Research On The Weak Solution Of Hyperbolic Conservation Laws

Despite of non-uniqueness of solutions of conservations laws, conservation laws are deduced by physical phenomena, which determine the uniqueness. So how to select the physical solutions is the hot spot. In practical, there are two methods: One is to use entropy inequity, and the other to add vanishing terms. The method of entropy inequity is reflected by dissipation of energy, that is, if external energy doesn't flow into the internal, the total energy can't be increased. The method of adding vanishing terms is reflected by considering the ignored ingredients during modeling the physical phenomena, that is, by adding physical vanishing terms, we can find the solutions of conservation laws by analyzing the solutions of augment equation. Firstly, the third chapter of thesis considers a special vanishing terms. We study the traveling waves of augment equations under the prototype flux and general flux, and analyze how the coefficients of varnishing terms affect the traveling waves. By traveling waves, the weak solutions of conservation laws are constructed.Secondly, since muti-dimensional conservation laws are also the hot spot. By studying the degradation conservation laws, we find the local linear mapping, and use it properly to construct the weak solutions of muti-dimensional conservation laws in the forth chapter. Through a given example, we find that the solutions constructed by our method are identical to that by Yang's method. But restriction of our method is less than yang's and can cope with more general conservation laws.