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This accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the Hilbert–Schmidt theorem, including orthonormal bases, the Riesz representation theorem, and the basics of spectral theory. The material on Banach spaces and their duals includes the Hahn–Banach theorem, the Krein–Milman theorem, and results based on the Baire category theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully-worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and beginning graduate-level courses.
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.
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
$$\normVT{f}_{L^6(\mathbb{R}^3)}\le c\normVT{\nabla f}_{L^2(\mathbb{R}^3)}$$
.
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
$$\Delta u= V u$$
.
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.
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 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.
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.
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.
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
$$C^0 (J, L^r (\Omega))$$
,
$$1 < r < \infty$$
.
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.
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
$$u\in L^3(0,T;W^{\alpha,3}(\R^3))$$
,
$$\alpha>1/3$$
.
The Euler and Navier–Stokes equations are the fundamental mathematical models of fluid mechanics, and their study remains central in the modern theory of partial differential equations. This volume of articles, derived from the workshop 'PDEs in Fluid Mechanics' held at the University of Warwick in 2016, serves to consolidate, survey and further advance research in this area. It contains reviews of recent progress and classical results, as well as cutting-edge research articles. Topics include Onsager's conjecture for energy conservation in the Euler equations, weak-strong uniqueness in fluid models and several chapters address the Navier–Stokes equations directly; in particular, a retelling of Leray's formative 1934 paper in modern mathematical language. The book also covers more general PDE methods with applications in fluid mechanics and beyond. This collection will serve as a helpful overview of current research for graduate students new to the area and for more established researchers.
In a previous paper, we presented results from a 12-week study of a Psychomotor DANCe Therapy INtervention (DANCIN) based on Danzón Latin Ballroom that involves motor, emotional-affective, and cognitive domains, using a multiple-baseline single-case design in three care homes. This paper reports the results of a complementary process evaluation to elicit the attitudes and beliefs of home care staff, participating residents, and family members with the aim of refining the content of DANCIN in dementia care.
Methods:
An external researcher collected bespoke questionnaires from ten participating residents, 32 care home staff, and three participants’ family members who provided impromptu feedback in one of the care homes. The Behavior Change Technique Taxonomy v1 (BCTTv1) provided a methodological tool for identifying active components of the DANCIN approach warranting further exploration, development, and implementation.
Results:
Ten residents found DANCIN beneficial in terms of mood and socialization in the care home. Overall, 78% of the staff thought DANCIN led to improvements in residents’ mood; 75% agreed that there were improvements in behavior; 56% reported increased job satisfaction; 78% of staff were enthusiastic about receiving further training. Based on participants’ responses, four BCTTv1 labels–Social support (emotional), Focus on past success and verbal persuasion to boost self-efficacy, Restructuring the social environment and Habit formation–were identified to describe the intervention. Residents and staff recommended including additional musical genres and extending the session length. Discussions of implementing a supervision system to sustain DANCIN regularly regardless of management or staff turnover were suggested.
Conclusions:
Care home residents with mild to moderate dementia wanted to continue DANCIN as part of their routine care and staff and family members were largely supportive of this approach. This study argues in favor of further dissemination of DANCIN in care homes. We provide recommendations for the future development of DANCIN based on the views of key stakeholder groups.