Research

Research Interests

Analysis of PDE, mainly in hyperbolic conservation law, nonlinear dispersive equations, and fluid mechanics.

• Vortex patch/front problems for a family of two-dimensional incompressible fluids.

• Shallow water equations: Camassa-Holm equations, kdv equations, etc. 

• Compressible Euler equations, shock waves theory. 

• Variational wave equations from liquid crystal. 

Publications

1. Initial-boundary value problems for Poiseuille flow of nematic liquid crystal via full Ericksen-Leslie model,  (Submitted) (with Geng Chen, Yanbo Hu).

2. Global solutions of quasi-linear Hamiltonian mKdV equation, (Submitted) (with Fangchi Yan).

3. Global solutions of quasi-geostrophic shallow-water fronts, (Submitted) (with Fangchi Yan).

4. Global solutions for a family of GSQG front equations, (submitted) (with John K. Hunter, Jingyang Shu).

5. Double shock solution for 3-dimensional irrotational isentropic flow, to appear on Journal of Hyperbolic Differential Equations. (with Dening Li). 

6. Shock Waves with Irrotational Rankine-Hugoniot Conditions, Quarterly of Applied Mathematics. (with Dening Li). 

7. On the approximation of vorticity fronts by the Burgers-Hilbert equation,  Asymptotic Analysis, 129(2), (2022) 141-177. (with  John K. Hunter, Ryan C. Moreno-Vasquez, Jingyang Shu).

8. Contour Dynamics for Surface Quasi-Geostrophic Fronts, Nonlinearity, 33, (2020) no. 9. (with John K. Hunter, Jingyang Shu).

9. Two-front Solutions of the SQG Equation and its Generalizations, Commun. Math. Sci. 18 (2020), no. 6, 1685-1741 . (with John K. Hunter, Jingyang Shu).

10. Global solutions of a surface quasi-geostrophic front equation, Pure and Applied Analysis, Vol. 3 (2021), No. 3, 403–472. (with John K. Hunter, Jingyang Shu).

11. Local well-posedness of an approximate equation for SQG fronts, J. Math. Fluid Mech., 20 (2018), no. 4, 1967–1984.  (with John K. Hunter, Jingyang Shu).

12. On finite time BV blow-up for the p-system,  Comm. Partial Differential Equations, 43 (2018), no. 8, 1242–1280. (with Alberto Bressan and Geng Chen)

13. Generic regularity of conservative solutions to Camassa-Holm type equations,  SIAM J. Math. Anal.,  49 (2017), no. 4, 2920-2949. (with Mingjie Li)

14. Uniqueness of conservative solutions to the two-component Camassa-Holm System via characteristics, Commun. Math. Sci. , 14 (2016), no. 6, 1581--1597. (with Mingjie Li)

15. Global well-posedness of cubic Camassa-Holm equations, Nonlinear Anal. 133 (2016), 61-73.

16. No BV bounds for approximate solutions to p-system with general pressure law, J. Hyperbolic Differ. Equ. 12 (2015), no. 4, 799-816. (with Alberto Bressan, Geng Chen and Shengguo Zhu)

17. Unique conservative solutions to a variational wave equation, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 1069-1101. (with Alberto Bressan and Geng Chen)

18. Uniqueness of conservative Solutions to the Camassa-Holm equation via characteristics, Discrete Contin. Dyn. Syst. 35 (2015) 25-42. (with Alberto Bressan and Geng Chen)

19. Lack of BV bounds for approximate solutions to the p-system with large data, J. Differential Equations 256 (2014), 3067-3085. (with Alberto Bressan and Geng Chen)

20. A blow-up criterion for 3-D compressible visco-elasticity, Commun. Math. Sci. 12 (2014), no. 3, 473-484. (with Yi Du and Chun Liu)

21. Singularities of solutions to compressible Euler equations with vacuum, Math. Res. Lett. 20 (2013), no. 1, 55-64. (with Zhen Lei and Yi Du)

22. Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ. 10 (2013), no. 1, 149-172. (with Geng Chen and Robin Young)