Research

Research Interests

My research interests lie in the mathematical analysis of nonlinear partial differential equations arising from fluid dynamics. In particular, I am intersted in the well-posedness, ill-posedness, long-time dynamics, and finite-time singularity formation of PDEs. Recently, I am also interested in the stochastic PDEs, which take randomness into consideration, as well as the application of machine learning in solving different PDEs.

Publications/Preprints

1. On the Three-dimensional Nernst-Planck-Boussinesq System (with Elie Abdo and Ruimeng Hu), submitted, arXiv: 2405.02152 (2024).

2. Global existence for the stochastic Boussinesq equations with transport noise and small rough data (with Rongchang Liu and Weinan Wang), SIAM J. Math. Anal. (2024). https://doi.org/10.1137/23M155953

3. Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise (with Rongchang Liu and Weinan Wang), submitted, arXiv:2401.11318 (2024).

4. Accuracy Analysis of Physics-Informed Neural Networks for Approximating the Critical SQG Equation (Elie Abdo and Ruimeng Hu), preprint, arXiv:2401.10879 (2024).

5. On the Long-time Dynamics and Ergodicity of the Stochastic Nernst-Planck-Navier-Stokes System (with Elie Abdo and Ruimeng Hu), submitted, arXiv: 2310.20484 (2023).

6. Anisotropic Viscosities Estimation for the Stochastic Primitive Equations (with Igor Cialenco and Ruimeng Hu), submitted, arXiv: 2309.06952 (2023).

7. Pathwise Solutions for Stochastic Hydrostatic Euler Equations and Hydrostatic Navier-Stokes Equations Under the Local Rayleigh Condition (with Ruimeng Hu), submitted, arXiv:2301.07810 (2023).

8. Higher-Order Error estimates for the Physics-Informed Neural Networks Approximating the Primitive Equations (with Ruimeng Hu, Alan Raydan, and Sui Tang), Partial Differ. Equ. Appl. (2023). https://doi.org/10.1007/s42985-023-00254-y

9. Stable Singularity Formation for the Inviscid Primitive Equations (with Charles Collot and Slim Ibrahim), Annales de l'Institut Henri Poincaré C, Analyse non linéaire (2023). https://doi.org/10.4171/AIHPC/87

10. On the Effect of Fast Rotation and Vertical Viscosity on the Lifespan of the 3D Primitive Equations (with Xin Liu and Edriss S. Titi), J. Math. Fluid Mech. (2022). https://link.springer.com/article/10.1007/s00021-022-00705-3 

11. Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations (with Ruimeng Hu), Stoch PDE: Anal Comp. (2022). https://doi.org/10.1007/s40072-022-00266-6 

12. On the Effect of Rotation on the Life-span of Analytic Solutions to the 3D Inviscid Primitive Equations (with Tej-Eddine Ghoul, Slim Ibrahim, and Edriss S. Titi), Arch. Ration. Mech. Anal. (2022). https://doi.org/10.1007/s00205-021-01748-y

13. Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation (with Slim Ibrahim and Edriss S. Titi), J. Differ. Equ.  (2021). https://doi.org/10.1016/j.jde.2021.03.037

14. On the Well-posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation (with Chongsheng Cao and Edriss S. Titi),  J. Math. Fluid Mech. (2020). https://doi.org/10.1007/s00021-020-00495-6