Actively engaged in mathematical research and dedicated to a close combination of basic theory with practical applications, Professor Li has successfully achieved a large number of important results in his research:
1. Systematic study on the local solvability and the global regularity with applications for general quasilinear hyperbolic systems in two independent variables.
A . Mainly motivated by the shock phenomenon in gas dynamics, he has established a complete theory on the local solvability for classical solutions and classical discontinuous solutions to the general quasilinear hyperbolic system in two independent variables (Li Tatsien, Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985).
(1). He initiated a systematic study on free boundary problems with shock waves and contact discontinuities regarded as instances of free boundaries for the quasilinear hyperbolic system. A unified framework is suggested and a simple necessary and sufficient algebraic condition is offered to solve various types of boundary value problems and free boundary problems for the general quasilinear hyperbolic system.
(2). Based on the model of centered rarefaction waves in gas dynamics, an effective way is proposed to overcome the difficulties brought about by multi-valued singularities and consequently an integrated theory is put forward for the centered wave solution to the general quasilinear hyperbolic system.
(3). As a result, a well-known result on the Riemann problem, given by P. D. Lax, is extended to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws, thus the local structure of discontinuous solutions is thoroughly revealed.
B. He has made an essential contribution on global classical solutions and global classical discontinuous solutions to quasilinear hyperbolic systems (Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Recherches en Math¨¦matiques Appliqu¨¦es 32 , Masson/John Wiley, 1994; Li Tatsien, Wang Libin, Global Propagation of Regular Nonlinear Hyperbolic Waves, Progress in Nonlinear Differential Equations and Their Applications, Vol. 76, Birkhauser, 2009).
(1). By introducing the concept of ¡°weak linear degeneracy¡± and the method of ¡°normalized coordinates¡±, the existence of global classical solutions is successfully discussed and a sharp estimate is obtained for the upper and lower bounds of the life-span of classical solutions for the Cauchy problem of the general quasilinear hyperbolic system with small and decaying initial data. This complete theory with applications in many important physical situations covers and essentially improves all the classic results previously obtained by F. John, L. Hörmander and Tai-Ping Liu under certain special hypotheses. A related open problem proposed by A. Majda in his monograph is then solved.
(2). Professor Li (with J. M. Greenberg) has shown the positive effect of the presence of boundary dissipation on the global regularity of the solution to the quasilinear hyperbolic system. This result promotes a series of successive work.
(3). The study of the global classical solution is extended form the Cauchy problem and mixed initial-boundary value problems with fixed boundaries to problems with moving boundaries and free boundaries, which are more important and much more difficult. A systematic theory is set up for the first time and a large class of nontrivial global classical discontinuous solutions is constructed for the problems including shock waves and (or) contact discontinuities.
(4). The singularity caused by eigenvectors as a new concept is proposed and the mechanism of the formation of singularities is deeply studied.
2. Complete result and unified framework on the global existence and the life-span of classical solutions to fully nonlinear wave equations (Li Tatsien, Chen Yunwei, Global Classical Solutions for Nonlinear Evolution Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics 45, Longman Scientific & Technical, 1992).
To Study the global existence and the life-span of classical solutions to fully nonlinear wave equations, Professor Li proposed a simple and unified framework------the global iteration method. With the help of some fine estimates on the solution to the linear wave equation, a complete result is derived on the global existence and the life-span of the classical solution for any space dimension n(¡Ý1) and for any integer order p(¡Ý2) of the nonlinear right-hand side . All the previous results established with various methods by F. John, L. Hörmander, S. Klainerman and D. Christodoulou etc. under special hypotheses are covered and improved so successfully that the problem can be virtually regarded as perfectly resolved.
3. Efficient mathematical model and methods on the resistivity well-logging and the spontaneous potential well-logging with applications in petroleum exploitation.
A. A unified basic theoretical and numerical framework has been established for various kinds of resistivity well-logging (Li Tatsien et al., Applications of the Finite Element Method in Electric Well-Loggings, Oil Industry Press, 1980 (in Chinese); Li Tatsien et al., Boundary Value Problems with Equivalued Surface and Resistivity Well-Logging, Pittman Research Notes in Mathematics Series 382, Longman, 1998).
(1) Based on a series of physical and mechanical models and especially on the resistivity well-logging in petroleum exploitation, Professor Li has described a new type of boundary value problem, namely, the boundary value problem with equivalued surface, and established an integrated theory.
(2) With the practical problem of resistivity well-logging by the patched electrode as the background, the idea and theory on the homogenization of boundary conditions is introduced and developed, resulting in a great reduction of computation complexity.
(3) A unified mathematical model illustrated by boundary value problems with equivalued surface is formed and highly efficient numerical scheme is designed for various types of resistivity well-loggings. Moreover, a convenient treatment of the corresponding problem concerning the patched electrode is suggested by applying the above-mentioned theory on homogenization of boundary conditions. This result which provides a basic theoretical and numerical framework becomes now a classic in the area of resistivity well-loggings. The instrument of mirospheric focusing well-logging made according to this framework has been actually employed by more than 10 domestic oil fields for more than 25 years up to now, bringing about better geological interpretation and considerable economic benefits.
B. A basic theoretical and numerical framework has been first established for the spontaneous potentil well-logging (Li Tatsien et al., Mathematical Model of Spontaneous Potential Well-Logging and Its Numerical Solutions, to be published in Spring Briefs in Mathematics).
(1) Establish the mathematical model for the spontaneous potential well-logging, which contains discontinuous interface conditions. The well-posedness of the model has been shown by means of Soblev spaces with fractional power.
(2) Propose several methods such that the finite element method can be applied to get the corresponding numerical solution with very good accuracy.
(3) This framework has been successfully applied to practical applications in petroleum explaintation.
4. Complete theory and efficient constructive framework on the exact controllability and observability for 1-D quasilinear hyperbolic systems (Li Tatsien, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS on Applied Mathematics 3, American Institute of Mathematical Sciences (AIMS) & Higher Education Press, 2010).
A. By establishing the theory on the semi-global classical solution to quasilinear hyperbolic systems, Prof. Li has presented a simple and clear constructive framework with modular structure and establish a systematic and complete theory in the quite open situation --- the quasilinear case on the exact boundary controllability and observability for general 1-D quasilinear hyperbolic systems.
B. The methods and results have been generalized to the case of tree-like network with general topology so that a complete theory has also been established.
C. The related methods and results have been applied to give the concept and theory of exact boundary controllability of nodal profile, stimulated by the demand of applications.