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. AlbanezH. 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. BelenkayaS. 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. ChenC. FoiasD. HolmE. OlsonE. 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. DullinR. 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. FriedlanderF. GancedoW. 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. FriedlanderW. 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. FriedlanderM. 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. FoiasD. 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. GuoC. 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. HolmJ. 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. HolmJ. 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. LatushkinY. 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.

show all references

References:
[1]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag New York, 1998.,

[2]

D. AlbanezH. 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. BelenkayaS. 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. ChenC. FoiasD. HolmE. OlsonE. 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. DullinR. 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. FriedlanderF. GancedoW. 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. FriedlanderW. 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. FriedlanderM. 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. FoiasD. 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. GuoC. 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. HolmJ. 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. HolmJ. 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. LatushkinY. 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.

Figure 1.  $ {\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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