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

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

2011-04-19

arxiv.org/abs/1104.3687v1

Physics

Mathematical Physics

6 pages, Key Words: Euler Equations, Navier-Stokes Equations, Analytical Solutions, Elliptic Symmetry, Makino's Solutions, Sel

Scientific paper

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: {| 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} with arbitrary constants {\xi}, a_{i0} and a_{i1}. Some blowup phenomena or global existences of the solutions obtained could be shown.

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