Self-Similar Solutions with Elliptic Symmetry for the Density-Dependent Navier-Stokes Equations in R^{N}

Details Self-Similar Solutions with Elliptic Symmetry for the Density-Dependent Navier-Stokes Equations in R^{N} Self-Similar Solutions with Elliptic Symmetry for the Density-Dependent Navier-Stokes Equations in R^{N}

2011-05-03

arxiv.org/abs/1105.0582v1

Physics

Mathematical Physics

7 Pages, Key Words: Navier-Stokes Equations, Reduction of Equations, Analytical Solutions, Elliptic Symmetry, Yuen's Solutions

Scientific paper

Based on Yuen's solutions with radially symmetry of the pressureless density-dependent Navier-Stokes in $R^{N}$, the corresponding ones with elliptic symmetry are constructed by the separation method. In detail, we successfully reduce the pressureless Navier-Stokes equations with density-dependent viscosity into $1+N$ differential functional equations. In particular for $\kappa_{1}>0$ and $\kappa_{2}=0$, the velocity is built by the new Emden dynamical system with force-force interaction:%\{{array} [c]{c}% \ddot{a}_{i}(t)=\frac{-\xi(\sum_{k=1}^{N}\frac{\dot{a}_{k}(t)}% {a_{k}(t)})}{a_{i}(t)(\underset{k=1}{\overset{N}{\Pi}}% a_{k}(t)) ^{\theta-1}}\text{for}i=1,2,...,N\ a_{i}(0)=a_{i0}>0,\text{}\dot{a}_{i}(0)=a_{i1}% {array}. with arbitrary constants $\xi$, $a_{i0}$ and $a_{i1}$. We can show some blowup phenomena or global existences for the obtained solutions. Based on the complication of the deduced Emden dynamical systems, the author conjectures there exist limit cycles or chaos for this kind of flows. Numerical simulation or mathematical proofs for the Emden dynamical systems are expected in the future.

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