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)=(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.

6 pages, Key Words: Euler Equations, Navier-Stokes Equations, Analytical Solutions, Elliptic Symmetry, Makino's Solutions, Self-Similar, Drift Phenomena, Emden Equation, Blowup, Global Solutions

References in corpus (3)

• Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars
• Perturbational Blowup Solutions to the 1-dimensional Compressible Euler Equations
• Some Exact Blowup Solutions to the Pressureless Euler Equations in R^N

Cited by in corpus (6)

• Vortical and Self-similar Flows of 2D Compressible Euler Equations
• Rotational and Self-similar Solutions for the Compressible Euler Equations in R^3
• Exact, Rotational, Infinite Energy, Blowup Solutions to the 3-Dimensional Euler Equations
• Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations
• Rigidity results in generalized isothermal fluids
• Drifting Solutions with Elliptic Symmetry for the Compressible Navier-Stokes Equations with Density-dependent Viscosity