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We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

We consider the motion of a large number of heavy particles in a Newtonian fluid occupying a bounded spatial domain. When we say "heavy", we mean a particle with a mass density that approaches infinity at an appropriate rate as its radius vanishes. We show that the collective effect of heavy particles on the fluid motion is similar to the Brinkman perturbation of the Navier-Stokes system identified in the homogenization process.

In this paper, we study the global solvability of the density-dependent incompressible Euler equations, supplemented with a damping term of the form $ \mathfrak{D}_{\alpha}^{\gamma}(\rho, u) = \alpha \rho^{\gamma} u $, where $\alpha>0$ and $ \gamma \in \{0,1\} $. To some extent, this system can be seen as a simplified model describing the mean dynamics in the ocean; from this perspective, the damping term can be interpreted as a term encoding the effects of the celebrated Ekman pumping in the system. On the one hand, in the general case of space dimension $d\geq 2$, we establish global well-posedness in the Besov spaces framework, under a non-linear smallness condition involving the size of the initial velocity field $u_0$, of the initial non-homogeneity $\rho_0-1$ and of the damping coefficient $\alpha$. On the other hand, in the specific situation of planar motions and damping term with $\gamma=1$, we exhibit a second smallness condition implying global existence, which in particular yields global well-posedness for arbitrarily large initial velocity fields, provided the initial density variations $\rho_0-1$ are small enough. The formulated smallness conditions rely only on the endpoint Besov norm $B^1_{\infty,1}$ of the initial datum, whereas, as a byproduct of our analysis, we derive exponential decay of the velocity field and of the pressure gradient in the high regularity norms $B^s_{p,r}$., Comment: Submitted

X-ray dark-field or ultra-small angle scatter imaging has become increasingly important since the introduction of phase-based x-ray imaging and is having transformative impact in fields such as in vivo lung imaging and explosives detection. Here we show that dark-field images acquired with the edge-illumination method (either in its traditional double mask or simplified single mask implementation) provide a direct measurement of the scattering function, which is unaffected by system-specific parameters such as the autocorrelation length. We show that this is a consequence both of the specific measurement setup and of the mathematical approach followed to retrieve the dark-field images. We show agreement with theoretical models for datasets acquired both with synchrotron and laboratory x-ray sources. We also introduce a new contrast mechanism, the variance of refraction, which is extracted from the same dataset and provides a direct link with the size of the scattering centres. We show that this can also be described by the same theoretical models. We study the behaviour of both signals vs. key parameters such as x-ray energy and scatterer radius. We find this allows quantitative, direct, multi-scale scattering measurements during imaging, with implications in all fields where dark-field imaging is used., Comment: 18 pages, 5 main figures, 6 extended data figures

The Target Absorbers for Neutrals (TANs) represent one of the most radioactive regions in the Large Hadron Collider (LHC). Seven 40 cm long fused silica rods with different dopant specifications, manufactured by Heraeus, were irradiated in one of the TANs located around the ATLAS experiment by the Beam RAte of Neutrals (BRAN) detector group. This campaign took place during the Run 2 p+p data taking, which occurred between 2016 and 2018. This paper reports a complete characterization of optical transmission per unit length of irradiated fused silica materials as a function of wavelength (240 nm - 1500 nm), dose (up to 18 MGy), and level of OH and H$_2$ dopants introduced in the manufacturing process. The dose delivered to the rods was estimated using Monte Carlo simulations performed by the CERN FLUKA team., Comment: 29 pages, 15 figures, to be submitted to NIM-A

In this paper we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when the its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the case where the density of the small rigid body is uniformly bounded respect to its size.

In this paper we study the evolution of a small rigid body in a viscous incompressible fluid, in particular we show that a small particle is not accelerated by the fluid in the limit when its size converges to zero under a lower bound on its mass. This result is based on a new a priori estimate on the velocities of the centers of mass of rigid bodies that holds in the case when their masses are also allowed to decrease to zero.

In this paper we highlight a set of ad hoc test functions to study the homogenization of viscous compressible fluid in domains with tiny holes. This set of functions allows to improve previous results in dimensions two and three. As an application we show that the presence of a small obstacle does not influence the dynamics of a viscous compressible fluid in dimension two.

In this paper, we perform the fast rotation limit $\varepsilon\rightarrow0^+$ of the density-dependent incompressible Navier-Stokes-Coriolis system in a thin strip $\Omega_\varepsilon\,:=\,\mathbb{R}^2\times\,]-\ell_\varepsilon,\ell_\varepsilon[\,$, where $\varepsilon\in\,]0,1]$ is the size of the Rossby number and $\ell_\varepsilon>0$ for any $\varepsilon>0$. By letting $\ell_\varepsilon\longrightarrow0^+$ for $\varepsilon\rightarrow0^+$ and considering Navier-slip boundary conditions at the boundary of $\Omega_\varepsilon$, we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a $3$-D setting. In this respect, we remark that the case $\ell_\varepsilon\geq\ell>0$ for all $\varepsilon>0$ remains largely open at present., Comment: Submitted

The still unmet ability of following the fate of specific cells or molecules within the whole body would give an outstanding breakthrough in the comprehension of disease mechanisms and in the monitoring of therapeutic approaches. Our idea is to push forward the bio-nanotechnology to a level where it serves the most advanced 3D bio-medical imaging to provide a multi-scale imaging ranging from the whole organ down to the cellular level, enabling high-resolution visualization of disease-relevant cells within the whole disease-altered biological context. We present here the first proof-of-concept of a novel tomography procedure, Immuno-Histo-X-ray Phase Contrast Tomography (XPCT) that combines cutting-edge XPCT, which provides detailed image of the whole organ, with molecular imaging at the cellular level, identifying the relevant cells via an immunohistochemistry-based approach. We combine metal nanoparticles and a single-domain antibody that target relevant cells. Our results lay the foundation for a new generation of 3D-bio-medical X-ray imaging., Comment: 11 pages 5 figures

The Target Absorbers for Neutrals (TANs) are located in a high-intensity radiation environment inside the tunnel of the Large Hadron Collider (LHC). TANs are positioned about $140$ m downstream from the beam interaction points. Seven $40$ cm long fused silica rods with different dopant specifications were irradiated in the TAN by the Beam RAte of Neutrals (BRAN) detector group during $p$+$p$ data taking from 2016 to 2018 at the LHC. The peak dose delivered to the fused silica rods was $18$ MGy. We report measurements of the $^{22}$Na activation of the fused silica rods carried out at the University of Illinois at Urbana-Champaign and Argonne National Laboratory. At the end of the irradiation campaign, the maximum $^{22}$Na activity observed was $A=21$ kBq$/{\rm cm^3}$ corresponding to a density, $\rho= 2.5\times 10^{12} /{\rm cm^3}$, of $^{22}$Na nuclei. FLUKA Monte Carlo simulations have been performed by the CERN FLUKA team to estimate $^{22}$Na activities for the irradiated BRAN rod samples. The simulations reproduce the $^{22}$Na activity profile measured along the rods, with a 35% underestimation of the experimental measurement results., Comment: 11 pages, 14 figures, to be submitted to PRAB

Well-posedness for the two dimensional Euler system with given initial vorticity is known since the works of Judovi\v{c}. In this paper we show existence of solutions in the case where we allowed the fluid to enter in and exit from the boundaries and from some points of the fluid domain. In particular we derive the equations of the model as the limit when we replace the points by some small holes. To do that we extend the DiPerna-Lions theory with non-tangent velocity field on the boundary to the case of time-dependent domain, we extend the existence result for the two dimensional Euler system with in-out flow to time-dependent domain and finally we derive the system that models a fluid which is allowed to enter in and exit from the boundary and some points. The solutions are characterized by the presence of source, sink and vortex points.

In this note we study the initial value problem in a critical space for the one dimensional Schr\"odinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper we investigate the existence of weak solutions to this system by relying on \textit{a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p >\frac43$, in the renormalized sense in the case where $p >1$, and in a symmetrized sense in the case where $p =1$.

In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on the influence of a vanishing obstacle in a compressible fluid for $ \gamma \geq 6$ .

The performance and safe operation of a particle accelerator is closely connected to the transverse emittance of the beams it produces. For this reason many techniques have been developed over the years for monitoring the transverse distribution of particles along accelerator chains or over machine cycles. The definition of beam profiles is explained and the different techniques available for the detection of the particle distributions are explored. Examples of concrete applications of these techniques are given., Comment: 37 pages, 53 figures

This chapter defines the concept of transverse emittance and describes the techniques most frequently used to measure it., Comment: 14 pages, contribution to the CAS - CERN Accelerator School: Beam Instrumentation, 2-15 June 2018, Tuusula, Finland