Regularity Of The Weak Entropy Solutions To The Cauchy Problem For γ=3 Isentropic Gas Dynamics Equations

Nonlinear hyperbolic conservation law is one of the core problems in nonlinear PDEs, it has very important physics background. Because of the high nonlinearity of the equations and the complexity of its physics phenomena, the research to it is all the time one of the important research fields in international mathematics. Gas dynamics equations is a sophisticated model of nonlinear hyperbolic conservation law. The global weak entropy solutions' existence and uniqueness of polytropic gas isentropic dynamics equations' general Cauchy problem has been solved. But, to the structure, property, regularity of solutions which has more physics meaning, there are less results now. In this paper, we take theγ= 3 isentropic gas dynamics equations for example to study the regularity of nonlinear hyperbolic conservation law's global weak entropy solutions. The single conservation law is the basis of the equations researching, so in this paper we firstly discuss the global solutions' regularity of single conservation law. Making use of the method of characteristics, we obtain a kind of sufficiency condition of single hyperbolic conservation law Cauchy problem (including shock waves) piecewise smoothness solutions. This kind of method is constructive, and computable, so we also dosimulation in this paper. To the Cauchy problem ofγ=3 isentropic gas dynamics equations, we take the overtaking problem of shock wave and rarefaction wave for example, constructively obtain local piecewise smoothness of weak entropy solutions. And we give the simulation of the overtaking problem.