### [A1] Nonlocal Vector Calculus and Calculus of Variations

Compared with the traditional PDEs-based models,
the recently developed nonlocal models that
account for interaction occurring at a distance have been shown to more faithfully and effectively
model observed phenomena that involve possible singularities and other anomalies.
Mathematical developments, indispensable to nonlocal models' success, are still relatively nascent.
My work involves the development of nonlocal vector calculus,
nonlocal function spaces, variational problems, trace theorems and asymptotic analysis.
I am also interested in other developments of nonlocal theory such as nonlocal exterior calculus and geometry and nonlocal Wasserstein distance.
- Z. Han and X. Tian,
* Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications*,
Math. Models Methods Appl. Sci., 33(12),2507–2556, 2023.

- Q. Du, T. Mengesha and X. Tian,
* L^p compactness criteria with an application to variational convergence of some nonlocal energy functionals*,
Math. Eng., 5(6): 1-31, 2023.

- Q. Du, X. Tian and Z. Zhou,
* Nonlocal diffusion models with consistent local and fractional limits*,
A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models: Proceedings of the 50th John H. Barrett Memorial Lectures,175-213, 2023.

- Q. Du, X. Tian, C. Wright and Y. Yu,
* Nonlocal trace spaces and extension results for nonlocal calculus*,
J. Funct. Anal., 282(12), 109453, 2022.

- Q. Du, T. Mengesha and X. Tian,
* Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localizations*,
Anal. Appl., 20(3), 579-614, 2022.

- Q. Du and X. Tian,
* Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations*,
Foundations of Computational Mathematics, 20, 801-826, 2020.

- Q. Du and X. Tian,
*Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling *,
SIAM J. Appl. Math., 78(3), 1536-1552 , 2018.

- X. Tian and Q. Du,
*Trace theorems for some nonlocal function spaces with heterogeneous localization *,
SIAM J. Math. Anal., 49(2):1621-1644, 2017.

- X. Tian and Q. Du,
*A class of high order nonlocal operators *,
Arch Rat. Mech. Anal., 222(3)1521-1553, 2016.

- X. Tian and Q. Du,
* Nonconforming discontinuous Galerkin methods for nonlocal varational problems*,
SIAM J. Numer. Anal., 53 (2015), 762-781.