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

TL;DR

This paper constructs analytical line solutions for the Euler and Euler-Poisson equations with a complex pressure law involving multiple gamma exponents, extending the understanding of solutions for such systems.

Contribution

It introduces the first analytical line solutions for systems with mixed pressure functions involving multiple gamma laws, including potential extensions to damping effects.

Findings

Analytical line solutions for Euler and Euler-Poisson equations with multiple gamma-law pressure.

Solutions can be extended to systems with generalized damping and pressure functions.

Novel approach to handling systems with mixed pressure functions.

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.