arXiv subjects

David Meyer

Publications and source records attributed to David Meyer.

At least 19 recordsLinked to original sources

arXiv2026

A Uniqueness theorem for the Vortex-Wave system with non-constant vorticity near the point vortex

We study the uniqueness of the vortex wave system, i.e.\ the 2D incompressible Euler equations with vorticity consisting of a point vortex and a bounded part. We show that if the initial vorticity and point vortex location $(w_0,X_0)$ satisfy a constraint of the type $|w_0(x)-w_0(X_0)|=O(|x-X_0|^\alpha)$ for $\alpha> 2$ and the vorticity is in $L^\infty$, then solutions are unique, globally in time, even though the velocity is not log-Lipschitz.

math.AP↗

arXiv2026

Deep Holes in the Clifford Hierarchy

We determine the covering radius of the topological closure of the single-qubit Clifford hierarchy in $\SU(2)\cong S^3$. This closure is a union of $18$ great circles --- the Clifford--Pauli circles --- and we prove that its covering radius is $\arccos\sqrt{5/6}$. The extremal points, which we call \emph{deep holes}, form a single orbit of size $192$ under left and right multiplication by Clifford gates, and are described in closed form. Equivalently, the minimum over one-qubit unitaries of the all-level Clifford fidelity is $5/6$. The proof rests on two structures attached to the configuration of $18$ planes in $\R^4$: their centered rank-two projectors form an orthonormal basis of the irreducible $\SO(4)$-module $\Sym_0(4)$, and the projection profile of a unit quaternion is exactly its image under the double cover $\SU(2)\to\SO(3)$. These reduce the covering problem to a minimax statement for the $\ell^\infty$-norm on $\SO(3)$ which we solve exactly, classifying its equality cases.

quant-ph↗

arXiv2026

Vortex-sheet desingularization for three-dimensional ideal fluids

We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to 0 as $\varepsilon \to 0$. We show that, as $\varepsilon \to 0$, these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of $ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.

math.AP↗

arXiv2025

On the attainment of boundary data in variational problems with linear growth

It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in $BV$ or $W^{\alpha,p}$ with $\alpha p\geq 2$ without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.

math.AP↗

arXiv2025

Motion of a massive rigid loop in a 3D perfect incompressible flow

We consider the motion of a rigid body immersed in an inviscid incompressible fluid. In 2D, an important physical effect associated with this system is the famous Kutta-Joukowski effect. In the present paper, we identify a similar effect in the 3D case. For this, we first recast the Newtonian dynamics of the rigid body as a first-order nonlinear ODE for the $6$-component body velocity, in the body frame. Then, we focus on the particular case where the rigid body occupies a slender tubular domain with a smooth closed curve as the centerline and a circular cross-section, in the limit where the radius goes to zero, with fixed inertia and circulation around the curve. We establish that the dynamics of the limit massive rigid loop are given by a first-order nonlinear ODE with coefficients that depend only on the inertia, on the fluid vorticity, and on the limit curve through two $3$D vectors, which are involved in a skewsymmetric $6 \times 6$ matrix that appears in the limit force and torque, a structure which is reminiscent of the 2D Kutta-Joukowski effect. We also identify the limit fluid dynamics as, where, as in the case of the Euler equation alone, the vorticity evolves according to the usual transport equation with stretching, but with a velocity field that is due not only to the fluid vorticity but also to a vorticity filament associated with the circulation around the limit rigid loop. This result is in stark contrast with the case where the filament is made of fluid, with non-zero circulation, since in the latter, the filament velocity becomes infinite in the zero-radius limit. However, considering the inertia scaling that corresponds to a fixed density, we prove that there are solutions for which the solid velocity and its displacement tend to infinity over a time interval of size $\mathcal{O}(1)$.

math.AP↗

arXiv2025

Long time confinement of multiple concentrated vortices

We study the stability of multiple almost circular concentrated vortices in a fluid evolving according to the two-dimensional Euler equations. We show that, for general configurations, they must remain concentrated on time-scales much longer than previously known as long as they remain separated. We further prove a new stability estimate for the logarithmic interaction energy as part of the proof.

math.AP↗

arXiv2025

Desingularization of vortex sheets for the 2D Euler equations

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon \in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $\omega_\varepsilon(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff-Rott system with initial datum $\omega^0_{\mathrm{sing}}$. The regularization relies on a layer construction designed to exploit the key observation that the Kelvin-Helmholtz instability has a strongly anisotropic effect: while vorticities must be analytic in the "tangential" direction, the way layers can be arranged in the "normal" direction is essentially arbitrary.

math.AP↗

arXiv2025

Steady bubbles and drops in inviscid fluids

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill's vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall-Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill's spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.

math.AP↗

arXiv2025

Open-Source Manually Annotated Vocal Tract Database for Automatic Segmentation from 3D MRI Using Deep Learning: Benchmarking 2D and 3D Convolutional and Transformer Networks

Accurate segmentation of the vocal tract from magnetic resonance imaging (MRI) data is essential for various voice and speech applications. Manual segmentation is time intensive and susceptible to errors. This study aimed to evaluate the efficacy of deep learning algorithms for automatic vocal tract segmentation from 3D MRI.

cs.CV↗

arXiv2024

Steady Ring-Shaped Vortex Sheets

In this work, we construct traveling wave solutions to the two-phase Euler equations, featuring a vortex sheet at the interface between the two phases. The inner phase exhibits a uniform vorticity distribution and may represent a vacuum, forming what is known as a hollow vortex. These traveling waves take the form of ring-shaped vortices with a small cross-sectional radius, referred to as thin rings. Our construction is based on the implicit function theorem, which also guarantees local uniqueness of the solutions. Additionally, we derive asymptotics for the speed of the ring, generalizing the well-known Kelvin--Hicks formula to cases that include surface tension.

math.AP↗

arXiv2024

Optimal Regularity for the 2D Euler Equations in the Yudovich class

We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with BMO gradient. As an application, we study the 2D Euler equations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variation of Bahouri and Chemin's vortex patch example.

math.AP↗

arXiv2023

Total $\mathbb{A}$-variation flows

We study the $L^2$-gradient flows, $\partial_t u-\mathrm{div}(\mathrm{D}f(x,\mathbb{A}u))=0$, of functionals of the type $\int_{\Omega}f(x,\mathbb{A}u)\,\mathrm{d}x$, where $f$ is a convex function of linear growth and $\mathbb{A}$ is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space $\mathrm{BV}^{\mathbb{A}}\cap L^2$, identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with $p$-growth as $p\searrow 1$.

math.AP↗

arXiv2023

Energy and environmental impacts of air-to-air heat pumps in a mid-latitude city

Heat pumps (HPs) have emerged as a key technology for reducing energy use and greenhouse gas emissions. This study evaluates the potential switch to air-to-air HPs (AAHPs) in Toulouse, France, where conventional space heating is split between electric and gas sources. In this context, we find that AAHPs reduce heating energy consumption by 57% to 76%, with electric heating energy consumption decreasing by 6% to 47%, resulting in virtually no local heating-related CO$_{2}$ emissions. We observe a slight reduction in near-surface air temperature of up to 0.5 {\deg}C during cold spells, attributable to a reduction in sensible heat flux, which is unlikely to compromise AAHPs operational efficiency. While Toulouse's heating energy mix facilitates large energy savings, electric energy consumption may increase in cities where gas or other fossil fuel sources prevail. Furthermore, as AAHPs efficiency varies with internal and external conditions, their impact on the electrical grid is more complex than conventional heating systems. The results underscore the importance of matching heating system transitions with sustainable electricity generation to maximize environmental benefits. The study highlights the intricate balance between technological advancements in heating and their broader environmental and policy implications, offering key insights for urban energy policy and sustainability efforts.

physics.soc-ph↗

arXiv2023

A model for the approximation of vortex rings by almost rigid bodies

We consider a model that approximates vortex rings in the axisymmetric 3D Euler equation by the movement of almost rigid bodies described by Newtonian mechanics. We assume that the bodies have a circular cross-section and that the fluid is irrotational and interacts with the bodies through the pressure exerted at the boundary. We show that this kind of system can be described through an ODE in the positions of the bodies and that in the limit, where the bodies shrink to massless filaments, the system converges to an ODE system similar to the point vortex system. In particular, we can show that in a suitable set-up, the bodies perform a leapfrogging motion.

math.AP↗

arXiv2022

Prospectively accelerated dynamic speech MRI at 3 Tesla using a self-navigated spiral based manifold regularized scheme

This work proposes a self-navigated variable density spiral(VDS) based manifold regularization scheme to prospectively improve dynamic speech MRI at 3T. Short readout 1.3ms spirals were used to minimize off-resonance. A custom 16-channel speech coil was used for improved parallel imaging of vocal tract. The manifold model leveraged similarities between frames sharing similar speech postures without explicit motion binning. The self-navigating capability of VDS was leveraged to learn the Laplacian matrix of the manifold. Reconstruction was posed as a SENSE-based non-local soft weighted temporal regularization scheme. Our approach was compared against view-sharing, low-rank, finite difference, extra-dimension-based sparsity reconstruction constraints. Under-sampling experiments were conducted on five volunteers performing repetitive and arbitrary speaking tasks at different speaking rates. Quantitative evaluation in terms of mean square error over moving edges were performed in a retrospectively under-sampled data. For prospective under-sampling, blinded image quality evaluation in the categories of alias artifacts, spatial blurring, and temporal blurring were performed by three voice research experts. Region of interest analysis at articulator boundaries were performed to assess articulatory motion. Our scheme provided improved reconstruction over the others. With prospective under-sampling, a spatial resolution of 2.4mm2/pixel and a temporal resolution of 17.4 ms/frame for single slice imaging, and 52.2 ms/frame for 3-slice imaging were achieved. We demonstrated implicit motion binning by analyzing the mechanics of the Laplacian matrix. Our method demonstrated superior image quality scores in reducing spatial and temporal blurring. While it exhibited faint alias artifacts similar to temporal finite-difference, it provided statistically significant improvements over remaining constraints.

eess.IV↗

arXiv2022

Propagation of regularity for transport equations. A Littlewood-Paley approach

It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We propose a new approach to the study of regularity that is based on Littlewood-Paley theory, thus measuring regularity in terms of Besov norms. We recover the results that are available in the literature and extend these optimally to the diffusive setting. As a consequence, we derive sharp bounds on rates of convergence in the zero-diffusivity limit.

math.AP↗

arXiv2022

Machine Learning Emulation of 3D Cloud Radiative Effects

The treatment of cloud structure in numerical weather and climate models is often greatly simplified to make them computationally affordable. Here we propose to correct the European Centre for Medium-Range Weather Forecasts 1D radiation scheme ecRad for 3D cloud effects using computationally cheap neural networks. 3D cloud effects are learned as the difference between ecRad's fast 1D Tripleclouds solver that neglects them and its 3D SPARTACUS (SPeedy Algorithm for Radiative TrAnsfer through CloUd Sides) solver that includes them but is about five times more computationally expensive. With typical errors between 20 % and 30 % of the 3D signal, neural networks improve Tripleclouds' accuracy for about 1 % increase in runtime. Thus, rather than emulating the whole of SPARTACUS, we keep Tripleclouds unchanged for cloud-free parts of the atmosphere and 3D-correct it elsewhere. The focus on the comparably small 3D correction instead of the entire signal allows us to improve predictions significantly if we assume a similar signal-to-noise ratio for both.

cs.LG↗

arXiv2022

Machine Learning Emulation of Urban Land Surface Processes

Can we improve the modeling of urban land surface processes with machine learning (ML)? A prior comparison of urban land surface models (ULSMs) found that no single model is 'best' at predicting all common surface fluxes. Here, we develop an urban neural network (UNN) trained on the mean predicted fluxes from 22 ULSMs at one site. The UNN emulates the mean output of ULSMs accurately. When compared to a reference ULSM (Town Energy Balance; TEB), the UNN has greater accuracy relative to flux observations, less computational cost, and requires fewer input parameters. When coupled to the Weather Research Forecasting (WRF) model using TensorFlow bindings, WRF-UNN is stable and more accurate than the reference WRF-TEB. Although the application is currently constrained by the training data (1 site), we show a novel approach to improve the modeling of surface fluxes by combining the strengths of several ULSMs into one using ML.

cs.LG↗