**Dr.** **Haitao Fan**

**Professor**** of Mathmatics**

**Ph.D., University of Wisconsin-Madison, 1990**

My main research interests are in the area of nonlinear partial differential equations, including conservation laws of hyperbolic or mixed hyperbolic-elliptic types, systems which are combinations of conservation laws and reaction-diffusion and convection equations. These systems find wide applications in reactive flows and phase transitions. The model I proposed for studying flows involving liquid/vapor phase transitions can qualitatively explain almost all 1-d wave phenomenons observed in actual experiments for fluids with high molar heat capacities. It can even explain some multi-dimensional phenomena such as symmetry breaking and ring formation observed in shock tube experiments.

Some of my works are about the numerical analysis for the nonlinear partial differential equations mentioned above. I have worked in mathematical physics, in particular quantum inverse scattering transformation. I studied the hyperbolic conservation laws with white noise initial data. My recent works extending my interest list to the pricing theory of financial options.

1. Bethe Ansatz equations for the multicomponent nonlinear Schr\"odinger model with supermatrices, Nuclear Physics B 299 (1988) 52-66 ( joint work with F. C. Pu and B. H. Zhao ).

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,

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,

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)

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. J.
Differential Eqs, 189 (2003) 267-291.

24. Front Motion in Muti-dimensional Conservation Laws with Stiff Source Terms Driven by Mean Curvature and Front Thickness Joint work with Shi Jin. Quart. Appl. Math. 61 (2003), no. 4, 701--721.

25. Mathematical analysis of pricing of lookback
performance options, joint work with Der-Chen Chang
and Eric Chang. Appl. Anal. 82 (2003), no. 10, 937--959.

26. Pattern Formation, Wave Propagation and Stability in Conservation Laws with Slow Diffusion and Fast Reaction, joint work with Hailiang Liu , J. Hyperbolic Differ. Equ. 1 (2004), no. 4, 605--626.

27. Symmetry breaking, ring formation and other phase boundary structures in shock tube experiments on retrograde fluids. J. Fluid Mech. 513 (2004), 47--75

28. The Riemann problem for a three-phase flow. Joint work
with A. Corli.

29. The Riemann problem for reversible reactive flows with metastability. Joint work with Andrea Corli.

30. Mathematical analysis of the two-color partial rainbow options. Joint work with Der-Chen Chang, Eric Chang and Duy-Minh Nhieu. Appl. Anal. 84 (2005), no. 7, 737--757.

31. Convergence to equilibrium rarefaction waves for discontinuous solutions of shallow water wave equations with relaxation. Joint work with Tao Luo. Quart. Appl. Math. 63 (2005), no. 3, 575--600.

31. Conservation laws with a degenerate source: traveling waves, large-time behavior and zero relaxation limit. Joint work with Jorg Harterich. Nonlinear Anal. 63 (2005), no. 8, 1042--1069.

32. The heat kernel for Kolmogorov type operators and its applications, joint with DerChen Chang and Ovidiu Calin, . J. Fourier Ananlysis and Applications, online version in Jan 2009. Printed version will appear later.

33. Existence of Collapsing and Explosion Waves in Phase Transitions with Metastability, its Stability and Riemann Problems. Joint work with Xiaobiao Lin. To appear in Discrete and Continuous Dynamical Systems.

34.
Supersonic rarefaction phase boundaries, joint work with Xiaobiao
Lin. Oberwolfach Reports, Vol. 5, Issue
1. 2008.

35. A Dynamical System Approach to Traveling Wave Solutions for Liquid/vapor Phase Transition, Joint work with Xiaobiao Lin. To appear in J. of Dynamics and Diff. Eqs.

36. A critical threshold for the sudden change of the pattern of propagation of condensation. Preprint.

37. Spherically Symmetric
Standing Waves for a Liquid/vapor Phase Transition Model, joint work with Xiao-Biao Lin, preprint.