Some Studies On The Large Initial Problems For Nonlinear Hyperbolic Systems

Nonlinear hyperbolic conservation laws is a very important mathematical model,which can be used to describe many physical phenomenon from fluid mechanics,mechanics of elasticity,gas dynamics,biology,aeronautics and astronautics.The compensated compactness theorem is a very important method when studying the existence of global solutions for hyperbolic conservation laws.It can solve many problems which are very hard to other methods such as Glimm s scheme or front-wave tracking algorithm.By applying compensated compactness theorem,the global existence results for several hyperbolic systems are obtained.These systems include a weakly coupled system,linear and nonlinear Keyftiz-Kranzer system with geometrical optics,a generalized system of quadratic type and LeRoux type.The main contributions are as follows.We study a class of weakly coupled hyperbolic system in L? and BV space respectively.The well-posed analysis is also obtained through a technique based on homotopy.The main difficulty lies in obtaining a uniform prior L? or BV estimates on the viscosity solutions.By carefully analyzing the interaction of source terms,we got these uniform estimates under very weak assumptions,and the global existence results is then also obtained.The existence of global solutions for symmetrical and non-symmetrical systems of Keyfitz-Kranzer type with geometrical optics are considered respectively.An obstacle is how to handle the singularity on the region ?=0 and u1=0.By studying the Hloc-1 compactness for sev-eral pairs of functions which is not entropy-entropy flux,we avoided the analysis of singularity mentioned above,and got the global existence results directly by applying compensated com-pactness method.We also consider the existence of global solutions for a generalized systems of quadratic and LeRoux type respectively.The emergence of more linearly degenerate fields brings more difficulties.By carefully analyzing the propagation and cancelation of initial oscillations along the linearly degenerate fields,we got the uniform BV estimates of viscosity solutions,and then the global existence results.A rescaling framework for large BV initial data Cauchy problem is divided into two parts.The first part is devoted to a compactness framework for vanishing viscosity solutions of a non-isentropic gas dynamics in Euler coordinate.The main difficulty is how to get the pointwise convergence of vanishing viscosity solutions without using entropy-entropy flux.By carefully analyzing the BV bounds of s?(·,t),using the kinetic formulation for isentropic gas and the?-perturbation technique,we apply Div-Curl to several pairs of functions which are not entropy-entropy flux and get the desired result.In the second part,we find a new rescale relationship between small initial data and big initial data,showing that the existence of global solutions for small initial data indicates the corresponding results for large initial data.