Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in RN

► We constructe the solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations. ► We reduce the Euler and Navier-Stokes equations into 1 + N differential functional equations. ► The constructed velocity is constructed by the novel Emden dynamical system. ► Some blowup phenomena or global existences of the solutions can be shown. ► Computing simulation or rigorous mathematical proofs for the Emden dynamical system, are expected to be followed.