On Research Of Formulas For Weak Solution And Relating Problems Of Hyperbolic Conservation Law

In this paper, we firstly consider1-D homogeneous and non-homogeneous Hyperbolic Con-servation Law, and construct several explicit formulas for the weak solution of Cauchy problem; then, we research Riemann problem of n-D non-homogeneous Conservation Law, and find the explicit expression of n-D non-selfsimilar basic wave solution; at last, we deal with Riemann problem of2-D Conservation Law, give more general global structure for its non-selfsimilar so-lution, and propose a new method for classification of global structure of non-selfsimilar solution based on the concepts of envelope.In chapter3, we consider the Cauchy problem of1-D Scalar Conservation Law,and give the explicit formula for weak solution of Cauchy problem. The basic assumptions on the flux pair (F(), G(u)) are that F’(u) is positive on R, and ((?))’≥0has only finite zero points; our initial data can be unbounded, but necessarily satisfying a restriction on its growth at infinite.It has an important corollary of Chapter3as follow:when F(u)=u, G{u)=f(u),the equation becomes ut+f(u)x=0, we only need the flux function f(u) satisfying f"(u)≥0(having finite zero points) and the initial data being local bounded, then the formula for weak solution which we obtain generalizes the famous Lax-Oleinik formula for1-D homogeneous scalar conservation law.In chapter4, we consider Cauchy problem of1-D Scalar Conservation Law with a linear source term(4.1.1). For general convex flux f(u) and bounded initial data, we define a real func-tion by the characteristics, which is used to define the weak solution of Cauchy problem, and then we get the formula for weak solution of Cauchy problem, we do not find any explicit formula for the weak solution of non-homogeneous conservation law in open mathematics journals before.In chapter5, we consider Riemann problem of Multi-dimensional Conservation Law(5.1.1). The initial data has two pieces of constant states, which are separated into two unbounded regions by a curved initial discontinuity, Since the source term exists, the characteristics are no longer lines, and then the values on both sides of the Shock surface must be non-constant, it is more complicate; when Condition ((?)) holds along the initial discontinuity, we construct the n-D Shock wave and n-D Rarefaction wave, which are the basic two cases of initial data problem. Then we explicitly calculate two examples, one of which is the case of a period source term existed, and discover essential difference between the structure of n-D basic waves of non-homogeneous case and homogeneous case.At last, we research the non-selfsimilar solution of Riemann problem of2-D Scalar Con-servation Law, The initial data has two pieces of constant states, which are separated into two unbounded regions by a curved initial discontinuity, but Condition ((?)) does not hold along the initial discontinuity, so the characteristics counterchange together, from analysis and classification of envelope, we propose the concepts of plus envelope, minus envelope and mixed envelope, then we classify some global structure for2-D non-selfsimilar solution based on the type of envelope, and discovere some new structures and evolution phenomena.