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-

The adjoint Rayleigh and Orr-Sommerfeld equations: Green function and eigenmodes

The Rayleigh and Orr-Sommerfeld equations are ODEs which arise from the linearized Euler and Navier-Stokes equation around a shear flow. In this paper, we consider the adjoints of the Rayleigh and Orr-Sommerfeld equations on $[0,\infty)$ with respect to the complex $L^2$ product. In the viscous case, we consider a family of viscosity-dependent Navier boundary conditions, which in the limit corresponds to the no-slip condition. We rigorously establish existence and asymptotic properties of their eigenvalues, eigenmodes and Green functions away from critical layers. The adjoint operators prove useful because they also allow us to deduce properties about the kernels and images of the original operators

Aggregation equations with fractional diffusion : preventing concentration by mixing

We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advection-aggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of [Kiselev & Xu, Arch. Rat. Mech. Anal., 222(2):1077-1112, 2016], and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller–Segel model devised by Kiselev and Xu also applies to the fractional Keller–Segel model (where Δ is replaced by −(−Δ)γ2) requiring only that γ>1. In addition, we remove the restriction to dimension d<4. As a by-product, a characterisation of the class of relaxation enhancing flows on the d-torus is extended to the case of fractional dissipation

Black-hole horizons as probes of black-hole dynamics II: geometrical insights

In a companion paper [1], we have presented a cross-correlation approach to near-horizon physics in which bulk dynamics is probed through the correlation of quantities defined at inner and outer spacetime hypersurfaces acting as test screens. More specifically, dynamical horizons provide appropriate inner screens in a 3+1 setting and, in this context, we have shown that an effective-curvature vector measured at the common horizon produced in a head-on collision merger can be correlated with the flux of linear Bondi-momentum at null infinity. In this paper we provide a more sound geometric basis to this picture. First, we show that a rigidity property of dynamical horizons, namely foliation uniqueness, leads to a preferred class of null tetrads and Weyl scalars on these hypersurfaces. Second, we identify a heuristic horizon news-like function, depending only on the geometry of spatial sections of the horizon. Fluxes constructed from this function offer refined geometric quantities to be correlated with Bondi fluxes at infinity, as well as a contact with the discussion of quasi-local 4-momentum on dynamical horizons. Third, we highlight the importance of tracking the internal horizon dual to the apparent horizon in spatial 3-slices when integrating fluxes along the horizon. Finally, we discuss the link between the dissipation of the non-stationary part of the horizon's geometry with the viscous-fluid analogy for black holes, introducing a geometric prescription for a "slowness parameter" in black-hole recoil dynamics.Comment: Final version published on PR

Energy conservation for the Euler equations on T2×R+\mathbb{T}^2\times \mathbb{R}_+ for weak solutions defined without reference to the pressure

We study weak solutions of the incompressible Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the pressure. We prove energy conservation under the assumptions that $u\in L^3(0,T;L^3(\mathbb{T}^2\times \mathbb{R}_+))$, $$\lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{T}^2}\int^\infty_{x_3>|y|} |u(x+y)-u(x)|^3\mathrm{d} x\, \mathrm{d} t=0,$$ and an additional continuity condition near the boundary: for some $\delta>0$ we require $u\in L^3(0,T;C^0(\mathbb{T}^2\times [0,\delta])))$. We note that all our conditions are satisfied whenever $u(x,t)\in C^\alpha$, for some $\alpha>1/3$, with H\"older constant $C(x,t)\in L^3(\mathbb{T}^2\times\mathbb{R}^+\times(0,T))$.Comment: 21 page

Controle da podridão cinzenta da maçã por produtos naturais biologicamente ativos

Indexación: Web of Science; ScieloBiorend SC (chitosan), BC-1000 EC (grapefruit extract plus bioflavonoids) and ECO-100 SC (bioflavonoids plus organic acids, citric phytoalexins, fatty acids, glycerides and sugars), respectively, suppressed grey rot of apple caused by B. cinerea by 80.1%, 79.0% and 76.5% when used as post-harvest treatments under controlled conditions. When applied as combined pre- and post-harvest treatments Biorend SC inhibited fruit rot by 49.9 %, while BC-1000 EC and ECO-100 SC were ineffective. None of the products inhibited fruit rot when applied as pre-harvest treatments under controlled conditions or as post-harvest treatments under commercial conditions. The algal polysaccharide ulvan used in post-harvest treatments suppressed grey rot by 56.0% under controlled conditions, but had no inhibitory effect on combined pre- and post-harvest treatments. The inability of products to activate defense mechanisms (chitinase and peroxidase) of fruits was consistent with the unsuccessful control of rot by pre-harvest treatment. The results suggest that the natural products used have potential for use in integrated management of Botrytis rot when applied after harvest.Biorend SC (quitosana), BC-1000 EC (extrato de toranja mais flavonóides), e ECO-100 SC (bioflavonóides mais ácidos orgânicos, fitoalexinas cítricas, ácidos graxos glicerídeos e açúcares) inibiram em 80,1%, 79,0% e 76,5%, respectivamente, a podridão causada por Botrytis cinerea quando utilizados no tratamento pós-colheita de frutos de maçã sob condições controladas.Tratamento combinado de Biorend SC, com aplicação tanto em pré como no pós-colheita, proporcionou 49,9% de inibição da podridão, enquanto BC-1000 e ECO-100 EC não foram efetivos. Nenhum desses produtos inibiu a podridão cinzenta, quando utilizados em tratamento de pré-colheita em condições controladas ou em tratamento de pós-colheita em condições comerciais. O polissacarídeo algal ulvana, utilizado nos tratamentos de pós-colheita, reduziu em 56% a podridão cinzenta das maçãs em condições controladas, mas não teve efeito inibitório nos tratamentos combinados de pré e pós-colheita. A incapacidade dos produtos em ativar mecanismos de defesa (quitinases e peroxidases) nos frutos, após o tratamento em pré-colheita, foi consistente com a falta de controle da podridão nesse tipo de ensaio. Pelos resultados, sugere-se que os produtos naturais utilizados apresentam potencial para a utilização no manejo integrado da podridão de Botrytis quando aplicados em pós-colheita.http://ref.scielo.org/kcxb9