My research interests are in the area of nonlinear partial differential equations, in particular conservation laws of hyperbolic or mixed hyperbolic-elliptic type and its applications in fluid dynamics and phase transitions. I am also working on the numerical analysis for these nonlinear partial differential equations. My previous research interests also lies in mathematical physics, in particular quantum inverse scattering transformation.
2. Quantum inverse scattering method for the nonlinear Schr\"odinger model of fermions with attractive interaction, Journal of Physics A: Mathematical and General. 22 (1989) 4835-4851 ( joint work with F. C. Pu and B. H. Zhao ).
3. The structure of the solutions of the gas dynamics equations and the formation of the vacuum state, Quarterly of Applied Mathematics 49 (1991) 29-48.
4. A limiting `viscosity' approach to the Riemann problem for the materials exhibiting change of phase (II), Archive for Rational Mechanics and Analysis, 116 (1991), 317-337.
5. The uniqueness and stability of solution of the Riemann problem for a system of conservation laws of mixed ty pe, Transaction of American Mathematical Society, 333 (1992) 913-938.
6. The Riemann problem for systems of conservation laws of mixed type, in the Conference Proceedings on Shock Induced Transitions and Phase Structure in General Media, Institute of Mathematics and its Applications, Minneapolis, Oct. 1990, p61-91, jroint work with M. Slemrod.
7. A vanishing viscosity approach on the dynamics of phase transitions in van der Waals fluids, Journal of Differential Equations 103 (1993) 179-204.
8. Large time behavior of Inhomogeneous conservation laws on $S^1$, Archive for Rational Mechanics and Analysis, 125 (1993) 201-216, ( joint work with Jack Hale ).
9. Global versus local admissibility criteria in dynamic phase transitions, Proceedings of Royal Society of Edinburgh 123 (1993) 927-944.
10. One phase Riemann problem and wave interactions in systems of conservation laws of mixed type, SIAM (Society of Industrial and Applied Mathematics) Journal of Mathematical Analysis 24 (1993) 840-865.
11. Attractors in inhomogeneous conservation laws and parabolic regularizations, ( joint work with Jack Hale ), Transaction of American Mathematics Society 347 (1995) 1239-1254.
12. Self-similar solutions for a modified Broadwell model and its hydrodynamic limits, SIAM J. Math. Anal., 28 (1997) 831-851.
13. Large time behavior of elementary waves of Burgers' equation under white noise perturbation, Comm PDE, 20 (1995) 1699-1723
14. Existence of discrete traveling waves and error estimates for Godunov scheme of conservation laws, Math Comp., 67 (1998) 87-109.
15. Existence of discrete traveling waves for a class of monotonicity preserving schemes for conservation laws, Math. Comp. 70 (2000) 1043-1096
16. Traveling waves, Riemann problems and computations of a model of the
dynamics of liquid/vapor phase transitions,
J. Diff. Eqs., 150 (1998)
385-437.
17. Zero reaction limit for hyperbolic conservation laws with source terms, joint work with Shi Jin and Zhenhuan Teng, J. Diff. Eqs. 168 (2000) 270-294.
18. On a model of phase transitions in shock tubes, SIAM J. Appl
Math. 60 (2000), no. 4, 1270--1301.
19. Convergence towards traveling waves in two models arising from the dynamics of phase transitions, J. Diff. Eqs. 168 (2000 ) 102-128.
20. Hyperbolic conservation laws with stiff reaction terms of
monostable type, Trans
AMS. 353 (2001), no. 10, 4139--4154 (electronic)
21. Conservation laws with a degenerate source: Traveling waves, large-time behavior and zero relaxation limit, joint work with Jorg Harterich, submitted to Nonlinear Anal.. (compressed ps file)
22. Dynamic Flows with Liquid/Vapor Phase Transitions, Joint work with M. Slemrod, Invited contribution to the Handbook of Fluid Dynamics. (compressed ps file)
23. Wave patterns, stability and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms, joint work with Shi Jin and J. Miller, to appear in J. Diff. Eqs.. (compressed ps file)
24. Front Motion in Muti-dimensional Conservation Laws with Stiff Source Terms Driven by Mean Curvature and Front Thickness. to appear in Q. Appl. Math. (compressed ps file)
25. Pattern Formation, Wave Propagation and Stability in Conservation Laws with Slow Diffusion and Fast Reaction, joint work with Hailiang Liu (pdf file), submitted to SIAM J. Math. Anal.
26. Symmetry breaking and
ring formation in closed end shock tube experiments on retrograde fluids,
submitted to J. Fluid Mech. (compressed
ps file)