $$\TT^2\times\R_+$$ , where the boundary is both flat and has finite measure; in this geometry we do not require any estimates on the pressure, unlike the proof in general bounded domains due to Bardos & Titi (2018). However, first we study the equations on domains without boundary (the whole space $$\R^3$$ , the torus $$\mathbb{T}^3$$ , and the hybrid space $$\TT^2\times\R$$ ). We make use of some arguments due to Duchon & Robert (2000) to prove energy conservation under the assumption that $$u\in L^3(0,T;L^3(\R^3))$$ and $${|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\R^3} |u(x+y)-u(x)|^3\,\d x\,\d t=0$$ or $$\int_0^T\int_{\R^3}\int_{\R^3}\frac{|u(x)-u(y)|^3}{|x-y|^{4+\delta}}\,\d x\,\d y\,\d t<\infty,\qquad\delta>0$$ , the second of which is equivalent to $$u\in L^3(0,T;W^{\alpha,3}(\R^3))$$ , $$\alpha>1/3$$ .

This article offers a modern perspective that exposes the many contributions of Leray in his celebrated work on the three-dimensional incompressible Navier-Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth or belongs to $$H^1$$ or $$L^2 \cap L^p$$ (with $$p \in (3,\infty]$$ ), as well as lower bounds on the norms $$\| \nabla u (t) \|_2$$ and $$\| u(t) \|_p$$ ( $$p\in(3,\infty]$$ ) as t approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most 1/2. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.

We consider the inhomogeneous heat and Stokes equations on the half space and prove an instantaneous space-time analytic regularization result, uniformly up to the boundary of the half space.