Line Solutions for the Euler and Euler-Poisson Equations with Multiple Gamma Law

Abstract

In this paper, we study the Euler and Euler-Poisson equations in $R^{N}$, with multiple $\gamma$-law for pressure function: \begin{equation} P(\rho)=e^{s}\sum_{j=1}^{m}\rho^{\gamma_{j}}, \end{equation} where all $\gamma_{i+1}>\gamma_{i}\geq1$, is the constants. The analytical line solutions are constructed for the systems. It is novel to discover the analytical solutions to handle the systems with mixed pressure function. And our solutions can be extended to the systems with the generalized multiple damping and pressure function.