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Current Research Projects:
Some of the Previous Research Results: 1. Mathematical theory of shock reflection/diffraction and von Nuemann’s conjectures, including the solution to the sonic conjecture and detachment conjecture for shock reflection-diffraction for potential flow (with Mikhail Feldman). 2. Multidimensional
transonic shock waves, free boundary problems, and nonlinear PDEs of mixed
hyperbolic-elliptic type. 3. Compactness and continuity of nonlinear partial differential equations, including the isentropic Euler equations, compensated compactness, and related problems in nonlinear conservation laws.
4. Relaxation
theory via entropy for hyperbolic conservation laws with stiff relaxation
terms (with David Levermore & Tai-Ping Liu). 5. Mathematical theory of divergence-measure fields and their underlying connections with and applications to nonlinear conservation laws, including the new notions of normal traces, product rules, and Gauss-Green formulas
for divergence-measure fields over general open domains. 6.
Well-posedness theory and large-time asymptotic behavior of solutions for anisotropic degenerate parabolic-hyperbolic
equations (with Benoit Perthame): 7.
Isometric embedding and weak continuity of the Gauss-Codazzi-Gauss equations; connections with the Euler
equations and related PDEs in continuum mechanics (fluid dynamics,
elasticity, and materials science). 8.
Theoretical analysis of numerical methods, including
the first convergence proof of the Lax-Friedrichs scheme and Godunov scheme
for the system of isentropic Euler equations. 9.
Vanishing viscosity solutions of the
compressible Euler equations with spherical symmetry and large initial data. 10. Global finite-energy
solutions of the compressible Euler-Poisson equations for general pressure
laws with large initial data of spherical symmetry. *Please also see the article: Some
Contributions of Gui-Qiang G. Chen to Nonlinear Conservation Laws and Partial
Differential Equations (by Mikhail Feldman, Xianpeng Hu, Dehua Wang, Wei Xiang, and Tong Yang), published in Communications in
Mathematical Analysis and Applications, Vol. 3(2), 121–135 (2024),
for some of Gui-Qiang G. Chen’s scientific research contributions to the
areas. --Under Construction
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