Gradient Of Solution Research Articles

Derivation of expressions for interdiffusion and intrinsic diffusion flux in presence of chemical potential gradient in a multicomponent system with composition dependent molar volume

Abstract It has been generally accepted and often mentioned in the text books that gradient in chemical potential of a species is the fundamental driving force for its diffusion. However, a general derivation of the interrelation between the diffusion flux of a component and chemical potential gradients in a non-ideal solution is lacking. Although there have been various studies in the literature reporting such interrelations for a binary system, they all assume constant molar volume. In a non-ideal system though molar volume changes with composition. Hence, in the present work, kinetic theory is used to derive a relation between diffusion flux and the chemical potential gradients for a multicomponent system with composition dependent molar volume. It is shown that the velocity of the marker as measured in a diffusion couple experiment should consist of the drift velocity (UN) due to change in molar volume accompanied by diffusion as well as the Kirkendall velocity caused by vacancy equilibration process. For the assumption of volume change occurring only in the direction of diffusion, the Kirkendall velocity is same as the marker velocity measured in a diffusion couple. However, if the lattice is allowed to relax in all directions, the contribution of UN to the marker velocity can be significant. This is shown to be as high as 20% for Cu in a Cu-Ni diffusion couple.

Symmetry results of solutions for elliptic systems with linear couplings

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1].

A Miniature Sucrose Gradient for Polysome Profiling.

Polysome profiling by sucrose density gradient centrifugation is commonly used to study the overall degree of translation (messenger RNA to protein synthesis). Traditionally, the method begins with synthesis of a 5-10 mL sucrose gradient onto which 0.5-1 mL of cell extract is layered and centrifuged at high speed for 3-4 h in a floor-model ultracentrifuge. After centrifugation, the gradient solution is passed through an absorbance recorder to generate a polysome profile. Ten to twelve fractions (0.8-1 mL each) are collected for isolating different RNA and protein populations. The overall method is tedious and lengthy (6-9 h), requires access to a suitable ultracentrifuge rotor and centrifuge, and requires a substantial amount of tissue material, which can be a limiting factor. Moreover, there is often a dilemma over the quality of RNA and protein populations in the individual fractions due to the extended experiment times. To overcome these challenges, here we describe a miniature sucrose gradient for polysome profiling using Arabidopsis thaliana seedlings that takes ~1 h centrifugation time in a tabletop ultracentrifuge, reduced gradient synthesis time, and also less tissue material. The protocol described here can be easily adapted to a wide variety of organisms and polysome profiling of organelles, such as chloroplasts and mitochondria. Key Features • Mini sucrose gradient for polysome profiling that requires less than half the processing time vs. traditional methods. • Reduced starting tissue material and sample volume for sucrose gradients. • Feasibility of RNA and protein isolation from polysome fractions. • Protocol can be easily modified to a wide variety of organisms (and even polysome profiling of organelles, such as chloroplast and mitochondria). Graphical Overview.

A note on closed vector fields

<abstract><p>Special vector fields, such as conformal vector fields and Killing vector fields, are commonly used in studying the geometry of a Riemannian manifold. Though there are Riemannian manifolds, which do not admit certain conformal vector fields or certain Killing vector fields, respectively. Closed vector fields exist in abundance on each Riemannian manifold. In this paper, we used closed vector fields to study the geometry of the Riemannian manifold. In the first result, we showed that a compact Riemannian manifold $ (M^{n}, g) $ admits a closed vector field $\boldsymbol{\omega }$ with $ \mathrm{div} \boldsymbol{\omega }$ non-constant and an eigenvector of the rough Laplace operator, the integral of the Ricci curvature $ Ric(\boldsymbol{\omega }, \boldsymbol{\omega }) $ has a suitable lower bound that is necessarily isometric to $ S^{n}(c) $ and that the converse holds. In the other result, we found a characterization of an Euclidean space using a closed vector field $\boldsymbol{\omega }$ with non-constant length that annihilates the rough Laplace operator and squared length of its covariant derivative that has a suitable upper bound. Finally, we used the closed vector field provided by the gradient of the non-trivial solution of the Fischer-Marsden equation on a complete and simply connected Riemannian manifold $ (M, g) $ and showed that it is necessary and sufficient for $ (M, g) $ to be isometric to a sphere and that the squared length of the covariant derivative of this closed vector field has a suitable upper bound.</p></abstract>

Global gradient estimates in directional homogenization

<abstract><p>In this research, we investigate a higher regularity result in periodic directional homogenization for divergence-form elliptic systems with discontinuous coefficients in a bounded nonsmooth domain. The coefficients are assumed to have small bounded mean oscillation (BMO) seminorms and the domain has the $ \delta $-Reifenberg property. Under these assumptions we derive global uniform Calderón-Zygmund estimates by proving that the gradient of the weak solution is as integrable as the given nonhomogeneous term.</p></abstract>

Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces

<abstract><p>We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > \frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in <sup>[<xref ref-type="bibr" rid="b13">13</xref>]</sup>, where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.</p></abstract>

Regularity of weak solutions to a class of fourth order parabolic variational inequality problems arising from swap option pricing

<abstract><p>This article concerns the regularity of weak solutions for a variational inequality problem constructed by a fourth-order parabolic operator which has received much attention recently. We first consider the internal regular estimate of weak solutions using the difference type test function. Then, the near edge regularity and global regularity of weak solutions are analyzed by using the finite cover principle. Since the quadratic gradient of the weak solution does not satisfy the conditions for a test function, we have constructed a test function using a spatial difference operator to complete the proof of regularity. The results show that the weak solution has a second order regularity and an $ {L^\infty }(0, T;{H^2}(\Omega)) $ estimation independent of the lower order norm of the weak one.</p></abstract>

Optimal estimates for the conductivity problem close to the boundary in dimension greater than two

<p style='text-indent:20px;'>In high-contrast composite materials, the electric field (or stress field) may blow up in the narrow region when a convex inclusion is closely to the boundary of the matrix domain. The gradient of solutions depends on the parameters–the distance between the inclusions and boundaries. An optimal upper bound was obtained for convex inclusions of arbitrary shape in dimension greater than two. Our results extend that in [<xref ref-type="bibr" rid="b3">3</xref>], where the disk inclusion case in 2D is considered.</p>

Existence and regularity estimates for quasilinear equations with measure data : the case 1 < p ≤ (3n − 2)∕(2n − 1)

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= \delta\, |\nabla u|^q +\mu$ in a bounded main $\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. In particular, in the case $\delta=1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Vibrations of a strip with delamination in the framework of the one-parameter Aifantis model of gradient elasticity theory

The problems on in-plane and anti-planar steady-state vibrations of an isotropic elastic strip with delamination at the lower boundary has been investigated. The goal of the study is to analyze the stress-strain state in the crack tips areas and to construct a crack opening function being the main mechanical characteristics in the crack theory problems. The problems under study have been solved in the framework of the nonclassical gradient elasticity theory (GET) on the basis of the one-parameter model proposed by Aifantis. The boundary integral equations (BIE) are obtained with respect to crack opening functions or their derivatives. The analysis of BIEs is carried out, regular and irregular parts are distinguished, the obtained BIEs with singular (e.g., with hypersingular, with cubic singularity) integrals are solved via collocation methods, approximating Chebyshev polynomials, quadrature formulas for singular integrals. For the in-plane problem solution, the simplified Ru-Aifantis method has been applied. The Ru-Aifantis method allows to divide the initial boundary value problem into two sub-problems - the classical linear elasticity theory (LTE) problem and the simplified boundary value problem for finding the gradient solution which includes the solution found via the classical theory. For each of the problems, semi-analytical expressions for the functions of crack opening have been constructed, and the analysis of the stress-strain state in the area of crack tips has been carried out. The problems have also been solved in the case of a crack with small relative length, the analysis of BIE depending on small parameters ratio has been carried out, and explicit expressions for the crack opening functions have been obtained. Numerical calculations have been performed; the applicability conditions for the asymptotic method are determined, and a comparative analysis of the results obtained on the basis of GET and LTE models, depending on the values of the gradient parameter and the delamination length, is realized.

An efficient and high-fidelity local multi-mesh finite volume method for heat transfer and fluid flow problems in metal additive manufacturing

The multi-scale and nonlinear feature of additive manufacturing problems poses great challenges to numerical algorithms regarding computational efficiency and fidelity. In this work, we propose a novel local multi-mesh finite volume method for heat transfer and fluid flow problems in the additive manufacturing process. In this method, two sets of meshes (coarse base mesh and finer overlay mesh) are used to resolve the macroscopic temperature field of the whole domain and the mesoscopic heat transfer and fluid flow problem within a relatively small region covering the melt pool, respectively. The interaction between the base mesh and the overlay mesh is handled by the Dirichlet boundary conditions on the interface following the refinement-by-superposition idea, such that the powder-scale simulation is available with high efficiency. The finer mesh with cells of sub-powder size uses the arbitrary Lagrangian–Eulerian description and moves with the heat source to keep coverage of the region with high solution gradients. According to the moving feature of the finer mesh, a particle-volume fraction conversion scheme is introduced to update the material distribution. The local multi-mesh finite volume method is further combined with the discrete element method as well as a surface reconstruction method to alternatively solve the powder layering process and the laser melting process. A set of numerical cases is presented, and the numerical results are compared with analytical and experimental data from the literature, with good agreement in cases where such data is available. Moreover, the computational efficiency of the proposed method is proved to be dozens of times (16∼50 for the given size ratios of the melt pool to the whole domain) of the finite volume method using a uniform structured mesh. It is expected to be a powerful tool for efficiently achieving powder-scale simulation of the additive manufacturing process and other similar multi-scale problems.

Thermosolutal Instability of Couple Stress Rivlin Ericksen Ferromagnetic Fluid with Rotation, Magnetic and Variable Gravity Field in Porous Medium

In this paper thermosolutal instability of a couple stress Rivlin-Ericksen ferromagnetic fluid layerwith rotation, magnetic and variable gravity field in porous medium is studied. Using normalmode analysis technique the dispersion relation has been obtained and solved analytically. Forstationary convention, it is found that couple stress, magnetic field and permeability of mediumhave stabilizing effect on thermal instability under certain conditions. Rotation has stabilizingeffect if λ&gt; 0 and destabilizing effect if λ&lt; 0. Solute gradients has stabilizing effect either λ &gt;0 or λ &lt; 0. The presence of solute gradient, viscoelastic parameter, rotation and magnetic fieldintroduces oscillatory modes, stability in the system and principle of exchange of stability is notsatisfied if λ &lt; 0 and system may be stable or unstable if λ &gt; 0.

Partial regularity for a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group: the superquadratic case

ABSTRACT In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used T = X i X n + i − X n + i X i .

The growth mechanism of diffused multilayer between electroplated nickel coating and nickel-aluminum bronze alloy

• 1. The calculated results of Fick’s second law prove the relationship: D Ni > D Cu > D Al . • 2. The larger D Ni and D Cu than D Al cause the preferential formation of Ni-Cu layer. • 3. The physicochemical approach is used to explain the ternary interdiffusion process. • 4. The formation of Ni-Al-Cu layer is at the expense of the depletion of Ni-Cu layer. The Ni-rich surface alloying modification has been implemented to enhance the corrosion resistance of nickel-aluminum bronze (NAB) alloy used in the marine environment, while the growth behavior of the Ni-rich diffused multilayer is still unclear. Herein, the theoretical analysis and morphological observation were applied to disclose the growth mechanism of the diffused multilayer, the results manifested that the diffusion coefficients of Ni and Cu were higher than that of Al, leading to the preferential interdiffusion process of Ni and Cu to generate the Ni-Cu solid solution gradient layer at the interface of the diffusion couple. With extending the heat treatment time, the Al diffused into the Ni-Cu layer to form the Ni-Al-Cu intermetallic layer between the Ni-Cu layer and NAB matrix, and the formation of Ni-Al-Cu layer was at the expense of the depletion of Ni-Cu layer.

Gradient estimates for non-uniformly elliptic problems with BMO nonlinearity

We provide a new approach to obtain Calderón-Zygmund type estimates for non-uniformly elliptic equations with discontinuous nonlinearities of double phase growth. This approach, which is based on a small higher integrability result for the gradient of weak solutions to the associated homogeneous problems together with extrapolation from Muckenhoupt weights, enables us to find a proper comparison estimate of approximation by imposing merely a small BMO assumption on the nonlinearity with respect to the x-variable. As a consequence, we are able to prove an optimal regularity theory for a larger class of double phase problems with discontinuous nonlinearities in the literature.

Influence of Oxygen on Chemoconvective Patterns in the Iodine Clock Reaction.

There is increasing interest in using chemical clock reactions to drive material formation; however, these reactions are often subject to chemoconvective effects, and control of such systems remains challenging. Here, we show how the transfer of oxygen at the air-water interface plays a crucial role in the spatiotemporal behavior of the iodine clock reaction with sulfite. A kinetic model was developed to demonstrate how the reaction of oxygen with sulfite can control a switch from a low-iodine to high-iodine state under well-stirred conditions and drive the formation of transient iodine gradients in unstirred solutions. In experiments in thin layers with optimal depths, the reaction couples with convective instability at the air-water interface forming an extended network-like structure of iodine at the surface that develops into a spotted pattern at the base of the layer. Thus, oxygen drives the spatial separation of iodine states essential for patterns in this system and may influence pattern selection in other clock reaction systems with sulfite.

CO2 transport and carbonate precipitation in the coupled diffusion-reaction process during CO2 storage

During geological CO2 storage and utilization, CO2 plume migration forms widespread static retention regions including the dead-end fractures and pores in the host rock, where the mass transfer is dominated by diffusion. A series of packed column experiments with natural olivine were conducted to investigate how the coupled CO2 diffusion–reaction process influences the CO2 transport and the temporal-spatial evolution of carbonate precipitation in diffusion-limited zones. The carbonate precipitations were mainly magnesite, and some dolomite due to the combined effect of carbonate precipitation kinetics and the low Ca content of the solution with Mg enrichment. Compared with the pure diffusion in porous media, the geochemical gradients in the pore solution resulting from the coupled diffusion–reaction interaction promoted mass transfer and broad migration, which led to an increase in effective diffusion coefficients of CO2 by approximately-one order of magnitude. The free water layer and pressure mode could affect the distribution of carbonate precipitation and the overall sequestration capacity. In particular, the coupled CO2 diffusion–reaction interaction under pressure decay mode could accelerate the carbonation in the deep region in the water-saturated column. This study advances the fundamental understanding on the transport and reaction of injected CO2 for mafic and ultramafic formations.

Existence of a Sign-Changing Weak Solution to Doubly Nonlinear Parabolic Equations

In this paper, assuming the initial-boundary datum belonging to suitable Sobolev and Lebesgue spaces, we prove the global existence result for a (possibly sign changing) weak solution to the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations of the form ∂t|u|q-1u-Δpu=0inΩ∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t\\left( |u|^{q-1}u\\right) -\\Delta _p u=0\\quad \ ext {in}\\,\\,\\,\\Omega _\\infty , \\end{aligned}$$\\end{document}where p>1 and q>0. This is a fair improvement of the preceding result by authors (Nonlinear Anal 175C :157–172, 2018). The key tools we employ are energy estimates for approximate equations of Rothe type and the integral strong convergence of gradients of approximate solutions.

Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$\begin{aligned} \partial _t u=[|D u|^q+a(x,t)|D u|^s]\left( \Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D u|}\right\rangle \right) , \end{aligned}$$ where $$1<p<\infty $$ , $$-1<q\le s<\infty $$ and $$a(x,t)\ge 0$$ . The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of q, s, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when $$q=p-2$$ and $$q<s$$ , it will encompass the parabolic p-Laplacian both in divergence form and in non-divergence form. We aim to explore the $$L^\infty $$ to $$C^{1,\alpha }$$ regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions.