The Qualitative Behavior Of Solutions For Systems Of Hyperbolic Balance Laws

Hyperbolic systems of balance laws is the natural framework of gas dynamics,and hyperbolic systems of conservation laws is the typical and the simplest example.Over the last one hundred and fifty years,there are plenty of works about hyperbolic systems of balance laws,especially about hyperbolic systems of conservation laws.Among them,the two major subjects are blow up theory of classical solutions and stability of weak solutions for the Cauchy problem.On basis of these works,this paper proves that large time asymptotic stability of a superposition of shock waves and contact discontinuities for the one dimensional Jin-Xin relaxation system when there is small perturbation on initial data,and finite time blow up of classical solutions for the one dimensional compressible Euler equations with general pressure law when there is no small compression on initial data.About the large time stability for the Jin-Xin relaxation systems,this paper proves that the global solution of the Cauchy problem for the one dimensional Jin-Xin relaxation system converges to a coupled solution of shock waves and contact discontinuities of the Riemann problem for its equilibrium system when there is small perturbation on initial data.Process of proof mainly depending on classical weighted energy estimate and a careful decay estimate on heat kernel.Particularly,one of two constraints imposed on the characteristic field of linearly degenerate,which is needed in proving the stability for a single contact wave,is removed.About the blow up theory for the Euler equations,this paper proves that the classical solution of the Cauchy problem for the one dimensional compressible Euler equations with general pressure law will blow up in finite time when there is no small compression on initial data.Process of proof relies on the uniform upper bound of the density and delicate estimate for the decoupled Riccati type ordinary differential equations constituted by derivatives of the solutions.It is worth noting that the uniform upper bound of the density is the key point of proving finite time blow up for large data.