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On the Dirichlet Fractional Laplacian and Applications to the SQG Equation on Bounded Domains
Elie Abdo, Quyuan Lin
We investigate new properties of the fractional Dirichlet Laplacian on smooth bounded domains and establish fractional product estimates and nonlinear Poincaré inequalities. We als…
Long Time Dynamics Of The Three-dimensional Nernst-Planck-Darcy Model
Elie Abdo, Đorđe Nikolić
We consider an electrodiffusion model describing the evolution of ionic species in a three-dimensional fluid flowing through a porous medium and forced by added body charges. W…
Error estimates of physics-informed neural networks for approximating Boltzmann equation
Elie Abdo, Lihui Chai, Ruimeng Hu +1
Motivated by the recent successful application of physics-informed neural networks (PINNs) to solve Boltzmann-type equations [S. Jin, Z. Ma, and K. Wu, J. Sci. Comput., 94 (2023),…
On the Three-dimensional Nernst-Planck-Boussinesq System
Elie Abdo, Ruimeng Hu, Quyuan Lin
In this paper, we analyze a three-dimensional Nernst-Planck-Boussinesq (NPB) system that describes ionic electrodiffusion in an incompressible viscous fluid. This new model incorpo…
Logarithmic Sobolev Inequalities for Bounded Domains and Applications to Drift-Diffusion Equations
Elie Abdo, Fizay-Noah Lee
We prove logarithmic Sobolev inequalities on higher-dimensional bounded smooth domains based on novel Gagliardo-Nirenberg type interpolation inequalities. Moreover, we use them to…
On the space-time analyticity of the Keller-Segel-Navier-Stokes system
Elie Abdo, Zhongtian Hu
In this paper, we study the coupled Keller-Segel-Navier-Stokes system, which models chemotaxis occuring in ambient viscous fluid. We consider this nonlinear, nonlocal system on a p…
A PDE Perspective on Approximating Nonlocal Periodic Operators with Applications on Neural Networks for Critical SQG Equations
Elie Abdo, Ruimeng Hu, Quyuan Lin
Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In…