Well-posedness for the diffusive 3D Burgers equations with initial data in H1/2H^{1/2}

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 201

Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace"

This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the 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 $H^1$, $L^2\cap L^p$ (with $p\in(3,\infty]$), as well as lower bounds on the norms $\| \nabla u (t) \|_2$, $\| u(t) \|_p$ ($p\in(3,\infty]$) 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 $\tfrac{1}{2}$. 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.Comment: 81 pages. All comments are welcom

Non-conservation of dimension in divergence-free solutions of passive and active scalar systems

For any $h\in(1,2]$, we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in $\mathbb{R}^2$ that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension $h$. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in $\mathbb{R}^2$ and $\mathbb{R}^3$ with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.Comment: 32 pages, 3 figures. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this article is published in Arch Rational Mech Anal, and is available online at https://doi.org/10.1007/s00205-021-01708-

An Eulerian–Lagrangian form for the Euler equations in Sobolev spaces

In 2000 Constantin showed that the incompressible Euler equations can be written in an “Eulerian–Lagrangian” form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Hölder spaces C1,μ . We review the Eulerian–Lagrangian formulation of the equations and prove that given initial data in H s for n≥2 and s>n2+1 , a unique local-in-time solution exists on the n-torus that is continuous into H s and C 1 into H s-1. These solutions automatically have C 1 trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian–Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory

On some alternative formulations of the Euler and Navier-Stokes equations

In this thesis we study well-posedness problems for certain reformulations and models of the Euler equations and the Navier{Stokes equations. We also prove several global well-posedness results for the diffusive Burgers equations. We discuss the Eulerian-Lagrangian formulation of the incompressible Euler equations considered by Constantin (2000). Using this formulation we give a new proof that the Euler equations are locally well-posed in Hs (Td ) for s > d/2 + 1. Constantin proved a local well-posedness result for this system in the Hӧlder spaces C1; for μ> 0, but an analysis in Sobolev spaces is perhaps more natural. After suggesting a possible Eulerian-Lagrangian formulation for the incompressible Navier{Stokes equations in which the back-to-labels map is not di used, we obtain the formulation written in terms of the so-called magnetization variables, as studied by Montgomery-Smith and Pokornẏ (2001). We give a rigorous analysis of the equivalence between this formulation and the classical one, in the context of weak solutions. Noting certain similarities between this formulation and the diffusive Burgers equations we begin a study of the latter. We prove that the diffusive Burgers equations are globally well-posed in Lp ∩ L2 (Ω ) for certain domains Rd , p > d, and d = 2 or d = 3. Moreover, we prove a global well-posedness result in H1= 2 (T3 ). Lastly, we consider a new model of the Navier{Stokes equations, obtained by modifying one of the nonlinear terms in the magnetization variables formulation. This new system admits a maximum principle and we prove a global well-posedness result in H1=2 (T3 ) following our analysis of the Burgers equations

Asymptotics for vortex filaments using a modified Biot-Savart kernel

We consider a family of approximations to the Euler equations obtained by adding to the non-locality in the Biot-Savart kernel together with a mollification (with parameter ε). We consider the evolution of a thin vortex tube. We show that the velocity on the filament (core of the tube) in the limit as is given where κ and B are the curvature and binormal of the curve, and C, are uniformly bounded

ASASSN-18am/SN 2018gk : An overluminous Type IIb supernova from a massive progenitor

ASASSN-18am/SN 2018gk is a newly discovered member of the rare group of luminous, hydrogen-rich supernovae (SNe) with a peak absolute magnitude of $M_V \approx -20$ mag that is in between normal core-collapse SNe and superluminous SNe. These SNe show no prominent spectroscopic signatures of ejecta interacting with circumstellar material (CSM), and their powering mechanism is debated. ASASSN-18am declines extremely rapidly for a Type II SN, with a photospheric-phase decline rate of $\sim6.0~\rm mag~(100 d)^{-1}$. Owing to the weakening of HI and the appearance of HeI in its later phases, ASASSN-18am is spectroscopically a Type IIb SN with a partially stripped envelope. However, its photometric and spectroscopic evolution show significant differences from typical SNe IIb. Using a radiative diffusion model, we find that the light curve requires a high synthesised $\rm ^{56}Ni$ mass $M_{\rm Ni} \sim0.4~M_\odot$ and ejecta with high kinetic energy $E_{\rm kin} = (7-10) \times10^{51}$ erg. Introducing a magnetar central engine still requires $M_{\rm Ni} \sim0.3~M_\odot$ and $E_{\rm kin}= 3\times10^{51}$ erg. The high $\rm ^{56}Ni$ mass is consistent with strong iron-group nebular lines in its spectra, which are also similar to several SNe Ic-BL with high $\rm ^{56}Ni$ yields. The earliest spectrum shows "flash ionisation" features, from which we estimate a mass-loss rate of $\dot{M}\approx 2\times10^{-4}~\rm M_\odot~yr^{-1}$. This wind density is too low to power the luminous light curve by ejecta-CSM interaction. We measure expansion velocities as high as $17,000$ km/s for $H_\alpha$, which is remarkably high compared to other SNe II. We estimate an oxygen core mass of $1.8-3.4$ $M_\odot$ using the [OI] luminosity measured from a nebular-phase spectrum, implying a progenitor with a zero-age main sequence mass of $19-26$ $M_\odot$

Moving beyond fan typologies: The impact of social integration on team loyalty in football

The purpose of this paper is to develop detailed insight into loyalty among football fans of Hibernian FC, moving beyond typologies to a more socially grounded approach. Issues explored include patterns of consumption, distinctions between fan groups, and antecedents of loyalty. The origins and development of the club are evaluated, and consumer fanaticism, football fan loyalty, consumption behaviour, and the sociological impact of fan communities are discussed. Data were collected using a variety of methods, including participant observation, in-depth interviews, and analysis of websites and fan forums. Key findings relate to the impact of family and community influences on loyalty, initial experiences of developing associations with the club, through to the impact of socialisation, and the lived experience of being a supporter. A supporter matrix is constructed as a portrayal of the loyalty found at the club. A range of theoretical implications is considered, and the matrix promoted as a tool for understanding loyalty in clubs with similar social structures and community connections

A short history of the 5-HT2C receptor: from the choroid plexus to depression, obesity and addiction treatment

This paper is a personal account on the discovery and characterization of the 5-HT2C receptor (first known as the 5- HT1C receptor) over 30 years ago and how it translated into a number of unsuspected features for a G protein-coupled receptor (GPCR) and a diversity of clinical applications. The 5-HT2C receptor is one of the most intriguing members of the GPCR superfamily. Initially referred to as 5-HT1CR, the 5-HT2CR was discovered while studying the pharmacological features and the distribution of [3H]mesulergine-labelled sites, primarily in the brain using radioligand binding and slice autoradiography. Mesulergine (SDZ CU-085), was, at the time, best defined as a ligand with serotonergic and dopaminergic properties. Autoradiographic studies showed remarkably strong [3H]mesulergine-labelling to the rat choroid plexus. [3H]mesulergine-labelled sites had pharmacological properties different from, at the time, known or purported 5-HT receptors. In spite of similarities with 5-HT2 binding, the new binding site was called 5-HT1C because of its very high affinity for 5-HT itself. Within the following 10 years, the 5-HT1CR (later named 5- HT2C) was extensively characterised pharmacologically, anatomically and functionally: it was one of the first 5-HT receptors to be sequenced and cloned. The 5-HT2CR is a GPCR, with a very complex gene structure. It constitutes a rarity in theGPCR family: many 5-HT2CR variants exist, especially in humans, due to RNA editing, in addition to a few 5-HT2CR splice variants. Intense research led to therapeutically active 5-HT2C receptor ligands, both antagonists (or inverse agonists) and agonists: keeping in mind that a number of antidepressants and antipsychotics are 5- HT2CR antagonists/inverse agonists. Agomelatine, a 5-HT2CR antagonist is registered for the treatment of major depression. The agonist Lorcaserin is registered for the treatment of aspects of obesity and has further potential in addiction, especially nicotine/ smoking. There is good evidence that the 5-HT2CR is involved in spinal cord injury-induced spasms of the lower limbs, which can be treated with 5-HT2CR antagonists/inverse agonists such as cyproheptadine or SB206553. The 5-HT2CR may play a role in schizophrenia and epilepsy. Vabicaserin, a 5-HT2CR agonist has been in development for the treatment of schizophrenia and obesity, but was stopped. As is common, there is potential for further indications for 5-HT2CR ligands, as suggested by a number of preclinical and/or genome-wide association studies (GWAS) on depression, suicide, sexual dysfunction, addictions and obesity. The 5-HT2CR is clearly affected by a number of established antidepressants/antipsychotics and may be one of the culprits in antipsychotic-induced weight gain