Abstract
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.
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Notes
A derivation of the equation analogous to (1.3) for the Euler equation (\(\nu =0\)) is given in Lemma 6.1 in the Appendix of [7]. The arguments therein can be adapted to derive (1.3) from (1.1). Since \(\omega =-\Delta \psi \), the Fourier modes of \(\omega \) and \(\psi \) satisfy the relation \(\omega _{{\mathbf {k}}}=\Vert {\mathbf {k}}\Vert ^{2}\psi _{{\mathbf {k}}}\). One then uses the formula \({\mathbf {u}} = (\psi _{y},-\psi _{x})\) and the relation \(\psi _{{\mathbf {k}}} = \Vert {\mathbf {k}}\Vert ^{-2}\omega _{{\mathbf {k}}}\) to obtain the Fourier modes of \({\mathbf {u}}\), see again, Lemma 6.1 in the Appendix of [7].
References
Albanez, D., Nussenzveig-Lopes, H.J., Titi, E.S.: Continuous data assimilation for the three-dimensional Navier–Stokes \(\alpha \)-model. Asymptot. Anal. 97(1–2), 139–164 (2016)
Beck, M., Wayne, C.E.: Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations. Proc. R. Soc. Edinb. Sect. A Math. 143, 905–927 (2013)
Belenkaya, L., Friedlander, S., Yudovich, V.: The unstable spectrum of oscillating shear flows. SIAM J. App. Math. 59(5), 1701–1715 (1999)
Berselli, L.C., Bisconti, L.: On the structural stability of the Euler–Voigt and Navier–Stokes–Voigt models. Nonlinear Anal. Theory Methods Appl. 75(1), 117–130 (2012)
Chen, S., Foias, C., Holm, D., Olson, E., Titi, E., Wynne, S.: Camassa–Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81(24), 5338–5341 (1998)
Coti Zelati, M., Elgindi, T.E., Widmayer, K.: Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2D Euler Equations.arXiv:2007.11547
Dullin, H., Latushkin, Y., Marangell, R., Vasudevan, S., Worthington, J.: Instability of the unidirectional flows for the 2D \(\alpha \)-Euler equations. Commun. Pure Appl. Anal. 19(4), 2051–2079 (2020)
Dullin, H.R., Marangell, R., Worthington, J.: Instability of equilibria for the 2D Euler equations on the torus. SIAM J. Appl. Math. 76(4), 1446–1470 (2016)
Dullin, H.R., Worthington, J.: Stability results for idealized shear flows on a rectangular periodic domain. J. Math. Fluid Mech. 20(2), 473–484 (2018)
Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes alpha model of fluid turbulence. Physica D 152–153, 505–519 (2001)
Frenkel, A.L.: Stability of an oscillating Kolmogorov flow. Phys. Fluids A 3(7), 1718–1729 (1991)
Frenkel, A.L., Zhang, X.: Large-scale instability of generalized oscillating Kolmogorov flows. SIAM J. Appl. Math. 58(2), 540–564 (1998)
Friedlander, S., Howard, L.: Instability in parallel flows revisited. Stud. Appl. Math. 101(1), 1–21 (1998)
Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. Poincare 14(2), 187–209 (1997)
Friedlander, S., Vishik, M., Yudovich, V.: Unstable eigenvalues associated with inviscid fluid flows. J. Math. Fluid Mech. 2(4), 365–380 (2000)
Gesztesy, F., Makarov, K.: (Modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels revisited. Integr. Equ. Oper. Theory 48, 561–602 (2004)
Gesztesy, F., Latushkin, Y., Makarov, K.: Evans functions, Jost functions, and Fredholm determinants. Arch. Ration. Mech. Anal. 186, 361–421 (2007)
Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. I. Birkhauser Verlag, Basel (1990)
Holm, D., Marsden, J., Ratiu, T.: The Euler Poincare equations and semidirect products with applications to continuum theories. Adv. Math. 137(1), 1–81 (1998)
Holm, D., Marsden, J., Ratiu, T.: Euler–Poincare models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80(19), 4173–4176 (1998)
Jones, W.B., Thron, W.J.: Continued Fractions: Analytic Theory and Applications. Cambridge University Press, Cambridge (1984)
Latushkin, Y., Li, Y.C., Stanislavova, M.: The spectrum of a linearized 2D Euler operator. Stud. Appl. Math. 112, 259–270 (2004)
Latushkin, Y., Vasudevan, S.: Stability criteria for the 2D \(\alpha \)-Euler equations. J. Math. Anal. Appl. 472(2), 1631–1659 (2019)
Latushkin, Y., Vasudevan, S.: Eigenvalues of the linearized 2D Euler equations via Birman–Schwinger and Lin’s operators. J. Math. Fluid Mech. 20(4), 1667–1680 (2018)
Li, Y.: On 2D Euler equations I On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations. J. Math. Phys. 41, 728–758 (2000)
Liu, X.L.: An example of instability for the Navier–Stokes equations on the 2-dimensional torus. Commun. Partial Differ. Equ. 17(11–12), 1995–2012 (1992)
Liu, X.L.: Instability for the Navier–Stokes equations on the 2-dimensional torus and a lower bound for the Hausdorff dimension of their global attractors. Commun. Math. Phys. 147(2), 217–230 (1992)
Lopes Filho, M., Lopes, H.N., Titi, E., Zang, A.: Approximation of 2D Euler equations by the second-grade fluid equations with Dirichlet boundary conditions. J. Math. Fluid Mech. 17, 327–340 (2015)
Meshalkin, L.D., Sinai, I.G.: Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech. 25, 1700–1705 (1961)
Yudovich, V.I.: Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech. 29(3), 527–544 (1965)
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Communicated by R Shvydkoy.
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The author is indebted to Professor Yuri Latushkin (University of Missouri-Columbia/Courant Institute of Mathematical Sciences) for several helpful discussions during the course of the preparation of this manuscript. The author also thanks Aleksei Seletskiy for making available the code from his forthcoming website and Yadugiri V. Tiruvaimozhi for help with the numerical illustrations. The author thanks the (anonymous) referees for their comments which improved the exposition of the paper.
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Vasudevan, S. Instability of Unidirectional Flows for the 2D Navier–Stokes Equations and Related \(\alpha \)-Models. J. Math. Fluid Mech. 23, 35 (2021). https://doi.org/10.1007/s00021-021-00568-0
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DOI: https://doi.org/10.1007/s00021-021-00568-0