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1. Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications

   https://doi.org/10.1142/S0218202523500549  

   Han, Zhaolong; Tian, Xiaochuan (November 2023, Mathematical Models and Methods in Applied Sciences)

   This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. 

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2. Nonlocal Effective Field Theory and Its Applications

   https://doi.org/10.3390/sym18010031  

   Wang, Ping; Gao, Zhengyang; He, Fangcheng; Ji, Chueng-Ryong; Melnitchouk, Wally; Salamu, Yusupujiang (January 2026, Symmetry)

   We review recent applications of nonlocal effective field theory, particularly focusing on nonlocal chiral effective theory and nonlocal quantum electrodynamics (QED), as well as an extension of nonlocal effective theory to curved spacetime. For the chiral effective theory, we discuss the calculation of generalized parton distributions (GPDs) of the nucleon at nonzero skewness, along with the corresponding gravitational (or mechanical) form factors, within the convolution framework. In the QED application, we extend the nonlocal formulation to construct the most general nonlocal QED interaction, in which both the propagator and fundamental QED vertex are modified due to the nonlocal Lagrangian, while preserving the Ward–Green–Takahashi identities. For consistency with the modified propagator, a solid quantization is proposed, and the nonlocal QED is applied to explain the lepton g−2 anomalies without the introduction of new particles beyond the standard model. Finally, with an extension of the chiral effective action to curved spacetime, we investigate the nonlocal energy–momentum tensor and gravitational form factors of the nucleon with a nonlocal pion–nucleon interaction. 

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3. $$ L^{p} $$ compactness criteria with an application to variational convergence of some nonlocal energy functionals

   https://doi.org/10.3934/mine.2023097  

   Du, Qiang; Mengesha, Tadele; Tian, Xiaochuan (January 2023, Mathematics in Engineering)

   Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $$ L^{p} $$ vector fields defined on a domain $$ \Omega $$ that is either a bounded domain in $$ \mathbb{R}^{d} $$ or $$ \mathbb{R}^{d} $$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $$ L^{p} $$ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. 

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4. Global regularity for critical SQG in bounded domains

   https://doi.org/10.1002/cpa.22221  

   Constantin, Peter; Ignatova, Mihaela; Nguyen, Quoc‐Hung (January 2025, Communications on Pure and Applied Mathematics)

   Abstract We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system. 

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5. Compactness results for a Dirichlet energy of nonlocal gradient with applications

   https://doi.org/10.1002/num.23149  

   Han, Zhaolong; Mengesha, Tadele; Tian, Xiaochuan (November 2024, Numerical Methods for Partial Differential Equations)

   Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

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