Related papers: Nonlocal vector calculus on the sphere

A direct analog of Hadamard's three-circle theorem is obtained for harmonic functions (in weighted L^2-norm) in case of (n-1)-dimensional non-concentric spheres in R^n. The result extends the concentric case to correlated non-concentric,…

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical…

We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an ``algebra tool kit'' to deal with limit and…

Gauss' and Stokes' theorems are fundamental results in vector calculus and important tools in physics and engineering. When students are asked to describe the meaning of Gauss' divergence theorem, they often use statements like this: "The…

Our work deals with the systematic study of the coupling between the nonlocal Stokes system and the Vlasov equation. The coupling is due to a drag force generated by the fluid-particles interaction. We establish the existence of global weak…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

In this article, discrete variants of several results from vector calculus are studied for classical finite difference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector…

Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on nonlocal gradient operators. In particular, we study a…

We propose a nonlocal operator method for solving partial differential equations (PDEs). The nonlocal operator is derived from the Taylor series expansion of the unknown field, and can be regarded as the integral form "equivalent" to the…

A general local Fourier analysis for overlapping block smoothers on triangular grids is presented. This analysis is explained in a general form for its application to problems with different discretizations. This tool is demonstrated for…

We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a…

We present a method for calculating the results of operation of differential operators operating on components of vector in generalized coordinates not restricted to orthogonal one. For this we use the relationships between covariant,…

In the series of this paper and the forthcoming papers [41,42] we study the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. We focus on the study of…

We study weak and strong solutions of nonlinear non-compact operator equations in abstract spaces of adapted random points. The main result of the paper is similar to Schauder's fixed-point theorem for compact operators. The illustrative…

We study the strong solvability of the nonstationary Stokes problem with non-zero divergence in a bounded domain.

In this short review main issues related to the non-Abelian Stokes theorem have been addressed. The two principal approaches to the non-Abelian Stokes theorem, operator and two variants (coherent-state and holomorphic) of the path-integral…

A generalization of fractional vector calculus as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency…

We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.

We propose and develop a new calculus for local variational differential operators. The main difference of the new formalism with the canonical differential calculus is that the image of higher order operators on local functionals does not…

In this paper we present a full discretization of the layer potentials and boundary integral operators for the elastic wave equation on a parametrizable smooth closed curve in the plane. The method can be understood as a non-conforming…