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

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.