I.The Basic Information of the Course
Course Number:202012420016
The English Name of the Course：Nonlinear Evolution Equation
The Chinese Name of the Course：非线性发展方程
In-class Hours and Allocation:total class hours:32 classroom teaching:27 class hours, classroom discussion:5 class hours
Credit(s):2
Semester:2
Applied Discipline(Professional Degree Category): Mathematics
Course Object Oriented：Academic Master,Academic Doctor
Evaluation Mode:Curriculum design
Teaching Method:Combination of teaching and discussion at classroom as well as extracurricular research reports
Course Opening Department:College of Mathematical Sciences
II. Prerequisite Course:Advanced mathematics or mathematical analysis, functional analysis, Sobolev space
III. The Objectives and Requirements of the Course
The Objectives of the Course.This course aims to enable the students to use functional analysis and the general methods of nonlinear equations to study the nature of nonlinear evolution equations, and to extend these basic methods to understand and study some special equations and more in-depth problems, further to get some new conclusions.
The Requirements of the Course.Through the study of this course, students are required to master the research background, research objectives and research methods of basic nonlinear equations. Furthermore, students are required to master the new methods and new progress in the research of development equations. So the students should basically have the ability of independent research.
IV. The Content of the Course
Chapter 1Nonlinear heat conduction equation
The Cauchy problem of the n-dimensional heat conduction equation, the decay estimate of the solution, some estimates of the product function and the composite function, the Cauchy problem of the n-dimensional nonlinear heat conduction equation, and the case where the nonlinear right end F does not contain u.
Chapter 2 Nonlinear wave equation
The Cauchy problem of n-dimensional wave equations, some estimates of solutions to the Cauchy problem of n-dimensional homogeneous wave equations, the Cauchy problem of n-dimensional non-homogeneous wave equations, some estimates of product functions and compound functions, Cauchy problem of quasi-linear, nonlinear wave equation.
Chapter 3 Semilinear wave equation
Some important invariant features and their functions in the equations, the introduction of the analysis foundation required for semi-linear wave equations including integrable spaces, differentiable spaces, and Sobolev spaces. The introduction of the theory of local well-posedness of nonlinear wave equations.
Chapter 4 Semilinear parabolic equation and nonlinear parabolic equation
Steady-state problems, basic properties of solutions, well-posedness of solutions with subcritical energy including an application of comparative principles and families of potential well, well-posedness of solutions with critical energy, well-posedness of solutions with supercritical energy.
Chapter 5 Nonlinear quasi-hyperbolic equation
Variational principle, potential well method, invariant set of nonlinear quasi-hyperbolic equations, existence and uniqueness of solutions of nonlinear quasi-hyperbolic equations.
Chapter 6 Strongly damped nonlinear wave equation
High-energy blow up problems for strong dissipative wave equations with weak damping term, strongly damped wave equations with fourth-order spatial dispersive terms, and fourth-order strongly dispersive-dissipative wave equations.
Chapter 7 Various high-order nonlinear evolution equations
The fourth-order generalized Boussinesq equations including the introduction of families of potential well, the global existence and finite time blow up of solutions with subcritical energy and critical energy.The sixth-order generalized Boussinesq equation including the introduction of potential well families, the global existence and finite time blow up of solutions with subcritical energy and critical energy as well as supercritical energy.
Chapter 8 Integral estimation method and energy estimation method
Calerkin approximation, Calerkin method and energy estimation.
Chapter 9 Compact method
Embedding theorem, embedding and compact embedding.
Chapter 10 Monotone operator method
The expansion of monotone nonlinear operators, the existence and uniqueness of generalized solutions of monotone operator equations, and the application of monotone operator methods on partial differential equations.
Chapter 11The initial boundary value problem and Cauchy problem of various nonlinear equations
The existence and uniqueness of the local solutions, the existence of global solution, the non-existence of global solutions and the bound of global solutions to the initial boundary value problems and Cauchy problems of various nonlinear equations, as well as some relevant open problems.
Chapter 12 Existence, uniqueness, non-existence, blow-up and asymptotic behavior of generalized and classical solutions
The connection and difference between the weak solution and the classical solution, the existence and uniqueness of solutions for various development equations, the blow-up of the solution with big initial data, and the asymptotic behavior of the solution.
Chapter 13Nonlinear evolution equation with non-positive definite energy
Characteristics of nonlinear evolution equations with non-positive definite energy, variational structure and dynamic behavior of nonlinear evolution equations with non-positive definite energy.
Course for Ideological and Political Education
In terms of curriculum ideological and politics, with the rapid development of modern science and technology and the deepening of understanding of nature, people began to use nonlinear evolution equation theory to study complex natural phenomena and solve complex engineering and technical problems. In the fields of aerospace, semiconductor materials science, astrophysics, theoretical physics, etc., many problems can be attributed to nonlinear evolution equations. The study of several types of development equations with a clear applied background and important theoretical significance is one of the mainstreams of research in the international mathematics community, and is the focus and hotspot of other natural science and engineering technology departments. Therefore, an in-depth understanding of the nonlinear evolution equation is helpful to deepen the understanding of the nonlinear phenomena in nature, and to help reveal the interaction between various nonlinearities and their influence on the nature of the solution, and thus promote the rigor and precision of various disciplines, and then promote the development of science and technology. At the same time, new ideas and methods will also be generated in the research process, which will promote the innovation and development of mathematics itself. Therefore, studying nonlinear evolution equations can not only solve problems theoretically, but also promote the development of aerospace, astrophysics, semiconductor materials and other undertakings.
This course uses the case of teaching methods, combined with the great achievements of the nonlinear evolution equations in the reform and opening up for more than 40 years in aerospace, astrophysics, semiconductor materials, etc., to stimulate students national pride and patriotic enthusiasm, and enhance students "four confidence" to allow students at HEU to make outstanding contributions to the defense industry at an early date.
The first example: Professor Zhou Yulin, an academician of the Chinese Academy of Sciences, is an expert in the field of nonlinear evolution equations in my country. In particular, he has laid the foundation for the research of parabolic equations in my country, and cultivated a large number of outstanding talents in this field. In the 1960s, Professor Zhou Yulin began to participate in the theoretical research of nuclear weapons in China and made outstanding contributions to the theoretical research of nuclear weapons in my country.
Professor Zhou Yulin was engaged in topology research in his youth. In order to speed up the research of partial differential equations in the motherland, Professor Zhou Yulin went to Moscow University to study in 1954, focusing on nonlinear partial differential equations. After obtaining a doctorate, he gave up the opportunity to be a professor at Moscow University without hesitation and returned to work in the Department of Mathematics, Peking University. In 1957, Professor Zhou Yulin opened a special class for nonlinear partial differential equations at Peking University. A number of high-level teaching and scientific research talents have emerged from the class, such as Professor Jiang Lishang, former president of Suzhou University; Professor Ye Qixiao, former head of the Department of Mathematics, Beijing Institute of Technology; Professor Ying Longan, former head of the Department of Mathematics, Peking University, etc. In the following thirty or forty years, these students were active at the forefront of nonlinear partial differential equation research. In 1960, for the motherland's national defense cause, Academician Zhou Yulin once again gave up the original research career, resolutely participated in China's nuclear weapons theoretical research work, engaged in the numerical simulation and fluid mechanics research in nuclear weapon theoretical research. Professor Zhou Yulin applied the theory of partial differential equation to the theoretical research of nuclear weapons, and solved many problems encountered in the research process of nuclear weapons.
The research results of the nonlinear equations involved in this course also gather the wisdom of a group of Chinese mathematicians. Therefore, through the introduction of Chinese mathematicians, students can understand the important results about nonlinear evolution equation in the national defense, learn the feelings of the older generation of mathematicians for the country and the people, learn the sacrifice spirit of the older generation of mathematicians for the development of the motherland, increase the patriotic feelings of young students, and establish correct values for young students.
The second example: Academician Gu Chaohao, an academician of the Chinese Academy of Sciences, as the representative of the second generation of Chinese modern mathematicians, has made outstanding contributions in the three areas of differential geometry, partial differential equations and mathematical physics known as the "Golden Triangle", especially the school of partial differential equation of Fudan University founded by him, which is internationally known. Three students, Li Daqian, Hong Jiaxing, and Chen Shuxing, have been awarded academicians of the Chinese Academy of Sciences. As the only second-generation mathematician who has won the highest national science and technology award to date, Gu Chaohao's outstanding mathematical achievements are recognized worldwide.
During his youth, Professor Gu Chaohao participated in the revolutionary cause while eagerly learning differential geometry with the famous mathematician Professor Su Buqing. While Professor Gu Chaohao's achievements in differential geometry are striking, he is keenly aware of the new demands placed on mathematics by the development of cutting-edge technology. In order to meet the needs of the national scientific development, he resolutely put his main energy into the new research field of partial differential equations. He is not only devoted to the theoretical research of partial differential equations, but also to applying mathematics to aerospace. The method he proposed has played a leading role in the calculation of the supersonic flow of blunt-headed objects in my country's missiles. He made important contributions to the national defense of the motherland. In addition, in the teaching career of more than 60 years, Professor Gu Chaohao has won the world. Among his students, 9 were elected as academicians of the Chinese Academy of Sciences or the Chinese Academy of Engineering.
The basic theory of PDEs involved in this course also contains the research results of Chinese mathematicians. Therefore, through the introduction of early Chinese mathematicians, the students can understand the research achievements of Chinese mathematicians in the field of partial differential equations, learn the spirit that the older generation of mathematicians will never give up, learn the spirit that mathematicians love the motherland, increase the patriotic faith of young students, and encourage young students to work hard.
V.Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1.徐润章，杨延冰.非线性发展方程的初值依赖问题.科学出版社，2017年.
2. Lawrence C.Evans. Partial Differential Equations. American Mathematical Sociey, 2003.
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