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Self-similar solutions with elliptic symmetry for the compressible Euler and Navier-Stokes equations in \(\mathbb R^N\). (English) Zbl 1431.35143
Commun. Nonlinear Sci. Numer. Simul. 17, No. 12, 4524-4528 (2012); supplement ibid. 18, No. 6, 1558-1561 (2013).
Summary: 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 \(\mathbb{R}^{N}\) \((N\geq 2)\). 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: \[ \begin{cases} \ddot a_i(t)= \frac{\xi}{a_i(t)\left(\prod^N a_k(t)\right)^{\gamma -1}}, \quad &\text{for } i=1,2,\dots, N\\ a_i(0) = a_{i0}>0, &\dot a(0)=a_{i1} \end{cases}\tag{1} \]
with arbitrary constants \(\xi\), \(a_{i0}\) and \(a_{i1}\). Some blowup phenomena or global existences of the solutions obtained can be shown. Computing simulation or rigorous mathematical proofs for the Emden dynamical system (1), are expected to be followed in the future research.

MSC:
35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:
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