Existence of a Hölder Continuous Extension on Embedded Balls of the 3-Torus for the Periodic Navier Stokes Equations

This article gives a general model using specific periodic special functions, which is degenerate elliptic Weierstrass P functions whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of cells of the 3-Torus. Satisfying a divergence-free vector field and periodic boundary conditions respectively with a general spatio-temporal forcing term which is smooth and spatially periodic, the existence of solutions which have finite time singularities can occur starting with the first derivative and higher with respect to time. The existence of a subspace of the solution space where v3 is continuous and {C, y1, y12}, is linearly independent in the additive argument of the solution in terms of the Lambert W function, (y12=y2, C∈R) together with the condition v2=-2y1v1. On this subspace, the Biot Savart Law holds exactly [see Section 2 (Equation (13))]. Also on this subspace, an expression X (part of PNS equations) vanishes which contains all the expressions in derivatives of v1 and v2 and the forcing terms in the plane which are related as with the cancellation of all such terms in governing PDE. The y3 component forcing term is arbitrarily small in ε ball where Weierstrass P functions touch the center of the ball both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3 only governing PDE resulting. With viscosity present as v changes from zero to the fully viscous case at v =1 the solution for v3 reaches a peak in the third component y3. Consequently, there exists a dipole which is not centered at the center of the cell of the Lattice. Hence since the dipole by definition has an equal in magnitude positive and negative peak in y3, then the dipole Riemann cut-off surface is covered by a closed surface which is the sphere and where a given cell of dimensions [-1, 1]3 is circumscribed on a sphere of radius 1. For such a closed surface containing a dipole it necessarily follows that the flux at the surface of the sphere of v3 wrt to surface normal n is zero including at the points where the surface of sphere touches the cube walls. At the finite time singularity on the sphere a rotation boundary condition is deduced. It is shown that v3 is spatially finite on the Riemann Sphere and the forcing is oscillatory in y3 component if the velocity v3 is. It is true that . A boundary condition on the sphere shows the rotation of a sphere of viscous fluid. Finally on the sphere a solution for v3 is obtained which is proven to be Hölder continuous and it is shown that it is possible to extend Hölder continuity on the sphere uniquely to all of the interior of the ball.

From Hölder Continuous Solutions of 3D Incompressible Navier-Stokes Equations to No-Finite Time Blowup on [ 0,∞ ]

This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.

HB-Continuous Mappings in L-Topological Space

In this paper, we introduce and study the notion of HB-closed sets in L-topological space. Then, HB-convergence theory for L-molecular nets and L-ideals is established in terms of HB-closedness. Finally, we give a new definition of fuzzy H-continuous [1] which is called HB-continuity on the basis of the notion of H-bounded L-subsets in L-topological space. Then we give characterizations and properties by making use of HB-converges theory of L-molecular nets and L-ideals.

Strong Feller Property for SDEs with Super-linear Drift and Hölder Diffusion Coefficients

In this paper,we investigate the strong Feller property of stochastic differential equations(SDEs)with super-linear drift and Hölder diffusion coefficients.By utilizing the Girsanov theorem,coupling method,truncation method and the Yamada-Watanabe approximation technique,we derived the strong Feller property of the solution.

Hölder不等式的两种新证法及Cauchy-Schwarz不等式的加细

Hölder不等式是一类重要的不等式。本文分别利用Cauchy-Schwarz不等式及关于凸函数的Jensen不等式,给出了积分型Hölder不等式的两种新的证明方法。据此,结合受控理论给出Cauchy-Schwarz不等式的一种加细及一个新的反向Hölder不等式。

Hölder不等式的一种新证法及推广

1引言Hölder不等式是非常重要的不等式,本文首先给出Hölder不等式的一个等价形式,然后利用函数的凸性给出了Hölder不等式的一种新的证明方法,并据此结合受控理论(theory of majorization)给出Cauchy-Schwarz不等式的一种加细及一个新的反向Hölder不等式.我们需要一些简单的符号和高等数学知识.

JOINT HOLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.

Changes in Soil Enzyme Activity and Nutrient Content in Different Years of Continuous Cropping Tobacco Fields

In order to find out changes in the main fertility indicators of continuous cropping tobacco fields in Henan Province with years of continuous cropping,5 typical tobacco fields( Jia County,Shaoling District,Fangcheng County,Xiangcheng County,and Song County) were studied. Indicators in question included activity of main enzymes( sucrase,urease,catalase,and acid phosphatase),p H value,organic matters,total nitrogen,available phosphorus and available potassium. The results indicate that in most tobacco fields,the activity of sucrase and urease increased in two years of continuous cropping,later,it showed a declining trend with increase in years of continuous cropping; in some fields,it directly declined with increase in years of continuous cropping; the activity of catalase increased in two years of continuous cropping,later,it showed a declining trend with increase in years of continuous cropping,and it became stable after three years of continuous cropping;generally,the activity of acid phosphatase declined with increase in the years of continuous cropping within 3 years continuous cropping,later,it became stable. Except Fangcheng County,the soil p H value of other districts( counties) generally declined with increase in years of continuous cropping,and it was in the range of 5. 6-6. 8. With the increase in continuous cropping,the soil organic matters( SOM) in tobacco fields of Jia County and Song County declined,changes of SOM in tobacco fields of Fangcheng County and Xiangcheng County were not obvious,and SOM showed the trend of first decline then rise in Shaoling District. In most tobacco fields( Jia County,Xiangcheng County,and Song County),the total nitrogen showed a declining trend; in Shaoling District,it showed a trend of first decline then rise; in Fangcheng County,it showed a rising trend. The available phosphorus firstly slightly declined then rose in tobacco fields of Shaoling District and Song County; it showed a trend of first rise then decline in Jia County; there were no obvious changes in Fangcheng County and Xiangcheng County. Changes in the available potassium were different in tobacco fields. In sum,there are certain rules for changes in soil enzyme activity and nutrient content in tobacco fields in Henan Province,but changes are not completely the same. It is concluded that the site conditions and farming activities exert a certain influence on soil enzyme activity,p H value,organic matters,and nitrogen,phosphorus and potassium content. Therefore,to solve obstacles in continuous cropping of tobacco fields,different regions should take different measures.

Hölder Derivative of the Koch Curve

In this paper, we introduce a K Hölder p-adic derivative that can be applied to fractal curves with different Hölder exponent K. We will show that the Koch curve satisfies the Hölder condition with exponent and has a 4-adic arithmetic-analytic representation. We will prove that the Koch curve has exact -Hölder 4-adic derivative.

Effects of prostaglandin E combined with continuous renal replacement therapy on septic acute kidney injury

BACKGROUND The effects of prostaglandin E(PGE)combined with continuous renal replacement therapy(CRRT)on renal function and inflammatory responses in patients with septic acute kidney injury(SAKI)remain unclear.AIM To investigate the effects of PGE combined with CRRT on urinary augmenter of liver regeneration(ALR),urinary Na+/H+exchanger 3(NHE3),and serum inflammatory cytokines in patients with SAKI.METHODS The clinical data of 114 patients with SAKI admitted to Yichang Second People's Hospital from May 2017 to January 2019 were collected.Fifty-three cases treated by CRRT alone were included in a control group,while the other 61 cases treated with PGE combined with CRRT were included in an experimental group.Their urinary ALR,urinary NHE3,serum inflammatory cytokines,renal function indices,and immune function indices were detected.Changes in disease recovery and the incidence of adverse reactions were observed.The 28-d survival curve was plotted.RESULTS Before treatment,urinary ALR,urinary NHE3,blood urea nitrogen(BUN),serum creatinine(SCr),CD3+T lymphocytes,CD4+T lymphocytes,and CD4+/CD8+T lymphocyte ratio in the control and experimental groups were approximately the same.After treatment,urinary ALR and NHE3 decreased,while BUN,SCr,CD3+T lymphocytes,CD4+T lymphocytes,and CD4+/CD8+T lymphocyte ratio increased in all subjects.Urinary ALR,urinary NHE3,BUN,and SCr in the experimental group were significantly lower than those in the control group,while CD3+T lymphocytes,CD4+T lymphocytes,and CD4+/CD8+T lymphocyte ratio were significantly higher than those in the control group(P<0.05).After treatment,the levels of tumor necrosis factor-α,interleukin-18,and high sensitivity C-reactive protein in the experimental group were significantly lower than those in the control group(P<0.05).The time for urine volume recovery and intensive care unit treatment in the experimental group was significantly shorter than that in the control group(P<0.05),although there was no statistically significant difference in hospital stays between the two groups.The total incidence of adverse reactions did not differ statistically between the two groups.The 28-d survival rate in the experimental group(80.33%)was significantly higher than that in the control group(66.04%).CONCLUSION PGE combined with CRRT is clinically effective for treating SAKI,and the combination therapy can significantly improve renal function and reduce inflammatory responses.

可积函数HÖlder不等式的等价不等式

HÖlder不等式在分析学和初等不等式理论中扮演着极其重要的角色。作为相关研究的一个重要方面, 它与其它不等式之间的等价关系的研究也越来越受到重视。本文证明了测度空间上积分形式的H¨older不等式与算术平均-几何平均不等式,以及它与喜平均不等式之间的等价关系。这些结果极大地推广了李勇涛等最近建立的离散不等式等价关系。

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

THE HLDER CONTINUITY FOR THE GRADIENT OF SOLUTIONS TO ONE-SIDED OBSTACLE PROBLEMS

The Hǒlder continuity is proved for the gradient of the solution Jo the one-sided obstacle problem of the following variational inequality in the case 1<p<2

Holder Continuity for Solutions of Degenerate Parabolic Equation with Two Degenerate Points

In this paper we discuss the quasilinear parabolic equation U_■=▽(u^u(1-u)~β.Vu)+■(x,t.u)▽u+C(x,t,u) which is degenerate at u =0 and u=1.Let u(x,t)be a weak solution of the equation satisfying 0<u(x,t)<1.Under some assumptions we establish H■lder continuity of u(x,t).

Directly electrospinning submillimeter continuous fibers on tubes to fabricate H2S detectors with fast and high response

The fast and high response detection of neurotoxic H_(2)S is of great importance for the environment.In this paper,directly electrospinning technology on the ceramic tube is developed to improve the response of H_(2)S detector based on superlong SnO_(2)fibers.The submillimeter continuous fibers are deposited directly on ceramic tubes by in-situ electrospinning method and can keep morphology of fibers during calcination.By employing this technology,CuO-doped SnO_(2)fiber H_(2)S detectors are fabricated,and 10%atom CuO-doped SnO_(2)H_(2)S detector shows the highest response of 40 toward 1 ppm H_(2)S at 150℃while the response is only 3.6 for the H_(2)S detector prepared in traditional route.In addition,the in-situ electrospinning H_(2)S detectors show faster response and recovery compared to the H_(2)S detectors fabricated by the conventional way.The high and fast response of H_(2)S detectors based on in-situ electrospinning can be ascribed to the continuous fiber structure and CuO modification.The present in-situ electrospinning technology may provide a new strategy for the development of other gasdetectors and bio-detectors with fast and high response.

Hlder continuity for solutions of elltptic equations involving measures

The Hoider continuity is proved to bounded solutions of degenerate elliptic e-quations involving measures. The structural conditions of the equation are more general and therestrictions on the structural coofficients are weaker.

一类Hlder连续的混沌系统分析和控制

构造了新的三维连续时间混沌系统.该系统含有两个参数和一个非线性项,其中的非线性项满足1/2阶Hlder连续但不满足Lipschitz连续性条件.通过理论分析和数值仿真研究了系统的动力学特性.此外,利用线性状态反馈对此混沌系统进行了控制,实现了系统的全局镇定.通过分段线性状态反馈方法对新混沌系统的同步进行了研究,实现了系统的全局同步.

度量空间中某类泛函极小的Hlder连续性

主要研究了牛顿空间中泛函F(u,gu)=∫f(u,gu)dμ的极小问题,其中对某常数c>0,泛函满足gup-c|u|p≤f(u,gu)≤gpu+c|u|p.牛顿空间是Sobolev空间在度是空间中的推广,具有更一般的性质.在该空间中研究偏微分方程,对建立更一般的偏微分方程理论具有指导和开拓意义.本文利用De Giorgi迭代的方法研究了该泛函极小的正则性,证明了该泛函极小的Hlder连续性.这使得我们在不求精确解的情况下,利用方程本身的条件可以对方程解进行数值估计.

从贝努利不等式到Hlder不等式的演变过程及应用

首先利用贝努利不等式给出几何平均算术平均不等式的证明,然后给出Young不等式和Young逆不等式的初等证明方法,进而给出Hlder不等式的初等证明,并将这些结果应用到一些重要不等式的证明.

A-调和方程弱解的逆Hlder不等式及其应用

本文给出A 调和方程弱解的逆H lder不等式及其若干应用 .