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).
transonic shock waves, free boundary problems, and nonlinear PDEs of mixed
3. Compactness and continuity of nonlinear partial differential equations, including the isentropic Euler equations, compensated compactness, and related problems in nonlinear conservation laws.
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
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.
Vanishing viscosity solutions of the compressible
Euler equations with spherical symmetry and large initial data.
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