Perturbational Blowup Solutions to the 1-dimensional Compressible Euler Equations

TL;DR

This paper develops a perturbational method to construct analytical solutions, including blowup and global solutions, for the 1D compressible Euler equations, extending previous results.

Contribution

It introduces a novel perturbational approach using a linear velocity perturbation and Hubble's transformation to find new analytical solutions.

Findings

Constructed a new class of blowup solutions.

Extended previous solutions by including non-zero perturbations.

Proved existence of solutions through functional differential equations.

Abstract

We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity:% \begin{equation} u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler equations. By comparing the coefficients of the polynomial, we could deduce the corresponding functional differential system of $(c(t),b(t),\rho^{\gamma-1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c(t)=\frac{\dot{a}(t)}{a(t)}, \end{equation} the functional differential equations can be simplified to be the system of $(a(t),b(t),\rho^{\gamma-1}(0,t))$. After proving the existence of the corresponding ordinary differential equations, a new class of blowup or global solutions can be shown. Here, our results fully cover the previous known ones by choosing $b(t)=0$.