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March  2018, 38(3): 1553-1565. doi: 10.3934/dcds.2018064

On the universality of the incompressible Euler equation on compact manifolds

Terence Tao

UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USA

Received  July 2017 Published  December 2017

The incompressible Euler equations on a compact Riemannian manifold
$(M,g)$
 take the form
$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$
We show that any quadratic ODE
$\partial_t y =B(y,y)$
, where
$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$
 is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold
$M$
 if and only if
$B$
 obeys the cancellation condition
$\langle B(y,y), y \rangle =0$
 for some positive definite inner product
$\langle,\rangle$
 on
$\mathbb{R}^n$
. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.
Keywords: Euler equation, quadratic ODE, universality, embedding, Riemannian manifolds.
Mathematics Subject Classification: 35Q35, 37N10, 76B99.
Citation: Terence Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1553-1565. doi: 10.3934/dcds.2018064
References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.

[2]

M. S. Ashbaugh, C. C. Chicone and R. H. Cushman, The twisting tennis racket, J. Dyn. Diff. Eq., 3 (1991), 67-85.  doi: 10.1007/BF01049489.

[3]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.

[4]

S. Bromberg and A. Medina, Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.  doi: 10.1007/BF02972671.

[5]

R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. of Diff. Equs., 7 (1990), 251-273.  doi: 10.1016/0022-0396(70)90110-5.

[6]

E. I. Dinaburg and Ya. G. Sinai, A quasilinear approximation for the three-dimensional Navier-Stokes system, Moscow Math. J., 1 (2001), 381-388. 

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.  doi: 10.2307/1970699.

[8]

S. Friedlander and N. Pavlovic, Blow-up in a three-dimensional vector model for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 705-725.  doi: 10.1002/cpa.20017.

[9]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995.

[10]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217. 

[11]

J. L. Kaplan and J. A. Yorke, Non associative real algebras and quadratic differential equations, Nonlinear Analysis, 3 (1979), 49-51.  doi: 10.1016/0362-546X(79)90033-6.

[12]

N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.  doi: 10.1090/S0002-9947-04-03532-9.

[13]

K. Okhitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 89 (1989), 329-341.  doi: 10.1143/PTP.81.329.

[14]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674. 

[15]

T. Tao, On the universality of potential well dynamics, Dynamics of Partial Differential Equations, 14 (2017), 219-238.  doi: 10.4310/DPDE.2017.v14.n3.a1.

show all references

References:
[1]

V. I. Arnold, Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.

[2]

M. S. Ashbaugh, C. C. Chicone and R. H. Cushman, The twisting tennis racket, J. Dyn. Diff. Eq., 3 (1991), 67-85.  doi: 10.1007/BF01049489.

[3]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, Dynamical Systems Approach to Turbulence, Cambridge University Press, 1998.

[4]

S. Bromberg and A. Medina, Completeness of homogeneous quadratic vector fields, Qual. Theory Dyn. Syst., 6 (2005), 181-185.  doi: 10.1007/BF02972671.

[5]

R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. of Diff. Equs., 7 (1990), 251-273.  doi: 10.1016/0022-0396(70)90110-5.

[6]

E. I. Dinaburg and Ya. G. Sinai, A quasilinear approximation for the three-dimensional Navier-Stokes system, Moscow Math. J., 1 (2001), 381-388. 

[7]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math.(2), 92 (1970), 102-163.  doi: 10.2307/1970699.

[8]

S. Friedlander and N. Pavlovic, Blow-up in a three-dimensional vector model for the Euler equations, Comm. Pure Appl. Math., 57 (2004), 705-725.  doi: 10.1002/cpa.20017.

[9]

U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995.

[10]

E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl., 18 (1973), 216-217. 

[11]

J. L. Kaplan and J. A. Yorke, Non associative real algebras and quadratic differential equations, Nonlinear Analysis, 3 (1979), 49-51.  doi: 10.1016/0362-546X(79)90033-6.

[12]

N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc., 357 (2005), 695-708.  doi: 10.1090/S0002-9947-04-03532-9.

[13]

K. Okhitani and M. Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully developed model of turbulence, Prog. Theor. Phys., 89 (1989), 329-341.  doi: 10.1143/PTP.81.329.

[14]

T. Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation, J. Amer. Math. Soc., 29 (2016), 601-674. 

[15]

T. Tao, On the universality of potential well dynamics, Dynamics of Partial Differential Equations, 14 (2017), 219-238.  doi: 10.4310/DPDE.2017.v14.n3.a1.

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