scholarly journals The Hardy–Morrey & Hardy–John–Nirenberg inequalities involving distance to the boundary

2016 ◽  
Vol 261 (6) ◽  
pp. 3107-3136 ◽  
Author(s):  
Stathis Filippas ◽  
Georgios Psaradakis
Keyword(s):  
Distance To The Boundary

Development of a directional resistivity logging-while-drilling tool using a joint-coil antenna and its geosteering applications

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. D165-D171
Author(s):  
Zhong Wang ◽  
Huaping Wang ◽  
Treston Davis ◽  
Jing Li ◽  
Suming Wu ◽  
...  
Keyword(s):  
Oil And Gas ◽  
Gas Production ◽  
Voltage Response ◽  
Drilling Tool ◽  
Detection Depth ◽  
Logging While Drilling ◽  
Resistivity Logging ◽  
Coil Antenna ◽  
Production Zone ◽  
Distance To The Boundary

Geosteering is a key technique to increase oil- and gas-production rates, especially within a thin reservoir layer. The purpose of geosteering in the production zone is to keep the drilling path in oil- and gas-bearing reservoirs. To keep the drilling system inside the production zone, downhole sensors must be able to detect bed boundaries, which include identifying the boundary location with respect to the sensor and the boundary distance from the sensor. We have developed a directional resistivity logging-while-drilling (LWD) tool for geosteering applications. The directional LWD tool is equipped with a joint-coil antenna composed of an axially polarized coil Rz connected in series with two transversely polarized coils Rx. During a revolution around the axis of the tool, the voltage of the axial coil VRz, voltage of the transverse coils VRx, and tool face angle [Formula: see text], which indicates the boundary direction, can be extracted through curve fitting the total voltage response of the joint-coil antenna. The distance to the boundary can be derived from a 1D inversion. The LWD tool has been tested in several reservoirs in China, and it has a demonstrated capability to provide reliable and accurate estimations of the boundary direction and distance. Field data indicate that the boundary detection depth can reach 2.1 and 1.7 m when the tool is in a sand and shale formation. Using wireline-logging data from surrounding wells as reference, deviations between the reference and the measured distance to the boundary are within 0.2 m.

A geometrical version of Hardy-Rellich type inequalities

2019 ◽  
Vol 69 (4) ◽  
pp. 785-800 ◽  
Author(s):  
Ramil Nasibullin
Keyword(s):  
Distance Function ◽  
Type Inequality ◽  
Second Order ◽  
Weight Functions ◽  
The Mean ◽  
Geometrical Version ◽  
Distance To The Boundary

Abstract We obtained a version of Hardy-Rellich type inequality in a domain Ω ∈ ℝn which involves the distance to the boundary, the diameter and the volume of Ω. Weight functions in the inequalities depend on the “mean-distance” function and on the distance function to the boundary of Ω. The proved inequalities connect function to first and second order derivatives.

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

Boundary layers of an anchored magnetic field in a highly conductive flow

2010 ◽  
Vol 466 (2123) ◽  
pp. 3153-3166 ◽  
Author(s):  
Manuel Núñez ◽  
Alberto Lastra
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Order Approximation ◽  
First Order ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
First Order Approximation ◽  
Distance To The Boundary

The effects of the flow of an electrically conducting fluid upon a magnetic field anchored at the boundary of a domain are studied. By taking the resistivity as a small parameter, the first-order approximation of an asymptotic analysis yields a boundary layer for the magnetic potential. This layer is analysed both in general and in three particular cases, showing that while in general its effects decrease exponentially with the distance to the boundary, several additional effects are highly relevant.

scholarly journals Sobolev and Lipschitz estimates for weighted Bergman projections

1997 ◽  
Vol 147 ◽  
pp. 147-178 ◽  
Author(s):  
Der-Chen Chang ◽  
Bao Qin Li
Keyword(s):  
Finite Type ◽  
Convex Domain ◽  
Functional Calculus ◽  
Smooth Boundary ◽  
Lipschitz Estimates ◽  
Pseudo Convex Domain ◽  
Pseudo Convex ◽  
Bergman Projections ◽  
Distance To The Boundary

AbstractLet Ω be a bounded, decoupled pseudo-convex domain of finite type in ℂn with smooth boundary. In this paper, we generalize results of Bonami-Grellier [BG] and Bonami-Chang-Grellier [BCG] to study weighted Bergman projections for weights which are a power of the distance to the boundary. We define a class of operators of Bergman type for which we develop a functional calculus. Then we may obtain Sobolev and Lipschitz estimates, both of isotropic and anisotropic type, for these projections.

scholarly journals An Upper Limit for Birefringence in the Boundary Layer of a Heat Conducting Gas

1985 ◽  
Vol 40 (2) ◽  
pp. 164-168
Author(s):  
H. van Houten ◽  
W. A. von Marinelli ◽  
J. J. M. Beenakker
Keyword(s):  
Boundary Layer ◽  
Angular Momentum ◽  
Boundary Layers ◽  
Qualitative Data ◽  
Heat Conducting ◽  
Polyatomic Gases ◽  
Upper Limit ◽  
Set Up ◽  
Distance To The Boundary

The Waldmann-Vestner theory for boundary layer effects in polyatomic gases in the nearhydrodynamic regime predicts that in a rarefield heat conducting gas the boundary layers will be birefringent due to the presence of second rank angular momentum polarization. An experiment has been set up in order to measure the birefringence as a function of the distance to the boundary. However, no effect could be detected for N2 and C02 within the sensitivity of the experiment. This corresponds to an upper limit for the birefringence, which is a factor 6 smaller than expected. These results are in contrast to the qualitative data reported earlier in the literature; no conclusions can be drawn, however, about the validity of the Waldmann-Vestner theory.

scholarly journals The Influence of Stratification and Nonlocal Turbulent Production on Estuarine Turbulence: An Assessment of Turbulence Closure with Field Observations

2011 ◽  
Vol 41 (1) ◽  
pp. 166-185 ◽  
Author(s):  
Malcolm E. Scully ◽  
W. Rocky Geyer ◽  
John H. Trowbridge
Keyword(s):  
Richardson Number ◽  
Turbulence Closure ◽  
Length Scale ◽  
Field Observations ◽  
Turbulent Length Scale ◽  
Stability Functions ◽  
Gradient Richardson Number ◽  
Simple Scaling ◽  
The Impact ◽  
Distance To The Boundary

Abstract Field observations of turbulent kinetic energy (TKE), dissipation rate ɛ, and turbulent length scale demonstrate the impact of both density stratification and nonlocal turbulent production on turbulent momentum flux. The data were collected in a highly stratified salt wedge estuary using the Mobile Array for Sensing Turbulence (MAST). Estimates of the dominant length scale of turbulent motions obtained from the vertical velocity spectra provide field confirmation of the theoretical limitation imposed by either the distance to the boundary or the Ozmidov scale, whichever is smaller. Under boundary-limited conditions, anisotropy generally increases with increasing shear and decreased distance to the boundary. Under Ozmidov-limited conditions, anisotropy increases rapidly when the gradient Richardson number exceeds 0.25. Both boundary-limited and Ozmidov-limited conditions demonstrate significant deviations from a local production–dissipation balance that are largely consistent with simple scaling relationships for the vertical divergence in TKE flux. Both the impact of stratification and deviation from equilibrium turbulence observed in the data are largely consistent with commonly used turbulence closure models that employ “nonequilibrium” stability functions. The data compare most favorably with the nonequilibrium version of the L. H. Kantha and C. A. Clayson stability functions. Not only is this approach more consistent with the observed critical gradient Richardson number of 0.25, but it also accounts for the large deviations from equilibrium turbulence in a manner consistent with the observations.

Dielectric conductivity of a bounded plasma and its rate of convergence towards its infinite-geometry value

2003 ◽  
Vol 69 (5) ◽  
pp. 449-463
Author(s):  
PASCAL OMNES
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
Rate Of Convergence ◽  
Dielectric Function ◽  
Bounded Geometry ◽  
First Order ◽  
First Order Correction ◽  
Time Periodic ◽  
Special Case ◽  
Distance To The Boundary

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known and independent of time. Then, a first-order Taylor expansion in the velocity variable is performed, thanks to an approximation of the exact characteristics by the unperturbed ones. The resulting first-order correction to the distribution function is finally integrated over velocities to yield the dielectric function. The special case of a plane wave for the electric field is examined and the results are compared with the more usual unbounded case: the integral does not present any singularity in the vicinity of resonant particles and the dielectric function depends on the distance to the boundary and tends to the usual infinite-geometry value when this distance tends to infinity, with a rate of convergence proportional to its inverse square root. Numerical examples are provided for illustration.

scholarly journals SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

2010 ◽  
Vol 20 (01) ◽  
pp. 95-120 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
FERNANDO LÓPEZ GARCÍA
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Basic Result ◽  
Weighted Sobolev Spaces ◽  
Mean Value ◽  
Well Posedness ◽  
Existence Of A Solution ◽  
Simply Connected ◽  
Distance To The Boundary

If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution [Formula: see text] for some r < 2 depending on the power of the cusp.

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