Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R^{N}

TL;DR

This paper extends radially symmetric solutions to elliptic symmetry for the compressible Euler and Navier-Stokes equations in multiple dimensions, introducing a novel Emden dynamical system to analyze solution behaviors including blowup and global existence.

Contribution

It develops new elliptic symmetric self-similar solutions for these equations using a separation method and a novel Emden dynamical system, expanding previous radially symmetric solutions.

Findings

Derived solutions exhibit blowup phenomena.

Identified conditions for global existence of solutions.

Extended Makino's solutions to elliptic symmetry.

Abstract

Based on Makino's solutions with radially symmetry, we extend the corresponding ones with elliptic symmetry for the compressible Euler and Navier-Stokes equations in R^{N} (N\geq2). By the separation method, we reduce the Euler and Navier-Stokes equations into 1+N differential functional equations. In detail, the velocity is constructed by the novel Emden dynamical system: {<K1.1/>| <K1.1 ilk="MATRIX" > a_{i}(t)=({\xi}/(a_{i}(t)({\Pi}a_{k}(t))^{{\gamma}-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} </K1.1> with arbitrary constants {\xi}, a_{i0} and a_{i1}. Some blowup phenomena or global existences of the solutions obtained could be shown.