R^{N}中可压缩Euler和纳维-斯托克斯方程的椭圆对称自相似解
Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R^{N}
摘要：
基于Makino的径向对称解，我们推广了可压缩Euler方程和纳维-斯托克斯方程在R^{N}（N \geq 2）中的相应的椭圆对称解.利用分离方法，将Euler方程和纳维-斯托克斯方程化为1+N微分泛函方程。详细地，速度由新颖的Emden动力系统构造：<K1.1/>|a_{i}<K1.1 ilk="MATRIX" >（t）=（ε/（a_{i}（t）（ε a_{k}（t））^{γ-1}）），其中i= 1，2，.，NPDMATHABMN（0）= a_{i0} >0，a_{i}（0）= a_{i1}</K1.1>，具有任意常数ε，a_{i0}和a_{i1}。证明了所得到的解的爆破现象或整体存在性。
英文摘要：
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)=(ξ/(a_{i}(t)(Πa_{k}(t))^{γ-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} </K1.1> with arbitrary constants ξ, a_{i0} and a_{i1}. Some blowup phenomena or global existences of the solutions obtained could be shown.
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