Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

Measuring^(222)Rn in aquatic environment via Pulsed Ionization Chamber Radon Detector

Radon(Rn)is a naturally occurring radioactive inert gas in nature,and^(222)Rn has been routinely used as a powerful tracer in various aquatic environmental research on timescales of hours to days,such as submarine groundwater discharge.Here we developed a new approach to measure^(222)Rn in discrete water samples with a wide range of^(222)Rn concentrations using a Pulsed Ionization Chamber(PIC)Radon Detector.The sensitivity of the new PIC system is evaluated at 6.06 counts per minute for 1 Bq/L when a 500 mL water sample volume is used.A robust logarithmic correlation between sample volumes,ranging from 250 mL to 5000 mL,and system sensitivity obtained in this study strongly suggests that this approach is suitable for measuring radon concentration levels in various natural waters.Compared to the currently available methods for measuring radon in grab samples,the PIC system is cheaper,easier to operate and does not require extra accessories(e.g.,drying tubes etc.)to maintain stable measurements throughout the counting procedure.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Air renewal times and ventilation rate calculations for underground workings using radioactive measurement

Potential alpha emitters are of prime concern to the ventilation engineer due to their rapid concentration increasing once radon released in the mine atmosphere, causing tissue irradiation and lung cancer.Studying of the time based variations of the natural ventilation in tunnels and their relationship to the external parameters contribute to the air circulation assessment. Due to the continuous and high fluctuation of the meteorological conditions affecting the air circulation and intensity through the underground workings, there is a difficulty in the natural ventilation assessment by only the ordinary meteorological measurements. So, in this paper, the possibility of using the radioactive measurements, allowing for the air aging and ventilation quality to be qualified, is investigated through three different underground structures. Referring to the most confined structure of them, results show that one structure has a better exchange rate by a factor 1.8 and the other has the best rate by a factor 2.1. This parameter can be linked to the operating costs and size of a future ventilation system.

DECOMPOSITION OF BV FUNCTIONS IN CARNOT-CARATHEODORY SPACES

The aim of this paper is to get the decomposition of distributional derivatives of functions with bounded variation in the framework of Carnot-Caratheodory spaces (C-C spaces in brievity) in which the vector fields are of Carnot type. For this purpose the approximate continuity of BV functions is discussed first, then approximate differentials of L1 functions are defined in the case that vector fields are of Carnot type and finally the decomposition Xu = (?)u ·Ln + X2 u is proved, where u ∈ BVx(?) and (Ω)u denotes the approximate differential of u.

A Class of Cauchy Problems for Porous Medium Equations with Strongly Nonlinear Sources and Convections

In this paper we study existence of solutions of a class of Cauchy problems for porous medium equations with strongly nonlinear sources or absorptions and convections when the initial trace is a Radon measure μ on RN.

Radonmeasurement in liquid scintillator with nitrogen stripping method

Purpose Liquid scintillator is the target material of neutrino in Jiangmen Underground Neutrino Observatory(JUNO)detector.it is necessary to control the radon at a very low level.Methods In this work,a device was set up to strip the radon from liquid scintillator with high-purity nitrogen bubbling,and the radon carrier nitrogen was collected into radon measurement chamber to measure radon concentration,then the stripping efficiency of radon in liquid scintillator can be measured.Results Under normal temperature and 2L/min gas flow rate,the stripping efficiency of radon in liquid scintillator is 3.85%.Conclusions The preliminary radon stripping results showthat the high-purity nitrogen stripping radon in the liquid scintillator has efficiency,and the device has the ability to measure radon concentration in the liquid scintillator,as well as the stripping device provides a basis for further low concentration radon measurement in the liquid scintillator.

ε-isometric Approximation Problem

In this paper, some problems for isometric approximation are discussed. It is shown that an almost isometry from the unit ball of C(X) into the unit ball of C(Y) is near to an isometry. We also give a counterexample to this problem on the unit ball of l1.

Potential Analysis on Carnot Groups,Part Ⅱ:Relationship between Hausdorff Measures and Capacities

In this paper,we establish the relationship between Hausdorff measures and Bessel capac- ities on any nilpotent stratified Lie group G (i.e.,Carnot group).In particular,as a special corollary of our much more general results,we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of G.Given any set E(?)G,B_(α,p)(E)=0 implies (?)^(Q-αp+(?))(E)=0 for all (?)>0.On the other hand,(?)^(Q-αp)(E)<∞ implies B_(α,p)(E)=0.Conse- quently,given any set E(?)G of Hausdorff dimension Q-d,where 0<d<Q,B_(α,p)(E)=0 holds if and only if αp(?)d. A version of O.Frostman's theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4).

Geochemical Characteristics and Origin of Nuanquanzi Geothermal Water in Yudaokou,Chengde,Hebei,North China

Study on the Nuanquanzi geothermal field in the Yanshan uplift is of great significance for understanding the origin of geothermal fluid in the intracontinental orogenic belt of the fault depression basin margin in North China.The geochemical characteristics and formation mechanism of the Nuanquanzi geothermal system were elucidated by classical hydrogeochemical analysis,multi-isotopes approach(δD,δ^(18)O,δ^(13)C,δ^(87)Sr/^(86)Sr),14CAMSdating,and integrated geophysical prospecting of surface-soil radon gas measurement and CSAMT inversion.The results show that the Nuanquanzi geothermal field is a medium-low temperature convection-fault semi-enclosed geothermal system.The hydrochemical type of thermal water is primarily HCO_(3)-Na,and rich in soluble SiO_(2),F^(-)and Cl^(-).The geothermal water primarily originated from the recharging meteoric water with a maximum circulation depth of 2400-3200 m,but affected by the mixing of endogenous sedimentary water.The reservoir temperature calculated by Na-K and quartz geothermometer of the Nuanquanzi geothermal system was determined to be 73.39-92.87℃.The conduction-cooling and shallow cold-water mixing processes occurred during the parent geothermal fluid ascent to surface,and the proportion of cold-water mixing during circulation was approximately 88.3%to 92.2%.The high-anomaly radon zones matched well to the low apparentresistance areas and hiding faults,indicating that the Nuanquanzi geothermal field was dominated by a graben basin restricted by multiple faults.

Steady Compressible Euler Equations of Concentration Layers for Hypersonic-limit Flows Passing Three-dimensional Bodies and Generalized Newton-Busemann Pressure Law

For stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space,the shock-front coincides with the upwind surface of the body,hence there is an infinite-thin layer of concentrated mass,in which all particles hitting the body move along its upwind surface.By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations,which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces,the authors derive the compressible Euler equations for flows in concentration layers,which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface.As a by-product,they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows,which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.

On the Kinetic Formulation of a Nonlinear Hyperbolic Variational Wave Equation

Motivated by [3], [4] and [5], we present the kinetic formulation of a nonlinear variational wave equation corresponding to some specific weak solutions . This equation arises from studies in nematic liquid crystals, long wave on a dipole chain and a few other fields.

A Note on Vanishing Mean Oscillation

A certain inequality for maximal operators is obtained. From this result, a necessary condition for a function f in VMO（R, dx） is generalized.

GENERALIZED BOCHNER THEOREM ON LOCALLY CONVEX SPACES

The purpose of this paper is to generalize the (classical) Bochner theorem to the case where Radon probability measures are defined on the weak dual spaces of locally convex spaces. We also compare our result with other topological descriptions of characteristic functionals of probability measures on other infinite dimensional spaces.