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 von Neumann's sonic conjecture and detachment conjecture for shock reflectiondiffraction for potential flow (with M. 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. Entropy
theory for hyperbolic conservation laws with stiff relaxation terms (with D. Levermore & T.P. Liu): 5. Mathematical
theory and applications of divergencemeasure fields and conservation laws,
including the GaussGreen theorem and normal traces for weakly differentiable
fields. 6.
Wellposedness theory and largetime asymptotic behabior of solutions for anisotropic degenerate
parabolichyperbolic equations (with Benoit Perthame): 7.
Isometric embedding and weak continuity of the GaussCodazziGauss equations 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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