Instability of Unidirectional Flows for the 2D Navier–Stokes Equations and Related $$\alpha $$-Models

Autor: Shibi Vasudevan
Rok vydání: 2021
Předmět:
Zdroj: Journal of Mathematical Fluid Mechanics. 23
ISSN: 1422-6952
1422-6928
DOI: 10.1007/s00021-021-00568-0
Popis: We study instability of unidirectional flows for the linearized 2D Navier–Stokes equations on the torus. Unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a single vector $${\mathbf {p}} \in {\mathbb {Z}}^{2}$$ . Using Fourier series and a geometric decomposition allows us to decompose the linearized operator $$L_{B}$$ acting on the space $$\ell ^{2}({\mathbb {Z}}^{2})$$ about this steady state as a direct sum of linear operators $$L_{B,{\mathbf {q}}}$$ acting on $$\ell ^{2}({\mathbb {Z}})$$ parametrized by some vectors $${\mathbf {q}}\in {\mathbb {Z}}^2$$ . Using the method of continued fractions we prove that the linearized operator $$L_{B,{\mathbf {q}}}$$ about this steady state has an eigenvalue with positive real part thereby implying exponential instability of the linearized equations about this steady state. We further obtain a characterization of unstable eigenvalues of $$L_{B,{\mathbf {q}}}$$ in terms of the zeros of a perturbation determinant (Fredholm determinant) associated with a trace class operator $$K_{\lambda }$$ . We also extend our main instability result to cover regularized variants (involving a parameter $$\alpha >0$$ ) of the Navier–Stokes equations, namely the second grade fluid model, the Navier–Stokes- $$\alpha $$ and the Navier–Stokes–Voigt models.
Databáze: OpenAIRE
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