Research

I am interested in applied mathematics and analysis of partial differential equations (PDEs). More precisely, I have mainly worked on the following topics:

Analysis of PDEs

Fluid dynamics, including stability theory and ill-posedness of the Navier-Stokes equations, the Oldroyd-B (visco-elastic) system, the porous medium equations, the fractional Boussinesq equations, and the magneto-hydrodyamics (MHD) equations

• On the well-posedness and stability for some fluid equations (with X. Xu**), in preparation, 2025. (** denotes a postdoc research mentee)

• Quantitative bounds of critically bounded solutions to the three-dimensional Navier-Stokes equations in Lorentz spaces (with W. Feng and J. He), submitted, 2025.

• Asymptotic stability for the 3D Navier-Stokes equations in $L^3$ and nearby spaces (with Z. Bradshaw), Proc. Amer. Math. Soc., 153 (2025), no. 9, pp. 3867-3881.

• Stability for a system of the 2D incompressible MHD equations with fractional dissipation (with W. Feng and J. Wu), J. Math. Fluid Mech., 26 (2024), no. 4, paper no. 57.

• On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions (with E. Abdo), Appl. Anal., 103 (2024), no. 16, 3017-3028.

• On the global well-posedness and analyticity of some electrodiffusion models in ideal fluids and porous media (with E. Abdo and F-N. Lee), SIAM J. Math. Anal., 55 (2023), no. 6, 6838-6866.

• Nonlinear stability for the 2D  incompressible MHD system with fractional dissipation in the horizontal direction (with W. Feng and J. Wu), J. Evol. Equ., 23 (2023), no. 2, paper No. 32, 37 pp.

• On the analyticity and Gevrey regularity of solutions to the three-dimensional inviscid Boussinesq equations in a half space, Commun. Math. Sci., 20 (2022), no. 2, 479-493.

• Global existence and stability for the 2D Oldroyd-B model with mixed partial dissipation (with W. Feng and J. Wu), Proc. Amer. Math. Soc., 150 (2022), no. 12, 5321-5334.

• On the global stability of large solutions for the 3D Boussinesq equations with Navier boundary conditions, Arch. Math. (Basel), 116 (2021), no. 4, 445-456.

• Norm inflation for the Boussinesq system (with Z. Li), Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), no. 10, 5449-5463.

• Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity (with I. Kukavica), Pure Appl. Funct. Anal., 5 (2020), no. 1, 27-45.

• Long time behavior of the solutions to the 2D Boussinesq equations without diffusivity (with I. Kukavica), J. Dynam. Differential Equations, 32 (2020), no. 4, 2061-2077.

• On the global regularity for a 3D Boussinesq model without thermal diffusion, Z. Angew. Math. Phys., 70 (2019), no. 6, paper 174.

Kinetic equations, including regularity, existence, and uniqueness of solutions to Boltzmann and Landau equations, and kinetic Fokker-Planck equations, especially with boundary effects

• A kinetic Nash inequality and precise boundary behavior of the kinetic Fokker-Planck equation (with C. Henderson and G. Lucertini), submitted, 2024.

• Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation (with C. Henderson), Discrete Contin. Dyn. Syst., 44 (2024), no. 4, 1026-1072.

• Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data (with C. Henderson), SIAM J. Math. Anal., 53 (2022), no. 3, 2845-2875.

Applied Mathematics & Interdisciplinary Research

Mathematical biology, incorporating social contagion theory, optimal control, human behavior, stochastic effects, and data-driven problems

• Optimal control in SIR models with noncompliance as a social contagion (with C. Ngo* and C. Parkinson), submitted, 2025. (* denotes an undergraduate author)

• Optimal control of a reaction-diffusion epidemic model with noncompliance (with M. Bongarti and C. Parkinson), European J. Appl. Math., Published online 2025: pp 1-26. https://doi.org/10.1017/S0956792525000130.

• A compartmental model for epidemiology with human behavior and stochastic effects (with C. Parkinson), under revision at Math. Biosci., 2025.

• Analysis of a reaction-diffusion SIR epidemic model with noncompliant behavior (with C. Parkinson), SIAM J. Appl. Math., 82 (2023), no. 5, 1969-2002.

Inverse problems, including geometric inverse problems, partial data problem, and unique continuation

• A Calderón type inverse problem for the active scalar equations with fractional dissipation (with L. Li), Inverse Probl. Imaging,  (Early Access) doi: 10.3934/ipi.2025044 , 2025.

• Quantitative unique continuation for Robin boundary value problems on C^{1, 1} domains (with Z. Li), Indiana Univ. Math. J., 72 (2023), no. 4, 1429-1460.

Applied probability and stochastic processes, including the study of transport-type noise, its regularization and smoothing effects, and random initial data in SPDEs

• Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise (with Q. Lin and R. Liu), Stochastic Process. Appl., 184 (2025), paper no. 104603, 22 pp.

• Global existence for the stochastic Boussinesq equations with transport noise and small rough data (with Q. Lin and R. Liu), SIAM J. Math. Anal., 56 (2024), no. 1, 501-528.

• Almost sure existence of global weak solutions to the Boussinesq equations (with H. Yue), Dyn. Partial. Differ. Equ., 17 (2020), no. 2, 165-183.