After graduating from college, Zhang Tong was assigned to the Institute of Mathematics of China Academy of Sciences to study partial differential equations, and the director of the research office was Wu Xinmou. Partial differential equation was a discipline with weak foundation in China at that time. 1 year later, when Soviet experts came to China to give lectures, Zhang Tong was ordered to respond to the call of state cadres and decentralize the rural labor force in 1957+02. Eight and a half months later, under the situation of the Great Leap Forward, he was transferred back to the office in advance. 1960, Wu Xinmou organized Ding and cooperated with two advanced teachers to complete a paper on the definition of ellipticity of constant coefficient equations. In fact, this paper only exerts a little matrix theory, but unexpectedly leads to the jade works of Hua, Lin Wei and Wu Ziqian. Unfortunately, this is the only mathematical research work that Zhang Tong participated in during the six years from 1956 to 1962.

After the Great Leap Forward, it entered a difficult period and all political movements were suspended. Previously, the research departments of the atomic bomb and the Three Gorges dam project respectively raised questions about shock waves and swells to the Institute of Mathematics. All these problems can be attributed to the study of discontinuous solutions of nonlinear hyperbolic conservation laws. According to the arrangement of the leaders, Zhang Tong and Guo Yufa, who were unable to participate in the national defense task in the political examination, formed a group, with Zhang Tong as the team leader, taught themselves the literature and independently selected topics to engage in relevant basic theoretical research. During the inspection, Zhang Tong was deeply moved by Gelfand's exposition of Riemann problem in Some Problems in the Theory of Quasi-linear Equations.

The basic equation describing gas motion is Euler equation, which consists of three conservation laws of mass, momentum and energy. Its biggest characteristic and difficulty lies in the discontinuity of solution, and shock wave is a kind of compression discontinuity. In 1858, Riemann firmly grasped the characteristics of discontinuity, and put forward and solved the simplest discontinuous initial value problem of Euler equation (that is, the initial value is a ladder function with arbitrary discontinuous points), which was later called Riemann problem. Riemann constructed its four kinds of solutions, which are composed of forward and backward evacuation waves (denoted as sum) and forward and backward shock waves (denoted as sum), that is, (or)+(or), and the discriminant conditions of these four kinds of solutions were given by using the phase plane analysis method. Riemann's work pioneered the concept of "generalized solution" of differential equations and the method of "phase plane analysis", which is very advanced. Riemann laid the first cornerstone for the mathematical theory of nonlinear hyperbolic conservation laws with keen insight and great originality. 1975 The biography of Riemann in the Dictionary of Scientific Biography published in the United States called this work "Riemann's best work in mathematical physics". But Riemann studies a simplified model (one-dimensional isentropic flow), which is not accepted by mechanics. Until the second world war, with the impetus of atomic bomb and supersonic flight research, applied mathematical authorities R. Courant and K. O. Friedrichs extended Riemann's results to one-dimensional non-isentropic flow, adding a contact discontinuity (J, the interface of gases with different densities) between the forward wave and the backward wave. Riemann solution is extended to (or) +J+ (or). R, S and J are collectively called the fundamental wave of Euler equation. From 65438 to 0962, Zhang Tong was deeply moved by the simplicity, beauty and profundity of this work, and decided to choose Riemann problem as his research direction, hoping to extend Riemann's work to more general equations and even high-dimensional situations. Without being understood and accepted by the people around him, he embarked on a challenging exploration road with great interest.

A few months later, Zhang Tong realized a geometrically intuitive idea about the unsolved problem put forward by Gelfand: by constructing convex hull, the convex function in Euler equation was extended to non-convex function, thus the Riemann result was skillfully extended to the case of non-convex equation, and the related entropy conditions were clarified. 1963, when directing the first graduation thesis of the Department of Mathematics of China University of Science and Technology, this idea evolved into the graduation thesis of four people, including Li and Xiao Ling. After this work was further popularized and improved, Zhang Tong and Xiao Ling jointly submitted the Journal of Mathematics at the end of 1963. Unfortunately, it was not until 1977 that this article was published in the Mathematical Journal in the form of a three-page abstract, and it was not until 198 1 that it was published in the American Mathematical Journal. Anal.Appl In 1984, Zhang Tong visited the University of Heidelberg for the first time.

At the same time, the graduation thesis group directed by Wu Xinmou tried to generalize Riemann's results to the case that the initial value contains two discontinuities. Its essence is to study the interaction of four fundamental waves contained in two Riemannian solutions, which can be divided into 16 sub-cases, of which only (+)+(+) sub-case is solved. On this basis, Zhang Tong and Guo Yufa found the general initial values of forward evacuation and backward compression, and proved the existence of the global solution and the invariance of the properties of the initial values of forward evacuation and backward compression to time by phase plane analysis. This is the first result to prove the existence of global discontinuous solution of Euler equation. After 1965 was published in the Journal of Mathematics, from 1967 to 1975, follow-up studies appeared in the United States, the Soviet Union and China, including the generalization of equations and initial values and the proof of uniqueness. Before the famous works of Ding, Chen Guiqiang and Luo appeared in 1985, this paper was considered as the most influential work in the study of China's discontinuous solutions, and was once called "the initial value of China" by P. Lax, an academician of the National Academy of Sciences.

The excellent working situation of 1963 was interrupted by the political movement again in 1964, and it was not until 1972 that it gradually recovered. From 1972 to 1979,, Xiao Ling, Ding, and Li have jointly completed six papers, covering convex and nonconvex Riemannian problems and wave interaction. In these works, they used hard analysis as a tool to deeply study the geometric properties of R lines and S lines on the phase plane, developed the phase plane analysis method and formed their own style. 1978 finally ushered in a new era of reform and opening up. A large number of researchers under the age of 45 from the Institute of Mathematics of the Chinese Academy of Sciences were sent abroad for further study, while Zhang Tong, then 46, continued to study Riemann in China and devoted himself to training the younger generation. He once went to the Graduate School of Chinese Academy of Sciences to teach mathematical equations, which was very popular. In order to help young scholars move to the forefront of research as soon as possible, the summer seminar on partial differential equations in Beijing Xiangshan Botanical Garden 1983 was jointly launched with Peking University Jiang. At that time, the National Foundation of China had not been established, and the funds were very difficult. Fortunately, with the strong support of Wang Guangyin, director of the Differential Equation Research Office of the Institute of Mathematics, young teachers and most graduate students of partial differential equations in China were able to participate at that time, with as many as 80 students. , Jiang, Wu Lancheng, and Xiao Ling offered three courses respectively to impart their professional knowledge. After the final exam, Chen Guiqiang, Xin and four outstanding students were selected to Peking University Institute of Mathematics and Peking University for free training. Later, most of them achieved excellent results and became internationally renowned scholars. In class, Zhang Tong is both the person in charge and the teacher. Due to overwork, occult blood appeared in the stool and was diagnosed as duodenal ulcer bleeding by the hospital. But he kept on working until the workshop ended smoothly (he was ill for many years and bled eight times until he was old). Due to the remarkable achievements, the workshop was held for five consecutive sessions at the request of many colleges and universities. Li Daqian of Fudan University has also participated in the joint organization since the fourth session. The last phase of Suzhou University (when Jiang was the president of Suzhou University) lasted for one semester and * * * offered eight courses. Many of these five students have been active in academic circles at home and abroad. Unexpectedly, five of them became Zhang Tong's students. Under his leadership, * * * initiated a systematic study of the two-dimensional Riemann problem.

On the basis of in-depth study of one-dimensional problems, Zhang Tong began to consider two-dimensional problems in 1984, and cooperated with student Chen Guiqiang to complete two papers, one of which clarified some basic concepts of two-dimensional nonlinear hyperbolic conservation laws, and the other gave the necessary and sufficient conditions for normal reflection in two-dimensional shock wave reflection problems (the mathematical accuracy of correlation discrimination proposed by J. von Neumann, two-dimensional shock wave,

The two-dimensional Riemann problem of Euler equation is a famous problem, and even its formulation needs to be clarified in the 1980s. 1985, Zhang Tong and student Zheng Yuxi studied the Riemann problem (single conservation law) of the following simplest two-dimensional model:

Initial value: u(x, y, t) When t=0, quadrants 1 to 4 on the (x, y) plane are arbitrary constants u 1, u2, u3 and u4, respectively.

When t >; 0, the discontinuity of the initial value on the four rays emitted from the origin will produce four plane fundamental waves R or S. The essence of this problem is to study how these four plane fundamental waves interact, and T > is to study the singularity of the initial value at the origin when T >; How to evolve at 0? By deeply analyzing the singularity of R and S caused by the origin, five solutions to the above problems are constructed. It is a substantial breakthrough in the study of two-dimensional Riemann problems, and it was published in 1989 Journal of the American Mathematical Society. Zhang Tong's cooperation with his students and colleagues came together from 1963 to 1986. Zhang Tong and Xiao Lingcong 1989 jointly published the monograph Riemann Problem and Wave Interaction in Gas Dynamics in the famous Pitman series of Longman Publishing House in Britain. Two years later, mathematicians and mechanics in the United States, the Netherlands and Germany published four book reviews in magazines such as Bulletin of the American Mathematical Society. J. Smoller wrote in the book review, "at first glance, people may be surprised that the whole book is devoted to such a very special problem (referring to the riemann problem)? The answer is: this problem is the most important problem in the whole field of nonlinear hyperbolic conservation laws. " Other book reviews also claim that this monograph can be regarded as a "valuable supplement" or "sequel" to the famous work Supersonic Flow and Shock Wave published by Kurante and Fried Riches in 1948.

After China's opening up, international peers spoke highly of Zhang Tong's work. Spring has finally arrived. In 1978, He Ding won the major achievement award of the National Science and Technology Conference. 1983, Zhang Tong and Xiao Ling won the second achievement award of China Academy of Sciences.

1985, Zhang Tong broke through the two-dimensional Riemann problem of single conservation law, and then turned to the two-dimensional Riemann problem of Euler equation:

Initial value: (p, u, p)(x, y, t) When t=0, quadrants 1 to 4 on the (x, y) plane are in a constant state respectively.

1986 In September, according to the agreement between China Academy of Sciences and the National Foundation of the United States, Zhang Tong visited Professor Liu Taiping at the University of Maryland for half a year upon the nomination of the United States. During Zhang Tong's visit to Berkeley, California with former student Zheng Yuxi, he put forward the following analysis and conjecture to solve the above problems:

The initial value of (1) is discontinuous on the four rays emitted by the origin, and each initial discontinuous line is at t >; 0, emitting three plane fundamental waves (or) +J+ (or). This 12 wave will interact in a cone with the origin as the vertex in (x, y, t) space. In order to simplify the problem without losing its essence, they introduced the assumption that t > each initial break line is at t >; 0, only one plane fundamental wave is emitted. In this way, the problem is reduced to the interaction of four basic waves.

(2) According to the different combinations of the four waves, the problem is divided into 16 categories. (3) Using their respective generalized characteristic analysis methods, the boundaries of interaction cones are found for each class, which are composed of several fixed boundaries (characteristic surface, sonic surface) and/or free boundaries (shock surface). The solution outside the cone is supersonic flow, which consists of four constant initial states and four plane fundamental waves. The transonic flow for which the solution is required is in a cone.

(4) A set of conjectures about the structure of solution in cone (how to distribute and combine shock wave, slip surface, sonic surface and vortex) are put forward, including gas expansion, shock wave reflection and vortex formation. Some new definite solutions of transonic flow are proposed in the conjecture.

In a word, the one-dimensional Riemann problem clarifies the basic waves of conservation law, while the two-dimensional Riemann problem reveals various basic flow field structures formed by the interaction of these basic waves in transonic flow.

This set of conjectures was announced at the relevant conference held in Berkeley in May 1987, which aroused strong repercussions. In a comprehensive article published the following year, J. Glimm, the moderator of the meeting and an academician of the National Academy of Sciences, said that "a complete set of conjectures has been formed". 1990 "conjecture" was published in the United States for 38 pages. Follow-up work has the following three aspects:

A. Perfect classification: According to the sign of vortex, the sliding surface is divided into (J+, J-) Schultz-ringtone, Lax and Liu Xudong, and Zhang Tong, Chen Guiqiang and Yang Shuli finally improved the classification to 19. The six main categories (four R, four S and four J) are all included in the original classification.

B. Numerical experiments: From 1993 to 2002, Schulz-Lynn, Collins and Glades, Graz and Liu Xudong, Zhang Tong, Chen Guiqiang and Yang Shuli, Kourganoff and Tadmor conducted numerical experiments with four completely different calculation formats respectively, and the calculation results were identical.

C. Strict proof: Strict proof is extremely difficult. Starting from 1986, Zhang Tong led his nine students to simplify the model and gradually approach the Euler equation. The main progress is as follows:

C 1。 The simplest model (single conservation law) has been completely solved. In addition to the previous work with Zheng Yuxi, Zhang Tong and Zhang Peng developed the generalized characteristic method when they further studied the initial values of three constants, and obtained the necessary and sufficient conditions for Mach-like reflection when shock waves interacted. /kloc-Mach reflection was discovered in the laboratory in the 0/9th century, and von Neumann once gave a criterion of Mach reflection. So far, there have been countless experimental and numerical simulation results, but there is no strict proof in mathematics. This result can be regarded as the first small step towards this difficult problem.

C2。 When Li Heceng studied the calculation method in 1985, according to the inspiration of mechanics, Euler equation was divided into zero pressure flow equation (reflecting inertia effect) and differential pressure flow equation (reflecting differential pressure effect):

Zhang Pingheng further clarified that the fundamental wave of the former is J, and the fundamental wave of the latter is R and S. The Riemann problem of the former has been completely solved by people such as Sheng Wancheng, Li Jiequan and Chen. The main results are included in 1999 series of special reports of American Mathematical Society. Significantly, in the interaction between J+ and J-, a new nonlinear wave-Dirac-Delta shock wave appears, which consists of the delta function of the density supported on the shock surface, and describes the phenomenon of mass concentration on the low-dimensional manifold. The most interesting thing is that this new phenomenon is also reflected in the numerical experiment of Euler equation. In subsonic flow, the delta shock wave is polished into a smooth delta wave. Previously, this new nonlinear wave was discovered by Zhang Tong, Tan Dechun and Yang Shuli when they were studying a set of non-physical conservation laws. It not only expounds its mathematical mechanism and propagation law, but also proves its stability to viscous disturbance. The research on Dirac-Delta shock wave has continued to this day.

Differential pressure flow is transonic flow, which is divided into 12 categories. In the range of supersonic smooth solution, Zhang Tong and Dai Zichang found a kind of "characteristic decomposition" of the equation, and then proved that the expansion of static gas to vacuum (the extreme case of the first case) has a supersonic solution without discontinuity, and the boundary of the wave interaction cone consists of characteristics and vacuum.

C3。 Euler Equation: In 1950s and 1960s, Soviet and American scholars considered the expansion of static gas into vacuum, but the problem was far from being solved. Li Jiequan, who was a student in Zhang Tong from 65438 to 0999, skillfully found a set of Riemann invariants after several years of exploration, and transformed the nonlinear hyperbolic equation into a linear degenerate hyperbolic equation, thus successfully solving the problem, and finally took the first step of strictly proving the "conjecture", which proved for the first time that the high-dimensional Euler equation had non-axisymmetric solutions in a large range.

It should also be mentioned that Zhang Tong and Zheng Yuxi extended the initial values of four constant states to infinite constant states, and considered the axisymmetric solution of Euler equation, which reduced the problem to a singular point connection problem of a three-dimensional dynamic system. Different from the classical theory, the orbit can be transited by discontinuous points (shock waves). After careful analysis, they completely solved the problem, constructed five solutions, including different combinations of vortex, vacuum, shock wave, evacuation wave and steady state, and found the exact solution of vortex.

1996 and yang Shuli jointly applied for the natural science award of China academy of sciences, and the president of the academy, long, wrote to solicit the opinions of J. glimm, an academician of the national academy of sciences of the United States (national science medal in 2004). Gliem wrote in his reply:

"Professor Zhang's work on the two-dimensional Riemann problem of gas dynamics has defined and led an extremely important research direction in the international academic community. I think the two-dimensional Riemann problem is the most important theoretical problem in the study of nonlinear conservation laws. Professor Zhang has made an authoritative contribution to this issue. His work was original and purely theoretical, and was later supplemented by numerical research. "

"Professor Zhang Tong's main achievement is to give a set of rich images of all two-dimensional Riemann solutions, which far exceeds people's previous understanding and has aroused widespread interest."

That year, Zhang Tong and Yang Shuli won the second prize of Natural Science Award of China Academy of Sciences. Since 1984, Zhang Tong has been invited to the United States, Germany, France, Japan, Taiwan, Hong Kong and Macao for cooperative research, academic visits and international conferences. 65438+ 1989 gave a 90-minute invitation report at the seminar on nonlinear partial differential equations and their applications held in Japan in August. 1998 12 gave a 45-minute invitation report at the first world congress of Chinese mathematicians held in Beijing.

Zhang Tong and his nine students' work from 1986 to 1998 was compiled into a monograph "Two-dimensional Riemann Problem in Gas Dynamics", which was also included in the famous Pitman series of Longman Publishing House in Britain and published in 1998. In 2000, the book review published in Mathematical Review of American Mathematical Society called this research group "China School".

In 2004, Zhang Tong, Li Jiequan and Zhang Peng won the first prize of Beijing Science and Technology Award.

1986, Zhang Tong and Zheng Yuxi completed the paper "Riemann Problem of Gas Combustion Theory", and the review opinion of American Journal of Differential Equations said: "This paper gives the first complete solution of Riemann problem of one-dimensional gas dynamic equation with reaction term. This problem was put forward by Courant and Friderichs in 1945 in basically the same form, and a special solution was given. Until the appearance of this article, people's understanding of this problem almost remained in that state. ..... This is a great progress, which will definitely bring a brand-new look to the research of this subject. ... an important issue ... is indeed a valuable effort. " Since then, Zhang Tong put forward a set of comprehensive ideas on this issue at the summer seminar jointly sponsored by the American Mathematical Society and the American Society of Industrial and Applied Mathematics at 1988: the reaction speed should be considered from finite (ZND) to infinite (CJ); From existence to non-existence, viscosity changes, and the equation ranges from the simplest model to gas dynamics. Through continuous efforts with his collaborators, he has finished six papers. The concept of the simplest model has been basically confirmed, and some substantial progress has been made in the gas dynamic equation.

Zhang Tong is cheerful and sincere; Indifferent to fame and fortune, willing to be lonely; Insist on exploration and originality. After 40 years of unremitting efforts, he led the students to finally create a new path on an important issue. He once said: "I like math as much as music, all in pursuit of the beauty of the world." He is always intoxicated with the beautiful realm of mathematics and music, happily doing his own mathematics with students and savoring life. He also said: "There are many people in China who are smarter than me and work harder than me. I admire them very much. Personally, I am just lucky to come to the treasure house of the Institute of Mathematics, lucky to choose a good subject, concentrate on this small field, and form a little feature. Its true value remains to be tested by time. " "The tough battle on this topic has just begun. My greatest wish is that young people can stick to the path we have found, and victory is beckoning to them. "