# American Institute of Mathematical Sciences

April  2020, 19(4): 2051-2079. doi: 10.3934/cpaa.2020091

## Instability of unidirectional flows for the 2D α-Euler equations

 1 School of Mathematics and Statistics, University of Sydney NSW 2006, Australia 2 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA 3 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 4 International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru, 560089, India 5 Cancer Research Division, Cancer Council NSW, Woolloomooloo, NSW 2011, Australia

*Corresponding author

Dedicated to Prof. Tomás Caraballo on the occasion of his 60-th birthday

Received  January 2019 Revised  July 2019 Published  January 2020

Fund Project: Partially supported by the USA NSF grant DMS-171098.

We study stability of unidirectional flows for the linearized 2D $\alpha$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $\mathbf p \in \mathbb Z^{2}$. We linearize the $\alpha$-Euler equation and write the linearized operator $L_{B}$ in $\ell^{2}(\mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,\mathbf q}$ in $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$ such that the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ covers the entire grid $\mathbb Z^{2}$. The set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ can have zero, one, or two points inside the disk of radius $\|\mathbf p\|$. We consider the case where the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ has exactly one point in the open disc of radius $\mathbf p$. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $L_{B, {\mathbf q}}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.

Citation: Holger Dullin, Yuri Latushkin, Robert Marangell, Shibi Vasudevan, Joachim Worthington. Instability of unidirectional flows for the 2D α-Euler equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2051-2079. doi: 10.3934/cpaa.2020091
##### References:
 [1] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag New York, 1998., [2] D. Albanez, H. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier-Stokes $\alpha$-model,, Asymptotic Analysis, 97 (2016), 139-164.  doi: 10.3233/ASY-151351. [3] M. Beck and C. E. Wayne, Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations, Proc. Royal Soc. Edinburgh Sect. A - Math., 143 (2013), 905-927.  doi: 10.1017/S0308210511001478. [4] L. Belenkaya, S. Friedlander and V. Yudovich, The unstable spectrum of oscillating shear flows, SIAM J. App. Math., 59 (1999), 1701-1715.  doi: 10.1137/S0036139997327575. [5] P. Butta and P. Negrini, On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus, Reg. Chaotic Dyn., 15 (2010), 637-645.  doi: 10.1134/S1560354710510143. [6] S. Chen, C. Foias, D. Holm, E. Olson, E. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338. [7] D. Coutand and S. Shkoller, Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-alpha) equations, Comm. Pure Applied Analysis, 3 (2004), 1-24.  doi: 10.3934/cpaa.2004.3.1. [8] H. R. Dullin, R. Marangell and J. Worthington, Instability of equilibria for the 2D Euler equations on the torus, SIAM J. Appl. Math., 76 (2016), 1446-1470.  doi: 10.1137/15M1043054. [9] H. R. Dullin and J. Worthington, Stability results for idealized shear flows on a rectangular periodic domain, J. Math. Fluid Mech., 20 (2018), 473-484.  doi: 10.1007/s00021-017-0329-2. [10] S. Friedlander, F. Gancedo, W. Sun and V. Vicol, On a singular incompressible porous media equation, J. Math. Phys., 53 (2012), 115-602.  doi: 10.1063/1.4725532. [11] S. Friedlander and L. Howard, Instability in parallel flows revisited, Studies Appl. Math., 101 (1998), 1-21.  doi: 10.1111/1467-9590.00083. [12] S. Friedlander and R. Shvydkoy, The unstable spectrum of the surface quasi-geostrophic equation, J. Math. Fluid Mechanics, 7 (2005), S81-S93.  doi: 10.1007/s00021-004-0129-3. [13] S. Friedlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. Poincare, 14 (1997), 187-209.  doi: 10.1016/S0294-1449(97)80144-8. [14] S. Friedlander and V. Vicol, On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations, Nonlinearity, 24 (2011), 3019-3042.  doi: 10.1088/0951-7715/24/11/001. [15] S. Friedlander, M. Vishik and V. Yudovich, Unstable eigenvalues associated with inviscid fluid flows, J. Math. Fluid. Mech., 2 (2000), 365-380.  doi: 10.1007/PL00000959. [16] C. Foias, D. D. Holm and E. S. Titi, The Navier Stokes alpha model of fluid turbulence, Physica D: Nonlinear Phenomena, 152–153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9. [17] Y. Guo, C. Hallstrom and D. Spirn, Dynamics near unstable, interfacial fluids, Comm. Math. Physics, 270 (2007), 635-689.  doi: 10.1007/s00220-006-0164-4. [18] D. Holm, J. Marsden and T. Ratiu, The Euler Poincare equations and semidirect products with applications to continuum theories, Advances in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [19] D. Holm, J. Marsden and T. Ratiu, Euler-Poincare models of ideal fluids with nonlinear dispersion, Phys. Rev. Letters, 80 (1998), 4173-4176. [20] W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, Cambridge University Press, 1984., [21] A. Kiselev and V. Sverak, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Annals of Math., 180 (2014), 1205-1220.  doi: 10.4007/annals.2014.180.3.9. [22] Y. Latushkin, On 2D Euler equations. I. On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations, J. Math. Phys.., 41 (2000), 728-758.  doi: 10.1063/1.533176. [23] Y. Latushkin, Y. C. Li and M. Stanislavova, The spectrum of a linearized 2D Euler operator, Studies Appl. Math., 112 (2004), 259-270.  doi: 10.1111/j.0022-2526.2004.01510.x. [24] L. D. Meshalkin and Ia. G. Sinai, 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 (1961), 1700-1705.  doi: 10.1016/0021-8928(62)90149-1. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1978. [26] R. Shvydkoy, The essential spectrum of advective equations, Comm. Math. Physics, 265 (2006), 507-545.  doi: 10.1007/s00220-006-1537-4.

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##### References:
 [1] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag New York, 1998., [2] D. Albanez, H. J. Nussenzveig Lopes and E. S. Titi, Continuous data assimilation for the three-dimensional Navier-Stokes $\alpha$-model,, Asymptotic Analysis, 97 (2016), 139-164.  doi: 10.3233/ASY-151351. [3] M. Beck and C. E. Wayne, Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations, Proc. Royal Soc. Edinburgh Sect. A - Math., 143 (2013), 905-927.  doi: 10.1017/S0308210511001478. [4] L. Belenkaya, S. Friedlander and V. Yudovich, The unstable spectrum of oscillating shear flows, SIAM J. App. Math., 59 (1999), 1701-1715.  doi: 10.1137/S0036139997327575. [5] P. Butta and P. Negrini, On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus, Reg. Chaotic Dyn., 15 (2010), 637-645.  doi: 10.1134/S1560354710510143. [6] S. Chen, C. Foias, D. Holm, E. Olson, E. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338. [7] D. Coutand and S. Shkoller, Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-alpha) equations, Comm. Pure Applied Analysis, 3 (2004), 1-24.  doi: 10.3934/cpaa.2004.3.1. [8] H. R. Dullin, R. Marangell and J. Worthington, Instability of equilibria for the 2D Euler equations on the torus, SIAM J. Appl. Math., 76 (2016), 1446-1470.  doi: 10.1137/15M1043054. [9] H. R. Dullin and J. Worthington, Stability results for idealized shear flows on a rectangular periodic domain, J. Math. Fluid Mech., 20 (2018), 473-484.  doi: 10.1007/s00021-017-0329-2. [10] S. Friedlander, F. Gancedo, W. Sun and V. Vicol, On a singular incompressible porous media equation, J. Math. Phys., 53 (2012), 115-602.  doi: 10.1063/1.4725532. [11] S. Friedlander and L. Howard, Instability in parallel flows revisited, Studies Appl. Math., 101 (1998), 1-21.  doi: 10.1111/1467-9590.00083. [12] S. Friedlander and R. Shvydkoy, The unstable spectrum of the surface quasi-geostrophic equation, J. Math. Fluid Mechanics, 7 (2005), S81-S93.  doi: 10.1007/s00021-004-0129-3. [13] S. Friedlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluid, Ann. Inst. Poincare, 14 (1997), 187-209.  doi: 10.1016/S0294-1449(97)80144-8. [14] S. Friedlander and V. Vicol, On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations, Nonlinearity, 24 (2011), 3019-3042.  doi: 10.1088/0951-7715/24/11/001. [15] S. Friedlander, M. Vishik and V. Yudovich, Unstable eigenvalues associated with inviscid fluid flows, J. Math. Fluid. Mech., 2 (2000), 365-380.  doi: 10.1007/PL00000959. [16] C. Foias, D. D. Holm and E. S. Titi, The Navier Stokes alpha model of fluid turbulence, Physica D: Nonlinear Phenomena, 152–153 (2001), 505-519.  doi: 10.1016/S0167-2789(01)00191-9. [17] Y. Guo, C. Hallstrom and D. Spirn, Dynamics near unstable, interfacial fluids, Comm. Math. Physics, 270 (2007), 635-689.  doi: 10.1007/s00220-006-0164-4. [18] D. Holm, J. Marsden and T. Ratiu, The Euler Poincare equations and semidirect products with applications to continuum theories, Advances in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [19] D. Holm, J. Marsden and T. Ratiu, Euler-Poincare models of ideal fluids with nonlinear dispersion, Phys. Rev. Letters, 80 (1998), 4173-4176. [20] W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, Cambridge University Press, 1984., [21] A. Kiselev and V. Sverak, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Annals of Math., 180 (2014), 1205-1220.  doi: 10.4007/annals.2014.180.3.9. [22] Y. Latushkin, On 2D Euler equations. I. On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations, J. Math. Phys.., 41 (2000), 728-758.  doi: 10.1063/1.533176. [23] Y. Latushkin, Y. C. Li and M. Stanislavova, The spectrum of a linearized 2D Euler operator, Studies Appl. Math., 112 (2004), 259-270.  doi: 10.1111/j.0022-2526.2004.01510.x. [24] L. D. Meshalkin and Ia. G. Sinai, 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 (1961), 1700-1705.  doi: 10.1016/0021-8928(62)90149-1. [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1978. [26] R. Shvydkoy, The essential spectrum of advective equations, Comm. Math. Physics, 265 (2006), 507-545.  doi: 10.1007/s00220-006-1537-4.
${\bf p} = (3,1)$; point ${\bf q}_1 = (-1,2)$ is a point of type $I_0$ (green $\Sigma_{{\bf q}_1}$), point ${\bf q}_2 = (-1,1)$ is a point of type $II$ (blue $\Sigma_{{\bf q}_2}$), point ${\bf q}_3 = (0,-2)$ is a point of type $I_+$ (red $\Sigma_{{\bf q}_3}$), and point ${\bf q}_4 = (2,-2)$ is a point of type $I_-$ (brown $\Sigma_{{\bf q}_4}$)
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