Scaling invariance and hyperbolic conservation laws

We consider two problems of hyperbolic conservation laws. One is the Riemann problem for the system of conservation laws of one dimensional isentropic gas dynamics in Eulerian coordinates and the other one is the Cauchy problem of Burgers equation. We study these problems using scaling invariance property. In the first example we construct solutions of the Riemann problem by the method of self-similar zero-viscosity limits, where the self-similar viscosity only appears in the equation for the conservation of momentum. No size restrictions on the data are imposed. The structure of the obtained solutions is also analyzed. In the second example we develop a method to study evolution and asymptotic behavior of the Burgers equation which is based on the scaling invariance variables. The Burgers equation is transformed to a conservation law with space dependent flux and we analyze the new equation. We also see the asymptotic behavior of the transformed equation numerically. In this way we can observe the long time behavior of the Burgers equation which was not possible with the original equation.