scholarly journals Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications

2020 ◽  
Vol 54 (1) ◽  
pp. 105-128 ◽  
Author(s):  
Hwi Lee ◽  
Qiang Du
Keyword(s):  
Linear Elasticity ◽  
Linear Models ◽  
Differential Operators ◽  
Stokes Equations ◽  
Integral Operators ◽  
Helmholtz Decomposition ◽  
Nonlocal Models ◽  
Extra Assumption

Nonlocal gradient operators are prototypical nonlocal differential operators that are very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on nonlocal gradient operators with a non-spherical interaction neighborhood. We show that the truncation of the spherical interaction neighborhood to a half sphere helps making nonlocal gradient operators well-defined and the associated nonlocal Dirichlet energies coercive. These become possible, unlike the case with full spherical neighborhoods, without any extra assumption on the strengths of the kernels near the origin. We then present some applications of the nonlocal gradient operators with non-spherical interaction neighborhoods. These include nonlocal linear models in mechanics such as nonlocal isotropic linear elasticity and nonlocal Stokes equations, and a nonlocal extension of the Helmholtz decomposition.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals Experimental validation of wind energy estimation

2020 ◽  
Vol 24 (6 Part A) ◽  
pp. 3795-3806
Author(s):  
Predrag Zivkovic ◽  
Mladen Tomic ◽  
Vukman Bakic
Keyword(s):  
Wind Speed ◽  
Linear Model ◽  
Linear Models ◽  
Stokes Equations ◽  
Momentum Conservation ◽  
Electricity Production ◽  
Navier Stokes ◽  
Energy Estimation ◽  
Navier Stokes Equations ◽  
Non Linear

Wind power assessment in complex terrain is a very demanding task. Modeling wind conditions with standard linear models does not sufficiently reproduce wind conditions in complex terrains, especially on leeward sides of terrain slopes, primarily due to the vorticity. A more complex non-linear model, based on Reynolds averaged Navier-Stokes equations has been used. Turbulence was modeled by modified two-equations k-? model for neutral atmospheric boundary-layer conditions, written in general curvelinear non-orthogonal co-ordinate system. The full set of mass and momentum conservation equations as well as turbulence model equations are numerically solved, using the as CFD technique. A comparison of the application of linear model and non-linear model is presented. Considerable discrepancies of estimated wind speed have been obtained using linear and non-linear models. Statistics of annual electricity production vary up to 30% of the model site. Even anemometer measurements directly at a wind turbine?s site do not necessarily deliver the results needed for prediction calculations, as extrapolations of wind speed to hub height is tricky. The results of the simulation are compared by means of the turbine type, quality and quantity of the wind data and capacity factor. Finally, the comparison of the estimated results with the measured data at 10, 30, and 50 m is shown.

scholarly journals Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1251-1298 ◽  
Author(s):  
Maria Eckardt ◽  
Kevin J. Painter ◽  
Christina Surulescu ◽  
Anna Zhigun
Keyword(s):  
Cell Migration ◽  
General Solution ◽  
Numerical Simulations ◽  
Integral Operators ◽  
Mathematical Framework ◽  
Local Models ◽  
Nonlocal Operators ◽  
Nonlocal Models ◽  
Limit Procedure

AbstractA rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

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