Smooth Solution And Its Blowup Of Multi-dimensional Non-homogeneous Conservation Law

The smooth solutions and the blowup of the smooth solutions of multi-dimensional non-homogeneous scalar conservation law are investigated in this master thesis.Although the Cauchy initial date is a smooth function,because the flux functions are nonlinear,the smooth solution of Cauchy problem may blow up at a finite time instead of global smooth and the singular structures appear,it is different from the case of linear flux functions.At the same time,due to the existence of source term,the characteristics are curves instead of straight lines.The smooth solutions and singular structure of the blowup of the smooth solutions are much more complex than those for the homogeneous case,so it is more difficult to characterize the smooth solutions and the blowup of the smooth solutions.Chapter 1 is the introduction of this paper,the research background of multidimensional conservation law and previous research results are introduced in Chapter 1,including the existence and uniqueness of the weak solutions,the existence and blowup of the smooth solutions and other related research results.In Chapter 2,we give the research object and corresponding concepts,the construction of formula for smooth solutions,the expression of main lemmas and the main theorem.In Chapter 3,we study the smooth solutions of multi-dimensional scalar conservation law with source term g(t,u).By solving the characteristic ordinary differential equations,we give the formula of the smooth solutions by using the implicit function theorem,and obtain the necessary and sufficient condition for the global existence of the smooth solutions,the necessary and sufficient condition for the blowup of the smooth solutions and the expression of the exact time of the blowup.In Chapter 4,we apply the formula of the smooth solutions and the main theorem in Chapter 3 to three examples of one-dimensional non-homogeneous conservation law.The necessary and sufficient condition for the global existence of the smooth solutions and the exact time of the blowup of the smooth solutions of Cauchy problem in examples are calculated.In Chapter 5,we summarize the research results of this paper,and give the direction of further research on this basis.