BV Estimates Research Articles

BV estimates on the transport density with Dirichlet region on the boundary

In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimensions under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is W1,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p}$$\\end{document}, for some p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}.

JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L^p and L^\infty norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher-order results, and in particular to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called “five gradients inequality” allows us to recover some estimates that can be deduced from the Bakry–Émery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller–Segel chemotaxis model.

Selection–mutation dynamics with asymmetrical reproduction kernels

We study a family of selection–mutation models of a sexual population structured by a phenotypical trait. The main feature of these models is the asymmetric trait heredity or fecundity between the parents: we assume that each individual inherits mostly its trait from the female or that the trait acts on the female fecundity but does not affect male. Following previous works inspired from principles of adaptive dynamics, we rescale time and assume that mutations have limited effects on the phenotype. Our goal is to study the asymptotic behavior of the population distribution. We derive non-extinction conditions and BV estimates on the total population. We also obtain Lipschitz estimates on the solutions of Hamilton–Jacobi equations that arise from the study of the population distribution concentration at fittest traits. Concentration results are obtained in some special cases by using a Lyapunov functional.

Critical review and meta-analysis of biological variation estimates for tumor markers.

Biological variation data (BV) can be used for different applications, but this depends on the availability of robust and relevant BV data. In this study, we aimed to summarize and appraise BV studies for tumor markers, to examine the influence of study population characteristics and concentrations on BV estimates and to discuss the applicability of BV data for tumor markers in clinical practice. Studies reporting BV data for tumor markers related to gastrointestinal, prostate, breast, ovarian, haematological, lung, and dermatological cancers were identified by a systematic literature search. Relevant studies were evaluated by the Biological Variation Data Critical Appraisal Checklist (BIVAC) and meta-analyses were performed for BIVAC compliant studies to deliver global estimates of within-subject (CVI) and between-subject (CVG) BV with 95% CI. The systematic review identified 49 studies delivering results for 22 tumor markers; four papers received BIVAC grade A, 3 B, 27 C and 15 D. Out of these, 29 CVI and 29 CVG estimates met the criteria to be included in the meta-analysis. Robust data are lacking to conclude on the relationship between BV and different disease states and tumor marker concentrations. This review identifies a lack of high-quality BV studies for many tumor markers and a need for delivery of BIVAC compliant studies, including in different disease states and tumor marker concentrations. As of yet, the state-of-the-art may still be the most appropriate model to establish analytical performance specifications for the majority of tumor markers.

Nonlocal scalar conservation laws with discontinuous flux

<abstract><p>We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with nonlocal flux. We approximate the problem adding a viscosity term and we provide $ {{\bf{L}}^\infty} $ and BV estimates for the approximate solutions. We use the doubling of variable technique to prove the stability with respect to the initial data from the entropy condition.</p></abstract>

Galaxy velocity bias in cosmological simulations: towards per cent-level calibration

ABSTRACT Galaxy cluster masses, rich with cosmological information, can be estimated from internal dark matter (DM) velocity dispersions, which in turn can be observationally inferred from satellite galaxy velocities. However, galaxies are biased tracers of the DM, and the bias can vary over host halo and galaxy properties as well as time. We precisely calibrate the velocity bias, bv – defined as the ratio of galaxy and DM velocity dispersions – as a function of redshift, host halo mass, and galaxy stellar mass threshold ($M_{\rm \star , sat}$), for massive haloes ($M_{\rm 200c}\gt 10^{13.5} \, {\rm M}_\odot$) from five cosmological simulations: IllustrisTNG, Magneticum, Bahamas + Macsis, The Three Hundred Project, and MultiDark Planck-2. We first compare scaling relations for galaxy and DM velocity dispersion across simulations; the former is estimated using a new ensemble velocity likelihood method that is unbiased for low galaxy counts per halo, while the latter uses a local linear regression. The simulations show consistent trends of bv increasing with M200c and decreasing with redshift and $M_{\rm \star , sat}$. The ensemble-estimated theoretical uncertainty in bv is 2–3 per cent, but becomes percent-level when considering only the three highest resolution simulations. We update the mass–richness normalization for an SDSS redMaPPer cluster sample, and find our improved bv estimates reduce the normalization uncertainty from 22 to 8 per cent, demonstrating that dynamical mass estimation is competitive with weak lensing mass estimation. We discuss necessary steps for further improving this precision. Our estimates for $b_v(M_{\rm 200c}, M_{\rm \star , sat}, z)$ are made publicly available.

The European Biological Variation Study (EuBIVAS): a summary report.

Biological variation (BV) data have many important applications in laboratory medicine. Concerns about quality of published BV data led the European Federation of Clinical Chemistry and Laboratory Medicine (EFLM) 1st Strategic Conference to indicate need for new studies to generate BV estimates of required quality. In response, the EFLM Working Group on BV delivered themulticenter European Biological Variation Study (EuBIVAS). This review summarises the EuBIVAS and its outcomes. Serum/plasma samples were taken from 91 ostensibly healthy individuals for 10 consecutive weeks at 6 European centres. Analysis was performed by Siemens ADVIA 2400 (clinical chemistry), Cobas Roche 8000, c702 and e801 (proteins and tumor markers/hormones respectively), ACL Top 750 (coagulation parameters), and IDS iSYS or DiaSorin Liaison (bone biomarkers). A strict preanalytical and analytical protocol was applied. To determine BV estimates with 95% CI, CV-ANOVA after analysis of outliers, homogeneity and trend analysis or a Bayesian model was applied. EuBIVAS has so far delivered BV estimates for 80 different measurands. Estimates for 10measurands (non-HDL cholesterol, S100-β protein, neuron-specific enolase, soluble transferrin receptor, intact fibroblast growth-factor-23, uncarboxylated-unphosphorylated matrix-Gla protein, human epididymis protein-4, free, conjugated and %free prostate-specific antigen), prior to EuBIVAS, have not been available. BV data for creatinine and troponin I were obtained using two analytical methods in each case. The EuBIVAS has delivered high-quality BV data for a wide range of measurands. The BV estimates are for many measurands lower than those previously reported, having an impact on the derived analytical performance specifications and reference change values.

Nonlocal approaches for multilane traffic models

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe the lane changing rate. The modelling framework includes the consideration of local and nonlocal flux functions. Based on a Godunov type numerical scheme, we provide BV estimates and a discrete entropy inequality. Together with the $L^1$-contractivity property, we prove existence and uniqueness of weak solutions. Numerical examples show the nonlocal impact compared to local flux functions and local sources.

BV estimates on Lax-Friedrichs' scheme for a radiating gas model with nonlinear radiative inhomogeneity

In this paper, we obtain BV estimates and L1 stability of solutions to a radiating gas model with nonlinear radiative inhomogeneity by using Lax-Friedrichs' scheme when a modified C-F-L (Courant-Friedrichs-Levy) condition is satisfied. Our result is a natural extension to that of the radiating gas model with linear radiative inhomogeneity so that it provides an understanding on the balance between nonlinear convection and nonlinear radiative inhomogeneity. The method for case of linear radiative inhomogeneity to obtain the key estimate on uniform bound does not work for the case of nonlinear radiative inhomogeneity. The difficulty is overcome by employing a simple but effective monotonic technic.

Micro-Macro Limit of a Nonlocal Generalized Aw-Rascle Type Model

We introduce a Follow-the-Leader approximation of a non-local generalized Aw-Rascle-Zhang (GARZ) model for traffic flow. We prove the convergence to weak solutions of the corresponding macroscopic equations deriving $L ∞$ and BV estimates. We also provide numerical simulations illustrating the micro-macro convergence and we investigate numerically the non-local to local limit for both the microscopic and macroscopic models.

Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-B{\'e}nilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound.

Stable s-minimal cones in ℝ3 are flat for s ~ 1

Abstract We prove that half spaces are the only stable nonlocal s-minimal cones in ℝ 3 {\mathbb{R}^{3}} , for s ∈ ( 0 , 1 ) {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s = 1 {s=1} . In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour

In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H1 estimates, L1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in Lloc1 towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We further provide a specific example of non-uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.

Non-local multi-class traffic flow models

We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform L ∞ and BV estimates for the sequence of approximate solutions. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition.

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.

Harmonization initiatives in the generation, reporting and application of biological variation data.

Abstract Biological variation (BV) data have many applications in laboratory medicine. However, concern has been raised that some BV estimates in use today may be irrelevant or of unacceptable quality. A number of initiatives have been launched by the European Federation of Clinical Chemistry and Laboratory Medicine (EFLM) and other parties to deliver a more harmonized practice in the generation, reporting and application of BV data. Resulting from a necessary focus upon the veracity of historical BV studies, critical appraisal and meta-analysis of published BV studies is possible through application of the Biological Variation Data Critical Appraisal Checklist (BIVAC), published in 2017. The BIVAC compliant large-scale European Biological Variation Study delivers updated high-quality BV data for a wide range of measurands. Other significant developments include the publication of a Medical Subject Heading term for BV and recommendations for common terminology for reporting of BV data. In the near future, global BV estimates derived from meta-analysis of BIVAC appraised publications will be accessible in a Biological Variation Database at the EFLM website. The availability of these high-quality data, which have many applications that impact on the quality and interpretation of clinical laboratory results, will afford improved patient care.

Existence of global [formula omitted] solutions to a system of conservation laws in chromatography with geometry effects

In this paper, we give the global existence of Lp bounded entropy solutions for the Cauchy problem of a system of conservation laws in chromatography with geometry effects. The main difficulty lies in establishing the Lp estimate of the viscosity solutions because the initial data may not be L∞ bounded and the principle of invariant regions developed by Conley–Chuey–Smoller is invalid here. We obtain the existence of the global weak solutions using the compensated compactness method and BV estimates on viscosity solutions.

Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel

We prove the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem by a Lax-Friedrichs scheme and we provide L∞ and BV estimates for the sequence of approximate solutions. Stability with respect to the initial data is obtained from the entropy condition through the doubling of variable technique. The limit model as the kernel support tends to infinity is also studied.