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 reflectiondiffraction for potential flow (with Mikhail Feldman). 2. Multidimensional
transonic shock waves, free boundary problems, and nonlinear PDEs of mixed
hyperbolicelliptic 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 & TaiPing Liu). 5. Mathematical theory of divergencemeasure fields and their underlying connections with and applications to nonlinear conservation laws, including the new notions of normal traces, product rules,
and GaussGreen formulas for divergencemeasure fields over general open
domains. 6.
Wellposedness theory and largetime asymptotic behavior of solutions for anisotropic degenerate
parabolichyperbolic equations (with Benoit Perthame): 7.
Isometric embedding and weak continuity of the GaussCodazziGauss 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 LaxFriedrichs
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. Under Construction

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